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7
Smith Chart and Impedance Matching

This chapter introduces the concept of the transmission line (TL) theory, impedance matching, and how the Smith Chart can be used for impedance matching.

7.1 Transmission Line

The sinusoidal steady‐state voltage and current of a TL (as depicted in Figure 7.1) at point of distance d from the load end can be written in cosine‐based phasor form as^[P-1]

(7.1.1)

(7.1.2)

where and are called the forward(incident)/reflected voltages and currents (each propagating from the source/load to the load/source), respectively. The characteristic impedance Z₀ and the propagation constant γ = α + jβ (α: attenuation constant, β: phase constant) are defined as

Diagram of a transmission line (a) and lumped-element equivalent model of a transmission line terminated in a load impedance ZL (b). — **Figure 7.1** A transmission line (TL) terminated in a load impedance Z_L.

(7.1.3)

(7.1.4)

(α: attenuation constant, β: phase constant, angular wave number, or spatial frequency)

with

(7.1.5)

Three graphs displaying the shapes of forward (incident) and reflected waves when the TL is open-ended (a), short-ended (b), and impedance-matched (c). — **Figure 7.2** The shapes of forward (incident) and reflected waves along the distance d from the load end.

If the forward (incident) and reflected voltage phasors are

(7.1.6)

the line voltage (with angular frequency ω) at point of distance d from the load can be expressed as

(7.1.7)

Figure 7.2(a), (b), and (c) show the shapes of forward (incident) and reflected waves when the TL is open‐ended, short‐ended, and impedance‐matched, respectively.

We divide V(d) (Eq. (7.1.1)) by I(d) (Eq. (7.1.2)) to get the input impedance Z_in of the TL at point of distance d from the load as

(7.1.8)

where the (input) voltage reflection coefficient Γ(d) is defined as the ratio of reflected voltage phasor to incident voltage phasor at a distance d from the load:

(7.1.9)

The voltage and current phasors at the load (d = 0) are related by the load impedance as

(7.1.10)

This can be solved for Γ(0) (the voltage reflection coefficient at the load, called the load [voltage] reflection coefficient) as

(7.1.11)

where the normalized load and input impedances are defined as

(7.1.12a,b) equation

As a measure of the mismatch of a lossless TL with γ = jβ (β: phase constant), we define the standing wave ratio (SWR) and attenuation ratio (ATR) as

(7.1.13a)

(7.1.13b)

where

(7.1.14)

We can substitute Eq. (7.1.9) into Eq. (7.1.12) to rewrite the (normalized) load impedance as

(7.1.15)

(7.1.16)

This can be written for a lossless TL (with an imaginary propagation constant γ = jβ = j2π/λ) terminated with a load impedance Z_L as

(7.1.17)

This implies that the input impedance has the same value at every one‐half wave length (λ/2), i.e. at d ± nλ/2 (n: an integer) on a lossless TL terminated with a load impedance Z_L since

At points of distance d = nλ/2 (n: an integer), the input impedance of a lossless TL terminated with a load impedance Z_L is the same as the load impedance:

(7.1.18)

Especially, at point of distance d = λ/4, the input impedance of a lossless TL terminated in a load impedance Z_L turns out to be

(7.1.19)

This can be used to determine the characteristic impedance of a λ/4‐length lossless line, called a quarter‐wave (λ/4) (impedance) transformer, with a desired input impedance Z_in,d as

(7.1.20)

In addition to the SWR, as another measure of discontinuity at the load in a TL, the return loss (RL) is defined as the ratio of the incident power to (negatively) reflected power if Γ(0) is real‐valued and negative, i.e. Γ(0) < 0:

(7.1.21a)

If Γ(0) is real‐valued and positive, i.e. Γ(0) > 0, the reflection loss (RFL) is defined as

(7.1.21b)

If Γ(0) = 0 (perfect impedance matching between the line and the load), we have RL = ∞ and RFL = 0. Figure 7.3 shows the lower part of Smith chart giving the SWR, RL/RFL, etc., where the values of the SWRs and RLs for Γ = −0.5 and Γ = 0.5 can be read as

(7.1.22a)

(7.1.22b)

(7.1.23a)

(7.1.23b)

To see how the voltage reflection coefficient is related with the power flow along the TL, let us rewrite Eqs. (7.1.1) and (7.1.2) for the total voltage and current as

(7.1.24)

(7.1.25)

Image described by caption and surrounding text. — **Figure 7.3** Lower part of Smith chart giving the Standing Wave Ratio (SWR), return/reflection loss, and reflection/transmission coefficients. (A part of the Smith chart from https://leleivre.com/rf_smith.html)

Then, the time‐average power flow along the TL at a point of distance d from the load can be written as

(7.1.26)

Here, |Γ(0)|² may well be called the power reflection coefficient.

7.2 Smith Chart

The Smith chart was developed by P. W. Smith as a graphical tool to analyze and design TL circuits, in 1939. It can be used for the conversions between reflection coefficients and normalized impedances based on Eq. (7.1.12):

(7.2.1)

This can be reduced to two real equations each corresponding to the real and imaginary parts:

(7.2.2a)

(7.2.2b)

where

(7.2.3)

Eqs. (7.2.2a,b) represent two families of circles, each called resistance circles for = constant and reactance circles for = constant, respectively, that constitute the Smith chart as plotted in Figure 7.4. Note that the center (, 0) of resistance circle gets closer to (1,0) as and the center (1,) of reactance circle moves up or down (depending on the sign of ) as .

Likewise, the normalized admittance is related with the reflection coefficient as

(7.2.4)

only with the reversed sign of Γ in comparison with Eq. (7.2.1). This implies that conductance circles for = constant and susceptance circles for = constant can be obtained by making symmetric transposes of the resistance and reactance circles w.r.t. the origin (Γ = 0) or rotating them by 180° so that the sign of Γ can be reversed. In the Smith chart in Figure 7.4, conductance and susceptance circles share the point (−1,0) while the resistance and reactance circles share the point (1,0).

Note that a point on the Γ‐plane (in the Smith chart) represents the (input) voltage reflection coefficient at a distance d from the load:

(7.2.5)

Remark 7.2 Some Observations about the Smith Chart

In view of Eq. (7.2.5), as d increases, i.e. an observation point (on a TL) moves away from the load toward the source (generator) so that the WTG (wavelength toward generator) increases, the corresponding point on the Γ‐plane (in the Smith chart) representing the (input) voltage reflection coefficient at the observation point makes a clockwise decreasing spiral movement toward the point corresponding to Z₀ (the characteristic impedance of the line), which is not always, but often, the origin on the Γ‐plane if the impedances are normalized to Z₀ as usually. Especially, the effect of increasing the length of a lossless TL (with α = 0) by d [m] can be described by rotating the observation point on the Γ‐plane clockwise by 2βd = 4π(d/λ) [rad] on the circle of radius |Γ| centered at the point corresponding to Z₀.
Since λ = c/f (c: light speed, f: frequency), increasing the frequency f has the same effect (on βd = 2π(d/λ)) with increasing the length d of TL.
In view of Eq. (7.2.5), as an observation point moves away from the load by 0.25λ (quarter WTG), the corresponding point on the Γ‐plane rotates clockwise by π[rad]. It makes a complete revolution as the observation point moves by each half‐wavelength, i.e. 0.5nλ (n: an integer) and that is why the circumference of the (unit) circle is said to be λ/2 regardless of its radius.
In view of Eqs. (7.2.1)/(7.2.4), the points Γ = −1 and Γ = 1 corresponds to = 0 (short‐circuit) and = 0 (open‐circuit), respectively.
Comparing Eqs. (7.1.13a) and (7.2.1), we can see that
- = (1 + Γ_R)/(1 − Γ_R) = SWR if Γ_R > 0 and Γ_I = 0
- = (1 + Γ_R)/(1 − Γ_R) = 1/SWR if Γ_R < 0 and Γ_I = 0
If Z_L = Z₀, i.e. the load impedance is perfectly matched to the characteristic impedance of the TL, the corresponding point on the Γ‐plane is at the origin Γ = 0 so that SWR = 1, implying no reflection, i.e. perfect absorption by the load (see Figure 7.2(c)).
In view of Eqs. (7.1.13) and (7.1.21), the constant SWR and constant RL (return loss) lines on the Smith chart are circles of radius |Γ| centered at the origin on the Γ‐plane.
The normalized impedance represented by point Γ₁= 0.2 + j0.4 can be found as 1 + j1 from the values of the (blue) resistance and reactance circles crossing the point and can also be computed using Eq. (7.2.1). The normalized admittance represented by the point can be found as 0.5 − j0.5 from the values of the (red) conductance and susceptance circles crossing the point and can also be computed using Eq. (7.2.4), which is just the reciprocal of Eq. (7.2.1).
The sign of reactance values for all the resistance curves above/below the real axis on the Γ‐plane is +/−, respectively, and the sign of susceptance values for all the conductance curves above/below the real axis on the Γ‐plane is −/+, respectively.
Connecting a capacitor or an inductor in shunt with a load rotates the point (representing the impedance) along a conductance circle clockwise or counterclockwise, respectively. Connecting an inductor or a capacitor in series with a load rotates the point (representing the impedance) along a resistance circle clockwise or counterclockwise, respectively.
The six dotted arcs on the Smith chart in Figure 7.4 are Q‐circles (also called constant Q circles) of quality factor Q = |X|/R = 1, 2, and 5, whose equations can be derived by dividing Eq. (7.2.2b) by Eq. (7.2.2a):
(7.2.6)

Example 7.1 Design of L‐type Impedance Transformers Using the Smith Chart

Find L‐type LC circuits transforming the normalized impedance Z₁ = 1 + j1 [Ω] or admittance Y₁ = 1/Z₁ = 0.5 − j0.5[S] represented by point Γ₁ into the normalized impedance Z₂ = 0.5 − j0.5[Ω] or admittance Y₂ = 1/Z₂ = 1 + j1[S] represented by point Γ₂ on the Smith chart of Figure 7.4.

There are three paths along resistance/conductance curves from point Γ₁ to Γ₂, each via Γ₃, Γ₄, and Γ₀. The first path consists of a section from Γ₁ to Γ₃ (corresponding to Z₃ = 0.5 − j0.866[Ω] or admittance Y₃ = 1/Z₃ = 0.5 + j0.866[S]) and one from Γ₃ to Γ₂ where
- the former section clockwise along a conductance circle corresponds to the shunt connection of a capacitor with susceptance B₁₃ = ωC₁₃ = 0.866 − (−0.5) = 1.366 Ω,
- and the latter section clockwise along a resistance circle corresponds to the series connection of an inductor with reactance X₃₂ = ωL₃₂ = −0.5 − (−0.866) = 0.366 Ω.
The resulting normalized impedance corresponds to Γ₂ as desired:

(E7.1.1)

The second path consists of a section from Γ₁ to Γ₄ (corresponding to Z₄ = 0.5 + j0.866[Ω] or admittance Y₄ = 1/Z₄ = 0.5 − j0.866[S]) and one from Γ₄ to Γ₂ where
- the former section counterclockwise along a conductance circle corresponds to the shunt connection of an inductor with susceptance B₁₄ = −1/ωL₁₄ = −0.866 − (−0.5) = −0.366 Ω,
- and the latter section counterclockwise along a resistance circle corresponds to the series connection of a capacitor with reactance X₄₂ = −1/ωC₄₂ = −0.5 − 0.866 = −1.366 Ω.
The resulting normalized impedance corresponds to Γ₂ as desired:

(E7.1.2)

The third path consists of a section from Γ₁ to Γ₀ and one from Γ₀ to Γ₂ where
- the former section counterclockwise along a resistance circle corresponds to the series connection of a capacitor with reactance X₁₀ = −1/ωC₁₀ = 0 − 1 = −1 Ω,
- and the latter section clockwise along a conductance circle corresponds to the shunt connection of a capacitor with susceptance B₀₂ = ωC₀₂ = 1 − 0 = 1 Ω.
The resulting normalized admittance and impedance correspond to Γ₂ as desired:

(E7.1.3a)

(E7.1.3b)
Noting that this problem is to find L‐type impedance matchers between load impedance of Z₁= 1 + j1[Ω] and source impedance = 0.5 + j0.5[Ω], we can use the MATLAB function ‘imp_matcher_L()’ (Section 6.3.2) as follows:
```
 >>Zs=0.5+0.5i; ZL=1+i; % Source/Load impedances
   fc=1/2/pi; % Faked carrier frequency
   [X11,X21,X12,X22]=imp_matcher_L(Zs,ZL,fc)
```
This results in
```
 L1-type impedance matcher with C1=1.000e+09[nF] and C2=1.000e+09[nF] %1
 L1-type impedance matcher with C1=1.000e+09[nF] and C2=1.000e+09[nF] %2
 L2-type impedance matcher with L1=3.660e+05[uH] and C2=1.366e+09[nF] %3
 L2-type impedance matcher with C1=7.321e+08[nF] and L2=2.732e+06[uH] %4
 X11 =  -1  -1 %1
 X21 =  -1  -1 %2
 X12 =   0.366025403784439 -1.366025403784439 %3
 X22 =  -0.732050807568877  2.732050807568878 %4
```
Here, we set the frequency as f_c = 1/2π so that ω_c = 2πf_c = 1 and consequently, the reactance values (ωL) of inductors and the susceptance values (ωC) of capacitors can be equal to their inductance and capacitance values, respectively, for simplicity. Noting that 1/0.7321 = 1.366 and 1/2.732 = 0.366, we see that the first and second results happened to be identical, corresponding to the third case in (a), the third one corresponds to the first case in (a), and the forth one corresponds to the second case in (a).
Let us use the Smith Chart software ‘Smith V4.1’ (developed by Professor Fritz Dellsperger) to find the LC filter implied by the path from Γ₁ to Γ₂ via Γ₃. To this end, take the following steps:
1. Start ‘Smith V4.1’ to open the main window as shown in Figure 7.5(a).
2. From the top menu bar, select Tools > Settings to open the Settings dialog box (Figure 7.5(b)), set the values of Z₀ (characteristic impedance) and frequency to 1[Ω] and 1/2π = 0.159155[Hz], respectively, and click OK button to close the dialog box.
3. From the toolbar (below the menu bar), select ‘Mouse’ and click on Γ₁ (the intersection of the 1 Ω‐circle and the 1 S‐circle). Alternatively, select Keyboard to open Data Point dialog box and insert the impedance Z₁ = 1 + j1[Ω] as a data point (Figure 7.5(c)).
4. Click OK button to close the Data Point dialog box. Then DP 1 will appear at Γ₁ on the Smith chart and its impedance value will appear in the Data Points pane at the right side as shown in Figure 7.6.
5. From the tool bar, click on ‘Insert parallel capacitor’ button and move the cross‐type mouse cursor to click on the point Γ₃ (the intersection of the 500 mΩ‐circle and 0.5 S‐circle). Then, TP 2 will appear at Γ₃ on the Smith chart and its impedance value will appear in the Data Points pane at the right side as shown in Figure 7.6(a).
  
  Figure 7.5 The main window, Tools/Setting dialog box, and Data Point dialog box of ‘Smith 4.1’.
  
  Figure 7.6 Main window with a design result of ‘Smith 4.1’.
6. From the tool bar, click on ‘Insert series inductor’ button and move the mouse cursor to click on the point Γ₂ (the intersection of the R = 500 mΩ‐circle, the G= 1 S‐circle, the B = 1 S‐circle, and also the X = −0.5 Ω‐circle) representing Z₂ = 0.5 − j0.5 [Ω], or equivalently, Y₂ = 1 + j1 [S]. Then, TP 3 will appear at Γ₂ on the Smith chart and its impedance value will appear in the Data Points pane at the right side as shown in Figure 7.6(a).
7. The designed network with the parameter values C = 1.4 F and L = 363.1 mH is shown in the Schematic pane at the upper‐right part of the window. It conforms with the first one (ωC₁₃ = 1.366 Ω and ωL₃₂ = 0.366 Ω with ω = 1) of the three design results obtained in (a).
8. If you want to read more digits of a numeric parameter value or change it, double‐click on it to open the Edit Element dialog box.
It would be a good practice if you try the other two paths from Γ₁ to Γ₂, one via Γ₄ and one via Γ₀, and check if the resulting parameter values are fairly close to those obtained in (a).

Figure 7.6(b) shows several insert functions (like connecting a resistor/inductor/capacitor, an open stub, or a short(‐circuited) stub in series or parallel with the load side) that can be activated by clicking on the corresponding button in the toolbar.

7.3 Impedance Matching Using Smith Chart

7.3.1 Reactance Effect of a Lossless Line

From Eq. (7.1.17), we see that connecting a lossless TL (with an imaginary propagation constant γ = jβ = j2π/λ and characteristic impedance Z₀) of length l in series with a load of impedance Z_L changes the impedance as

(7.3.1)

How about a short‐circuited stub (SS) with Z_L = 0 and an open‐circuited stub (OS) with Z_L = ∞? Their impedances will be

(7.3.2a,b)

like those of an inductor or a capacitor, respectively. That is why a short‐circuited or an open‐circuited stub (line) is used for impedance matching where stubs can easily be fabricated in microstrip or stripline form compared with the ‘equivalent’ lumped element. To feel how connecting a stub line of length l in series with an impedance Z_L affects the impedance in terms of its reflection coefficient Γ (on the Smith chart), let us run the following MATLAB script to get Figure 7.7. It shows that as the length of the stub increases, the reflection coefficient Γ rotates clockwise around the point Γ₀₁ corresponding to the characteristic impedance Z₀₁ of the stub line, which is not necessarily equal to the characteristic impedance Z₀ for normalization. Figure 7.7 also shows the effect of connecting a lumped element of reactance X in series with an impedance Z_L = R_L + jX_L is to rotate the reflection coefficient Γ clockwise/counterclockwise along the R_L‐circle depending on the sign of the reactance as |X| increases where Γ = 1 for X = ±∞ (see Eq. (7.2.3) with = (R + jX)/Z₀ = ±j∞ where Z₀ = R₀ > 0).

%plot_line_imp.m
clear, clf
Z0=50; % Characteristic impedance for normalization
Z01=100; % Characteristic impedance of the line
b=2*pi; % Phase constant
ZL=100+50i; % Load impedance
Z=@(l)Z01*(ZL+j*Z01*tan(b*l))./(Z01+j*ZL*tan(b*l)); % Eq. (7.3.1)
GammaZ=@(Z)(Z/Z0-1)./(Z/Z0+1); % Eq. (7.2.3)
% To see the effect of increasing the length of stub
ds=0:0.01:0.5; % Lengths of the stub
plot(GammaZ(ZL)+j*eps,'o'), hold on
axis('equal'), axis([-1 1 -1 1])
for n=1:length(ds)
    plot(GammaZ(Z(ds(n))),'r.'), pause(0.05)
end
% To see the effect of changing series reactance of lumped element
Xs=0.5*3.^[0:0.1:20]; % Values of the reactance
Xs=[-fliplr(Xs) 0 Xs]; %-1000:0.1:1000;
for n=1:length(Xs)
    plot(GammaZ(ZL+j*Xs(n))), pause(0.01)
end
% plot the characteristic impedance of the line
plot(GammaZ(Z01)+j*eps,'ro')
% To plot the Smith chart as a background
Rs=[12.5 25 50 100 200]; % Resistance values for Smith chart
Xs=[-50 -25 0 25 50]*2; % Reactance values for Smith chart
Gs=1./[12.5 25 50 100 200]; % Conductance values for Smith chart
Bs=1./[-50 -25 0 25 50]; % Susceptance values for Smith chart
plot_Smith_chart(Z0,Rs,Xs,Gs,Bs)

function plot_Smith_chart(Z0,Rs,Xs,Gs,Bs)
% plot the Smith chart.
ths=pi/180*[0:360];
if ~isempty(Rs)
 for n=1:length(Rs), [c,r]=Rcircle(Rs(n),Z0); plot(c+r*exp(j*ths)); end
end
if nargin>1&~isempty(Xs)
 for n=1:length(Xs), [c,r]=Xcircle(Xs(n),Z0); plot(c+r*exp(j*ths)); end
end
if nargin>2&~isempty(Gs)
 for n=1:length(Gs), [c,r]=Gcircle(Gs(n),Z0); plot(c+r*exp(j*ths)); end
end
if nargin>3&~isempty(Bs)
 for n=1:length(Bs), [c,r]=Bcircle(Bs(n),Z0); plot(c+r*exp(j*ths)); end
end
plot(exp(j*ths),'k:'); plot([-1 1],[0 0],'k', [0 0],[-1 1],'k')

function [c,r]=Rcircle(R,Z0)
% To find the radius r and center c (in complex value) of an R-circle
R0=R/Z0; c=R0/(1+R0)*exp(j*eps); r=1/(1+R0); % Eq. (7.2.2a)

function [c,r]=Xcircle(X,Z0)
% To find the radius r and center c (in complex value) of an X-circle
X0=X/Z0; c=1+i/X0; r=1/abs(X0); % Eq. (7.2.2b)

function [c,r]=Gcircle(G,Z0)
% To find the radius r and center c (in complex value) of a G-circle
G0=G*Z0; c=G0/(1+G0)*exp(j*pi); r=1/(1+G0);

function [c,r]=Bcircle(B,Z0)
% To find the radius r and center c (in complex value) of a B-circle
B0=B*Z0; c=-1-i/B0; r=1/abs(B0);

7.3.2 Single‐Stub Impedance Matching

The single‐stub impedance matching (tuning) is to connect a single open‐circuited or short‐circuited lossless stub line in shunt (parallel) or series with an existing lossless TL for matching a (possibly complex‐valued) load impedance Z_L to the TL of characteristic impedance R₀ as shown in Figure 7.8. The problem is how distant the stub should be from the load and how long it should be.

7.3.2.1 Shunt‐Connected Single Stub

To consider how to determine the distance d from the load and the length l of the shunt stub in Figure 7.8(a), let us write the normalized impedance and admittance of the load as

Schematic diagrams illustrating shut (parallel) connection (a) and series connection (b) of a stub. — **Figure 7.8** Single‐stub impedance matching.

(7.3.3a)

(7.3.3b)

where R₀ and G₀ = 1/R₀ are the characteristic impedance and admittance of the TL, respectively, that are used to normalize the impedances and admittances. Noting that

the impedance matching is achieved when the (parallel) combination of the line‐load and the shunt stub results in a unity normalized admittance/impedance corresponding to G₀/R₀, i.e.
(7.3.4)

and the stub (whose admittance y_s will be added to the admittance y_B of the line‐load to yield the input admittance y_i), short‐circuited or open‐circuited, is purely reactive/susceptive,

we should determine the distance d of the stub from the load so that the real part (conductance) of the normalized line‐load admittance can be unity:

(7.3.5)

where

(7.3.6)

If the stub has its own characteristic impedance R₀₁ ≠ R₀, y_B should be multiplied with R₀₁/R₀. Now we determine the length l of the stub such that the susceptance can be b_s = −Im{y_B} to make the imaginary part of the input admittance y_i = y_B + y_s zero, satisfying Eq. (7.3.4):

(7.3.7a)

(7.3.7b)

Note that OSs (open‐circuited stubs) are preferable for easy fabrication in microstrip or stripline form while SSs (short‐circuited stubs) are preferable for coax or waveguide.

function [ds,ls,Bs,Gammas]=imp_match_1stub_shunt(Z0,ZL,r,os)
% Input: Z0 = Characteristic impedances [Z0,Z01] of the TL and stub
%        ZL = Load impedance
%        r = a+bi = Propagation constant
%        os = 'o'/'s' for open/short(-circuited) stub
% Output: ds = Distance of the stub from load
%        ls = Length of the stub
if nargin<4, os='s'; end
if length(Z0)>1, Z01=Z0(2); Z0=Z0(1); else Z01=Z0; end
a=real(r); beta=imag(r); lambda=2*pi/beta;
yB=@(ZL,Z0,d)(Z0+ZL*tanh(r*d))./(ZL+Z0*tanh(r*d)); %Eq. (7.3.1)normalized
eq=@(d)real(yB(ZL,Z0,d))-1; % Eq. (7.3.5)
options=optimoptions('fsolve','TolX',1e-20,'TolFun',1e-20);
ds=real([fsolve(eq,0,options) fsolve(eq,0.25*lambda,options)]);
ds=mod(ds*beta,pi)/beta; % Possibly two solutions for d
bB=imag(yB(ZL,Z0,ds))*Z01/Z0; bs=-bB; Bs=bs/Z0; % Susceptances
ys=1+j*bs; Gammas=(1-ys)./(1+ys); % Intermediate points
if lower(os(1))=='s', ls=acot(bB)/beta; % Eq. (7.3.7a)
 else ls=atan(-bB)/beta; % Eq. (7.3.7b)
end
ls=mod(ls*b,pi)/beta;
% Usage: Z0=50; ZL=35-47.5i; lambda=1; r=2*pi*i/lambda;
% [ds,ls]=imp_match_1stub_shunt(Z0,ZL,r,'s')

Example 7.2 Design of Shunt Single Short(‐Circuited) Stubs for Impedance Matching

Determine the distances (from the load) and lengths of shunt single short‐circuited stubs in the unit of wavelength so that they can match a load impedance Z_L = 30 − j40[Ω] to the lossless transmission (or feed) line of characteristic impedance Z₀ = 50 Ω.

We can easily get the solution by using the MATLAB function ‘imp_matcher_1stub_shunt()’ (listed above) as
```
 >>Z0=50; ZL=30-40i; lambda=1; beta=2*pi/lambda; r=i*beta;
   [ds,ls]=imp_match_1stub_shunt(Z0,ZL,r,'s') % for SS
```
This yields
```
 ds =  0.0417   0.2083
 ls =  0.1136   0.3864
```
which means two solutions, one of length 0.1136λ and distance 0.0417λ from the load and the other of length 0.3864λ and distance 0.2083λ from the load.
To find the solutions using the Smith chart, we first should recall the following:
- Our target point Γ₃ = 0 (corresponding to both R₀ and G₀) is on the G = G₀ = 1/R₀‐circle.
- A lossless TL of length d connected to the load rotates the Γ‐point clockwise by 2βd = 4π(d/λ) [rad] along the |Γ_L|‐circle, i.e. the circle of radius |Γ_L| centered at the origin Γ= 0 where (Remark 7.2(1)).
- A shunt reactive element, a lumped one or a stub, can change only the susceptance, not the conductance, so that the Γ‐point can be rotated along a G‐circle.
That is why we first should rotate the Γ‐point clockwise along the |Γ_L|‐circle (passing through Z_L = 30 − j40[Ω]) to some point (like Γ₂ or Γ₄ in Figure 7.9) on the G = G₀ = 1/R₀ = 20 mS‐circle. Here, the distance d of a stub (to be shunt‐connected) from the load is to be determined from the angles by which the Γ‐point has been rotated. Then, we should determine the length l of the parallel (shunt) short‐circuited stub, which rotates the Γ‐point counterclockwise along the G‐circle to the target point Γ = 0. With this strategy in mind, let us use ‘Smith V4.1’ by taking the following steps:
1. From the top menu bar, select Tools > Settings to open the Settings dialog box (Figure 7.5(b)), set the values of Z₀ (characteristic impedance) and frequency to 50[Ω] and 1[GHz], respectively, and click OK button to close the dialog box. Here, the frequency does not matter because we will determine d and l just in the unit of wavelength.
2. From the toolbar (below the menu bar), select Keyboard to open Data Point dialog box, insert Z₁ = 30 − j40[Ω] as a data point, and click OK button to close the dialog box.
3. From the tool bar, click on ‘Insert serial line’ button and move the cross‐type mouse cursor to click on the point Γ₂ (see Figure 7.9).
4. From the tool bar, click on ‘Insert parallel line shorted’ button (denoted by SS: shorted stub) and move the mouse cursor to click on the point Γ₃ = 0.
  
  Figure 7.9 Using ‘Smith 4.1’ to design shunt single short‐stub impedance matchers in Example 7.2.
5. Then, the designed network with d₁ = 0.0415λ and l₁ = 0.1137λ will be shown in the Schematic pane at the upper‐right part of the window. It conforms with the results obtained using the MATLAB function ‘imp_matcher_1stub_ shunt()’ in (a).

7.3.2.2 Series‐Connected Single Stub

Similarly to the case of shunt‐connected single‐stub matching, note that

the impedance matching is achieved when the (series) combination of the line‐load and the series stub (Figure 7.8(b)) results in a unity normalized impedance corresponding to R₀, i.e.
(7.3.8)

and the stub (whose impedance z_s will be added to the impedance z_B of the line‐load to yield the input impedance z_i), short‐circuited or open‐circuited, is purely reactive.

We should determine the distance d of the stub from the load so that the real part (resistance) of the normalized line‐load impedance can be unity:

(7.3.9)

where

(7.3.10)

If the stub has a characteristic impedance R₀₁ ≠ R₀, z_B should be multiplied with R₀/R₀₁.

Now we determine the length l of the stub such that the reactance can be x_s = −Im{z_B} to make the imaginary part of the input impedance z_i = z_B + z_s zero, satisfying Eq. (7.3.8):

(7.3.11a)

(7.3.11b)

function [ds,ls,Xs,Gammas]=imp_match_1stub_series(Z0,ZL,r,os)
% Input:  Z0 = Characteristic impedances [Z0,Z01] of the TL and stub
%        ZL = Load impedance
%         r = alpha+beta*i = Propagation constant
%        os = 'o'/'s' for open/short(-circuited) stub
% Output: ds = Distances of the stub from load
%        ls = Lengths of the stub
if nargin<4, os='s'; end
if length(Z0)>1, Z01=Z0(2); Z0=Z0(1);
 else Z01=Z0;
end
beta=imag(r); lambda=2*pi/beta;
zB=@(ZL,Z0,d)(ZL+Z0*tanh(r*d))./(Z0+ZL*tanh(r*d)); %Eq. (7.3.1)normalized
eq=@(d)real(zB(ZL,Z0,d))-1; % Eq. (7.3.9,10)
options=optimoptions('fsolve', 'TolX',1e-20, 'TolFun',1e-20);
ds=real([fsolve(eq,0,options) fsolve(eq,0.25*lambda,options)]);
ds=mod(ds*b,pi)/beta;
xB=imag(zB(ZL,Z0,ds))*Z0/Z01; xs=-xB; Xs=xs*Z0; % Reactances
zs=1+j*xs; Xs=xs*Z0; Gammas=(zs-1)./(zs+1); % Intermediate points
if lower(os(1))=='o', ls=acot(xB)/beta; % Eq. (7.3.11b)
 else ls=atan(-xB)/beta; % Eq. (7.3.11a)
end
ls=mod(ls*b,pi)/beta;
% Usage: Z0=50; ZL=35-47.5i; lambda=1; r=2*pi*i/lambda; % [ds,ls]=imp_match_1stub_series(Z0,ZL,r,'o')

Example 7.3 Design of Series Single Stubs for Impedance Matching

Determine the distances (from the load) and lengths of series single open‐/short‐circuited stubs in the unit of wavelength so that they can match a load impedance Z_L = 30 − j40[Ω] to the lossless transmission (or feed) line of characteristic impedance Z₀ = 50 Ω.

We can easily get the solutions by using the MATLAB function ‘imp_matcher_1stub_shunt()’ as
```
 >>Z0=50; ZL=30-40i; lambda=1; beta=2*pi/lambda; r=i*beta;
   [dso,lso]=imp_match_1stub_series(Z0,ZL,r,'o') % for OSs
   [dss, lss]=imp_match_1stub_series(Z0,ZL,r,'s') % for SSs
```
This yields
```
 dso = 0.4583 0.2917   lso = 0.3864 0.1136
 dss = 0.4583 0.2917   lss = 0.1364 0.3636
```
which means two open stubs, one of length 0.3864λ and distance 0.4583λ (from the load) and the other of length 0.1136λ and distance 0.2917λ, and also two short stubs, one of length 0.1364λ and distance 0.4583λ (from the load) and the other of length 0.3636λ and distance 0.2917λ.

There are some observations to make about the comparison of the open and short stubs:
- It does not matter to the distance d (i.e. the point of attachment) of the stub whether the stub is open‐circuited or short‐circuited.
- Even the difference in the stub length is just 0.25λ (a quarter wavelength), corresponding to halfway round the Smith chart (by 2βl = 4π/λ × 0.25λ = π[rad]), which is not surprising.
- If either short(‐circuited) stub or open(‐circuited) stub can be freely chosen, we can keep the stub length in the range of 0∼0.25λ. In general, open stubs are preferred on microstrip for constructional reasons while short stubs are preferred on coax line or parallel wire line because they have less radiation from the ends.
To find the solutions using the Smith chart, we first should recall the following:
- Our target point Γ₃ = 0 (corresponding to both R₀ and G₀) is on the R = R₀‐circle.
- A lossless TL of length d connected to the load rotates the Γ‐point clockwise by 2βd = 4π(d/λ) [rad] along the circle of radius |Γ_L| centered at the origin Γ = 0 (Remark 7.2(1)).
- A series reactive element, a lumped one or a stub, can change only the reactance, not the resistance, so that the Γ‐point can be rotated along an R‐circle.
Figure 7.10 Using ‘Smith 4.1’ to design series single open‐stub impedance matchers in Example 7.3

That is why we first should rotate the Γ‐point clockwise along the |Γ_L|‐circle (passing through Z_L = 30 − j40 [Ω]) to some point (like Γ₂ or Γ₄ in Figure 7.10) on the R = R₀ = 50 Ω‐circle. Here, the distance d of a stub (to be series‐connected) from the load is to be determined from the angles by which the Γ‐point has been rotated. Then, we should determine the length l of the open‐circuited stub, which rotates the Γ‐point clockwise along the R‐circle to the target point Γ₃ = 0). With this strategy in mind, let us use ‘Smith V4.1’ by taking the following steps:
1. Take the first three steps as done in Example 7.2.
2. With the path Γ_L‐Γ₂‐Γ₃ in mind, click on ‘Insert serial line’ button from the tool bar and move the cross‐type mouse cursor to click on the point Γ₂, which is one of the intersections between the |Γ|‐circle crossing Γ_L and the R = R₀ = 50 Ω‐circle (see Figure 7.10).
3. To realize the section Γ₂‐Γ₃, you may look for ‘Insert serial line open’ button in the tool bar, but in vain, because there is nothing like that. Then, how can we move the Γ‐point to Γ₃(Γ = 0)? Instead of a series stub, a lumped element of inductor can be connected in series with the load to rotate the Γ‐point clockwise along the R‐circle and that is why we select the ‘Insert serial inductor’ button and move the mouse cursor to click on the point Γ₃ = 0).
4. Then, the designed network with d₁ = 0.4587λ and L₁ = 9.2(≈9.161) nH will be shown in the Schematic pane. Can we check if this value of inductance conforms with the open‐stub length l₁ = 0.3864λ obtained using MATLAB in (a)? Yes, we can be sure of the conformity by computing the reactance value of this inductor as X₁ = ω_cL₁ = 2πf_cL₁ = 2π × 10⁹ × 9.161 × 10⁻⁹ = 57.56 Ω and also using Eq. (7.3.2b) to find the reactance of the open stub of length 0.3864λ as
  (E7.3.1)
5. To find another series open‐stub impedance matcher corresponding to another path Γ_L‐Γ₄‐Γ₃, right‐click anywhere on the Smith chart two times to delete the existing two data points Γ₃ and Γ₂ (in LIFO order) except Γ_L, click on ‘Insert serial line’ button, and move the mouse cursor to click on the point Γ₄, which is another intersection between the |Γ|‐circle crossing Γ_L and the R = R₀ = 50 Ω‐circle (see Figure 7.10).
6. To rotate the Γ‐point to Γ₃ = 0 counterclockwise along the R = R₀ = 50 Ω‐circle, we select the ‘Insert serial capacitor’ button and move the mouse cursor to click on the point Γ₃ = 0.
7. Then, the designed network with d₂= 0.2919λ and C₂= 2.8(≈2.766) [pF] will be shown in the Schematic pane. To check if this value of capacitance conforms with the open‐stub length l₂= 0.1136λ obtained using MATLAB in (a), we compute the reactance value of this capacitor as X₂ = −1/ω_cC₂ = −1/2πf_cC₁ = −1/(2π×10⁹× 2.766×10⁻¹²) = −57.54 Ω, use Eq. (7.3.2b) to find the reactance of the open stub of length 0.1136λ as
  (E7.3.2)
  
  and see that these two reactance values are close enough.

7.3.3 Double‐Stub Impedance Matching

The single‐stub matching introduced in the previous section requires the stub to be attached to the main line at a specific point of precise distance from the load, presenting practical difficulties because the specified point may be undesirable to attach a stub and it is very difficult to build a variable‐length coaxial line with a constant characteristic impedance. An alternative is to attach two short‐circuited stubs at fixed positions, as shown in Figure 7.11 where the distance d₀ (of stub A from a load) and distance d (of stub B from stub A) can be arbitrarily chosen and fixed (say, as λ/16, λ/8, 3λ/8, etc.), and the lengths (l_A and l_B) of the two stubs are determined to match the load impedance Z_L0 to the characteristic impedance R₀ of the main lossless line. This scheme is the so‐called double‐stub matching (tuning).

For convenience, let the serial line of length d₀ be a part of the load so that the virtual load impedance can be obtained by using Eq. (7.1.17) as

(7.3.12)

Also, let us write the normalized impedance and admittance of the load as

(7.3.13a)

(7.3.13b)

where the lowercase/uppercase letters stand for normalized/unnormalized values. Then, to satisfy the impedance matching condition

(7.3.14)

the real part (conductance) of y_B (the input admittance at terminals B‐B' looking into the load excluding stub B) should be 1 since y_sB = jb_sB (the admittance of stub B) has only an imaginary part:

(7.3.15)

where

(7.3.16a)

function [lA,lB,bsA,bsB,bA]=imp_match_2stub(Z0,ZL,r,d)
% Input:  Z0 = Characteristic impedance
%        ZL = Load impedance
%         r = a+bi = Propagation constant
%         d = Distance of the stub B from A (load)
%   where d may be given as [d d0] with d0: Distance of A from load
% Output: lA = Length of stub A at (d0 from) the load
%        lB = Length of stub B at d from stub A
if numel(d)>1
 d0=d(2); d=d(1);
 ZL=Z0*(ZL+Z0*tanh(r*d0))/(Z0+ZL*tanh(r*d0)); % Eq. (7.1.16)/(7.3.12a)
end
if imag(r)<eps, r=r*i; end % Since a real-valued r must be beta.
RL=real(ZL); a=real(r); b=imag(r); lambda=2*pi/b;
yBgb=@(G,B)(G+j*B+j*tan(b*d))./(1+j*(G+j*B)*tan(b*d)); % Eq. (7.3.16a)
YL=1/ZL; yL=YL*Z0; gL=real(yL); bL=imag(yL);
eq=@(bA)real(yBgb(gL,bA))-1; % Eq. (7.3.15)
options=optimoptions('fsolve','TolX',1e-20, 'TolFun',1e-20);
bA=real([fsolve(eq,0,options) fsolve(eq,1,options)])
bsA=bA-bL; lA=acot(-bsA)/b; % Eq. (7.3.17a), Eq. (7.3.7a)
yB=yBgb(gL,bA); % Eq. (7.3.16a)
bsB=imag(yB); lB=acot(bsB)/b; % Eq. (7.3.17b), Eq. (7.3.7a)
lA=mod(lA*b,pi)/b; lB=mod(lB*b,pi)/b;

(7.3.16b)

and y_sB = jb_sB and y_sA = jb_sA denote the admittances of stubs B and A, respectively, that depend on their lengths as described by Eq. (7.3.2). Thus, we should first find the value of b_A such that Eq. (7.3.15) can be satisfied. Then, we can determine the susceptances {b_sA, b_sB} of stubs A and B as

(7.3.17a,b)

so that the susceptance value of y_B can be really b_sA and that of the overall input admittance can be cancelled out to be zero. Then, the stub lengths l_A and l_B are determined from Eq. (7.3.7). This solution process has been cast into the above MATLAB function ‘imp_match_2stub()’.

One of the drawbacks of the double‐stub matching is that it can be used for a certain range of load impedances satisfying^[M-1]

(7.3.18)

Still it is not so fatal because it is easy to make this condition satisfied by using a serial line (of a certain length d₀) between the load and the first stub as long as it is allowed.

Example 7.4 Design of Double Stubs for Impedance Matching

Determine the lengths of double‐stub matchers with d = λ/8 and d₀ = 0 in the unit of wavelength λ so that they can match a load impedance Z_L = 60 − j80 [Ω] to the lossless transmission (or feed) line of characteristic impedance Z₀ = 50 Ω. Note that z_L = Z_L/Z₀ = 1.2 − j1.6[Ω] and y_L = 1/z_L = 0.3 + j0.4[S].

We can easily get the solutions by using the MATLAB function ‘imp_matcher_2stub()’ as

 >>Z0=50; ZL=60-80i; lambda=1; r=2*pi*i/lambda; d=1/8; d0=0;
   [lA,lB]=imp_match_2stub(Z0,ZL,r,[d d0]) % for SSs

This yields

  lA = 0.2319   0.3965
  lB = 0.0998   0.4542

which means two double short stubs, {l_A = 0.2319λ, l_B = 0.0998λ} and {l_A = 0.3965, l_B = 0.4542λ}. If you want open stubs, add/subtract 0.25λ depending on whether the stub is shorter/longer than 0.25λ.

To find the solutions using the Smith chart, we first should recall the following:
- Our target point Γ = 0 (corresponding to both R₀ and G₀) is on the R = R₀‐circle.
- As a data point (just before the target point Γ = 0), we need a point, say, Γ_B, corresponding to y_B (the input admittance at terminals B–B' looking into the load excluding stub B) on the G = G₀ = 20 [mS]‐circle or g = g₀ = 1[S]‐circle (along which the Γ‐point rotates to reach Γ = 0 by the effect of the shunt stub B).
- The data point Γ_B should be the −2βd = −4π/λ × λ/8 = −π/2 [rad]‐rotated version (due to the serial line of length d = λ/8 according to Remark 7.2(1)) of another data point, say, Γ_A, corresponding to y_A (the input admittance at A‐A'), which should be on the G = G_L = 6 [mS]‐circle or g = g_L = 0.3 [S]‐circle because the shunt stub A changes only the susceptance, not the conductance.
- It is easier to rotate the G = G₀ = 20 [mS]‐circle (by +π/2 [rad] (counterclockwise)) than to rotate the G = G_L = 6 [mS]‐circle (by −π/2 [rad] (clockwise)).
Thus, we are going to rotate the G = G₀ = 20[mS]‐circle by +π/2[rad] (counterclockwise), find its intersection with the G = G_L=6 [mS]‐circle as Γ_A, and rotate Γ_A back by −π/2[rad] (clockwise) to get Γ_B. Then, we can get the length l_A of stub A from the difference between the susceptances b_A (corresponding to Γ_A) and b_L (of the load). Also, we can get the length l_B of stub B from the susceptance b_B (corresponding to Γ_B). This idea is implemented in the following MATLAB script “elec07e04.m,” which has been run to yield the same result as obtained in (a):
```
 GamA = -0.4382 - 0.7407i   0.4675 - 0.3227i
 lA = 0.3965   0.2319
 GamB = -0.7407 + 0.4382i  -0.3227 - 0.4675i
 lB = 0.4542   0.0998
```

 %elec07e04.m
Z0=50; Y0=1/Z0; % Characteristic impedance/admittance
R0=real(Z0); G0=real(Y0);
ZL=60-80i; % Load impedance
zL=ZL/Z0; rL=real(zL); xL=imag(zL);
yL=1/zL; gL=real(yL); bL=imag(yL);
lambda=1; b=2*pi/lambda; d=1/8;
% To use the Smith chart
[cG0,rG0]=Gcircle(G0,Z0); % G0-circle
[cGL,rGL]=Gcircle(gL,1); % GL-circle
cG0_r=cG0*exp(j*2*b*d); % Center of the rotated G0-circle
% To find the intersections of the GL-circle and the rotated G0-circle.
cG0_r_R=real(cG0_r); cG0_r_I=imag(cG0_r);
[GamA_R,GamA_I]=circcirc(cG0_r_R,cG0_r_I,rG0,real(cGL),imag(cGL),rGL);
GamA=GamA_R+i*GamA_I % Two GammaA points
G2Z=@(G,Z0)Z0*(1+G)./(1-G); % Eq. (7.1.12)
yA=Z0./G2Z(GamA,Z0); % Normalized admittance at A
bsA=imag(yA)-bL; % Eq. (7.3.17a): Susceptance of stub A
lA=acot(-bsA)/b; lA=mod(lA*b,pi)/b % Eq. (7.3.7a)
GamB=GamA*exp(-j*2*b*d) % Two GammaB points with the d[m]-serial line
yB=Z0./G2Z(GamB,Z0); % Normalized admittance at B
bsB=-imag(yB); % Eq. (7.3.17b): Susceptance of stub B
lB=acot(-bsB)/b; lB=mod(lB*b,pi)/b % Eq. (7.3.7a)

Additionally, we have got two intermediate points Γ_A = 0.4675 − j0.3227 and Γ_B = −0.3227 − j0.4675, that can be of a help for working with ‘Smith V4.1’.
Now, let us work with ‘Smith V4.1’ taking the following steps:
1. From the top menu bar, select Tools > Settings to open the Settings dialog box (Figure 7.5(b)), set the values of Z₀ (characteristic impedance) and frequency to 50[Ω] and 1[GHz], respectively, and click OK button.
2. From the toolbar (below the menu bar), select Keyboard to open Data Point dialog box, insert Z_L = 60 − j80 [Ω] as a data point, and click OK button to close the Data Point dialog box.
3. Select Keyboard again to open Data Point dialog box, insert Γ_A = 0.4675 − j0.3227 as a data point, and click OK button to close the Data Point dialog box. It seems to be difficult to draw auxiliary circles like a rotated version of standard constant‐parameter circle.
  
  Figure 7.12 Using ‘Smith 4.1’ to design a double‐stub tuner (impedance matcher) in Example 7.4.
4. From the tool bar, click on ‘Insert parallel line shorted’ button (denoted by SS: short stub) and move the cross‐type mouse cursor to click on the point Γ_A (DP2) (see Figure 7.12).
5. From the tool bar, click on ‘Insert serial line’ button and move the mouse cursor along the automatically appearing |Γ|‐circle by d = λ/8 to click on an intersection point Γ_B with the G = G₀ = 20 [mS]‐circle.
6. From the tool bar, click on ‘Insert parallel line shorted’ button and move the mouse cursor to click on Γ = 0.
7. Then, the designed double‐stub circuit with l_A = 0.2319λ and l_B = 0.0998λ will be shown in the Schematic pane.
Note that there is some forbidden region (Eq. (7.3.18)) in the sense that if the reflective coefficient of a load is in the region, no shunt element can move it to intersect the rotated G₀‐circle.

7.3.4 The Quarter‐Wave Transformer

As mentioned in Section 7.1, a real load impedance R_L can be matched to a feed line of characteristic impedance Z₀ by using a λ/4‐length lossless line (Figure 7.13(a)), called a QWT (quarter‐wave (λ/4) (impedance) transformer), with characteristic impedance Z₁

An N-section impedance transformer depicting the single section QWT (a) and N-section QWT (b). — **Figure 7.13** An N‐section impedance transformer.

(7.3.19)

which transforms the load impedance Z_L into

(7.3.20)

With this QWT, the reflection coefficient will be zero at β₀ = 2π/λ₀ (for a certain wavelength λ₀) so that there is no reflected wave beyond the point distant from the load by λ₀/4. However, at other frequency with βl ≠ π/2, the reflection coefficient response will be

(7.3.21a)

(7.3.21b)

Now, we consider a multisection transformer connected between a TL of characteristic impedance Z₀ and a load R_L as shown in Figure 7.13 where Z_n denotes the characteristic impedance of each section n. The impedance matching occurs if

(7.3.22)

where Z_in,n is the input impedance of the nth section that can be obtained from Eq. (7.1.17) with the input impedance of the (n + 1)th section as the load impedance as

(7.3.23)

There are two classical N‐section transformers, the binomial transformer and the Chebyshev transformer where the reflection coefficient of the former over the frequency βd is flat while that of the latter has equiripples as a result of trying to minimize the maximum deviation for better response.

The following MATLAB function ‘multi_line()’ returns the input impedance and reflection coefficient response of a QWT with characteristic impedance vector, etc., given as input arguments.

function [Zin,Gam,bb]=multi_line(Z0s,ls,Z0,ZL,b)
% To find the input reflection coefficient and impedance of a multi-line
% consisting of multisections each of impedance Z0s(m) and length ls(m).
% Input:
%  Z0s = Vector of characteristic impedances of each section of line
%  ls = Length(s) of each section, which is a scalar for equal length.
%  Z0 = Characteristic impedance of transmission line
%  ZL = Load resistance
%  b = Phase constant (a number or a vector with 2*pi*fc/c as 1st entry)
%     where fc=cutoff frequency, c=light velocity
% Output:
%  Zin = Input impedance, Gam = Reflection coefficient
N=numel(Z0s);
if numel(ls)<2, ls=repmat(ls,1,N); end
if numel(b)>1, bb=b(2:end); b=b(1); else bb=pi/90*[0:360]; end
Zl=@(ZL,Z0,bl)(ZL+j*Z0*tan(bl))./(Z0+j*ZL.*tan(bl))*Z0;
Zi(N+1)=ZL; Zibl=ZL;
for n=N:-1:1
  Zi(n)=Zl(Zi(n+1),Z0s(n),b*ls(n)); % Eq. (7.3.23)
  Z0s_(n)=sqrt(Zi(n)*Zi(n+1)); % Eq. (7.3.19)
  Zibl=Zl(Zibl,Z0s(n),bb*ls(n)); % Cascaded reflection response
end
Zin=Zi(1); Gam=(Zibl-Z0)./(Zibl+Z0); % Eq. (7.3.21b) Ref coef response
Mismatching = norm(Z0s-Z0s_)

7.3.4.1 Binomial Multisection QWT

According to the design method of binomial N‐section QWT introduced in [P-1], the characteristic impedance of each section can be approximately determined in turn from the first one as

(7.3.24)

This design process has been cast into the following MATLAB function ‘binomial_QWT()’.

function Z0s=binomial_QWT(Z0,ZL,N)
% To determine characteristic impedances of Nth-order binomial QWT
% Input : Z0 = Characteristic impedance of the transmission line
%         RL = Load resistance, N = Number of sections
% Output: Z0s = Characteristic impedances of each section of binomial QWT
A=2^-N*(ZL-Z0)/(ZL+Z0); Zp=Z0;
for n=N:-1:1
  logZn=log(Zp)+2^-N*nchoosek(N,N-n)*log(ZL/Z0); % Eq. (7.3.24)
  Zp=real(exp(logZn)); Z0s(N+1-n)=Zp;
end

7.3.4.2 Chebyshev Multisection QWT

The following MATLAB function ‘Cheby_QWT()’ uses the functions ‘chebtr2()’ and ‘chebtr3()’ implementing the Collin/Riblet algorithm (presented in [O-1]) to design Chebyshev QWTs. It regards the fourth input argument as a desired value of (minimum) fractional bandwidth or a required (minimum) attenuation [dB] in the reflectionless band (relative to unmatched case) depending on whether its value is less than 1 or not.

function Z0s=Cheby_QWT(Z0,ZL,N,A_fB)
% To determine the characteristic impedance of Nth-order Chebyshev QWT
% Input:
%  Z0   = Characteristic impedance of the transmission line
%  RL   = Load resistance
%  N    = Number of sections
%  A_fb = Attenuation or Fractional bandwidth
%        depending on whether A_fb >=1 or not.
% Output:
%  Z0s = Characteristic impedances of each section of Chebyshev QWT
% This uses the MATLAB functions chebtr2() and chebtr3() presented in
%  "Electromagnetic Waves and Antennas" by Sophocles J. Orfanidis,
%  which is available at <<http://www.ece.rutgers.edu/~orfanidi/ewa/>,
%  much appreciated.
if A_fB<1, Y0s=chebtr2(1/Z0,1/ZL,N,A_fb); % with A_fb=fb/f0 (fractional BW)
 else Y0s=chebtr3(1/Z0,1/ZL,N,A_fb); % with A_fb=As (attenuation)
end
Z0s=1./Y0s(2:end-1);

We can run the following MATLAB script “test_QWT.m” to use ‘binomial_QWT()’ and ‘Cheby_QWT()’ to design 1,2,3,4‐section binomial and Chebyshev QWTs and use ‘multi_line()’ to plot their reflection coefficient responses as shown in Figure 7.14, which shows the following:

Three graphs with curves labeled N = 1, N = 2, N = 3, and N = 4 representing the reflection responses for binomial QWT (a) and Chebyshev QWT (fractional BW specified) (b1) and Chebyshev QWT (attenuation specified) (b2). — **Figure 7.14** Reflection coefficient responses of binomial and Chebyshev QWTs.

Chebyshev QWTs ((b1) and (b2)) have wider bandwidth than binomial QWTs ((a)).
Chebyshev QWTs with the fractional bandwidth (BW) specified have smaller ripples as the number N of sections increases.
Chebyshev QWTs with the attenuation specified have larger attenuations paying the cost of larger equiripples.

%test_QWT.m
Z0=50; ZL=100; % Line characteristic impedance and Load impedance
fb=0.8; % Required fractional bandwidth (f2-f1)/fc where fc=(f1+f2)/2
A=10; % Attenuation[dB] of reflectionless band (relative to unmatched case)
b=2*pi; l=1/4; % Phase constant and Length of each section
for N=1:5
 Z0sb=binomial_QWT(Z0,ZL,N) % Binomial QWT
 [Zin,Gam,bb]=multi_line(Z0sb,l,Z0,ZL,b);
 subplot(231), plot(bb*l,abs(Gam)), hold on
 Z0sCf=Cheby_QWT(Z0,ZL,N,fb) % Chebyshev QWT with fractional BW specified
 [Zin,Gam,bb]=multi_line(Z0sCf,l,Z0,ZL,b);
 subplot(232), plot(bb*l,abs(Gam)), hold on
 Z0sCa=Cheby_QWT(Z0,ZL,N,A) % Chebyshev QWT with attenuation specified
 [Zin,Gam,bb]=multi_line(Z0sCa,l,Z0,ZL,b);
 subplot(233), plot(bb*l,abs(Gam)), hold on
end

Schematics of Richard’s transformation illustrating the replacement of L by a SS (short-circuited stub) (a) and replacement of C by an OS (open-circuited stub) (b). — **Figure 7.15** Richard's transformation.

7.3.5 Filter Implementation Using Stubs^[P-1]

The LC ladder realization using lumped elements discussed in Section 6.3.1 has two problems if they are going to be used at microwave frequencies. First, lumped elements such as inductors and capacitors are available only for a limited range of values. Second, the distances between components is not negligible at microwave frequencies. For these reasons, distributed components like short‐ or open‐circuited stubs (TL sections) are used. To implement an LC ladder using stubs, we use Richard's transformation to convert lumped elements to stubs and then use Kuroda's identities to transform series/parallel stubs into parallel/series stubs.

<Richard's Transformation>

To understand the Richard's transformation, we need to recall Eqs. (7.3.2a,b) that the impedances of short‐ and open‐circuited stubs with characteristic impedance Z₀ are

(7.3.25a,b)

and therefore, with characteristic impedance

(7.3.26a,b)

a short‐circuited stub (SS) and an open‐circuited stub (OS) of length l = λ₀/8 (λ₀ = v_p/f₀) can act as an inductor of impedance jΩL and a capacitor of impedance 1/jΩC, that is,

(7.3.27a,b)

respectively, at a frequency

(7.3.28)

Here v_p is the phase velocity of signal propagating on the line and f₀ is some reference frequency, which is often the cutoff frequency f_c of a filter implemented using the SSs and OSs, corresponding to

(7.3.29)

just like the unity cutoff frequency of a low‐pass filter (LPF) prototype. Since the tangent function is periodic in its argument with period π, the impedances of the SSs and OSs are periodic in f with period 4f₀. Here, replacing inductance L and capacitance C by SS (with characteristic impedance Z₀ = L) and OS (with characteristic impedance Z₀ = 1/C), respectively, through Eq. (7.3.26) based on Eq. (7.3.28) (see Figure 7.15) is referred to as the Richard's transformation where their impedances are regarded as Eq. (7.3.27a,b).

Note that in principle, the inductors and capacitors of a lumped‐element filter can be replaced with SSs and OSs, respectively, and if all the stubs are of equal length, they are said to be commensurate.

<Kuroda's Identities>

Figure 7.16(a1)‐(b4) shows the four Kuroda's identities that can be used to transform series/parallel stubs into parallel/series stubs, physically separate stubs, and change impractical characteristic impedances into more realizable ones without changing any two‐port network property such as the input and output impedances. To understand and get familiar with them, let us show the equivalence of each pair of two‐port networks in terms of the input impedance by running the following MATLAB script “show_Kuroda.m” and checking if Z_in,a = Z_in,b for the four identities where the ideal transformer with turns ratio of 1: n in Figure 7.16(b3) makes the load impedance Z_L (on the secondary [load] side) reflected onto the primary (source) side as Z_L/n²:

(7.3.30a,b)

(7.3.31a,b) equation

(7.3.32a,b)

(7.3.33a,b) equation

%show_Kuroda.m
%Zl=@(ZL,Z0,bl)(ZL+j*Z0*tan(bl))./(Z0+j*ZL.*tan(bl))*Z0;
Zu=@(ZL,Z0)(ZL+j*Z0)./(Z0+j*ZL)*Z0; % with bl=pi/4
syms Z1 Z2 ZL; n=1+Z2/Z1; %
% Kuroda Identity 1 (Table 8.7 of [P-1])
Zina=parallel_comb([-j*Z2 Zu(ZL,Z1)]) % Fig. 7.16(a1)
Zinb=Zu(j*Z1/n+ZL,Z2/n), simplify(Zina-Zinb) % Fig. 7.16(b1)
% Kuroda Identity 2 (Table 8.7 of [P-1])
Zina=j*Z1+Zu(ZL,Z2) % Fig. 7.16(a2)
Zinb=Zu(parallel_comb([-j*n*Z2 ZL]),n*Z1) % Fig. 7.16(b2)
simplify(Zina-Zinb)
% Kuroda Identity 3 (Table 8.7 of [P-1])
Zina=parallel_comb([j*Z1 Zu(ZL,Z2)]) % Fig. 7.16(a3)
Zinb=Zu(parallel_comb([j*Z1/n ZL/n^2]),Z2/n) % Fig. 7.16(b3)
simplify(Zina-Zinb)
% Kuroda Identity 4 (Table 8.7 of [P-1])
Zina=-j*Z2+Zu(ZL,Z1) % Fig. 7.16(a4)
Zinb=Zu(-j*n*Z2+n^2*ZL,n*Z1), simplify(Zina-Zinb) % Fig. 7.16(b4)

function [Z0,Z2,SPso]=Kuroda(Z1,SPso,Z0)
% SPso='Ss' for Series-to-Parallel of Z1(L) or Z1(short)
% SPso='So' for Series-to-Parallel of Z1(C) or Z1(open)
% SPso='Ps' for Parallel-to-Series of Z1(L) or Z1(short)
% SPso='Po' for Parallel-to-Series of Z1(C) or Z1(open)
% where Z0 is the characteristic impedance of a UE (unit element)
% Input: Z1 = Impedance (Reactance) of the other element than UE
%        Z0 = Characteristic impedance of UE
% Output: Z0 = Characteristic impedance of UE
%         Z2 = Impedance (Reactance) of the other element than UE
if nargin<3|~isnumeric(Z0)|Z0==0, Z0=1; end
if nargin<2, SPsp='Ss'; end
SPso=upper(SPso);
if SPso(1)=='S' % Series-to-Parallel conversion
  if SPso(2)=='S', SPso='Po'; n=1+Z0/Z1; Z2=-n*Z0; Z0=n*Z1; % Kuroda 2
  else SPso='Po'; Z1=abs(Z1); n=1+Z1/Z0; Z0=n*Z0; Z2=[-n*Z1 n]; % K-4
  end
  % The +/- sign of Z2 means L(short stub)/C(open stub), respectively.
else % Parallel-to-Series conversion
  if SPso(2)=='S', SPso='Ss'; n=1+Z0/Z1; Z0=Z0/n; Z2=[Z1/n 1/n]; % K-3
  else SPso='Ss'; Z1=abs(Z1); n=1+Z1/Z0; Z2=Z0/n; Z0=Z1/n; % K-1
 end
end

Figure 7.16(c) shows a unit element (UE) of characteristic impedance Z₀ and length l = λ₀/8 (λ₀ = v_p/f₀) or equivalently, electrical (phase) length βl = π/4 that has input impedances at ports a and b as

(7.3.34a,b) equation

This implies that the frequency characteristic of a TL is not affected at frequency f₀ = v_p/λ₀ by the (cascade) insertion of UE (a λ₀/8‐long TL) if only its characteristic impedance is equal to that of the TL. That is why UEs are inserted between a TL and a distributed or lumped element as a preparation for applying some Kuroda's identity.

Example 7.5 Design of LPF Using Stubs [P-1]

With source/load impedances R_s = 50 Ω/Z_L = 50 Ω, design a third‐order Chebyshev (equiripple) LPF with passband ripple R_p = 3 dB and f_c = 4 GHz for fabrication using microstrip lines with shunt stubs only.

To design such an LPF, let us take the following steps:

(Step 1) Determine the normalized LPF prototype element values by running the following MATLAB statements:

 >>Rs=50; % Source (line) characteristic impedance
   N=3; type='C'; Rp=3; % for 3rd order Chebyshev with Rp=3dB
   gg=LPF_ladder_coef(N,type,Rp)

This yields

      gg = 3.3489e+00 7.1167e-01 3.3489e+00 1.0000e+00

(Step 2) Determine the normalized LC values of the prototype LPF by running the following MATLAB statements:

 >>Rs0=1; pT='T'; wc=1; % for a T-type prototype filter
   [LCRs,LCRs_,Gs]=LC_ladder(Rs0,gg,pT,wc);
   LCRs, LCRs_, pretty(simplify(Gs))

This yields

  LCRs = 3.3489 0.7117 3.3489 1.0000
  LCRs_ = 'L(h)' 'C(v)' 'L(h)' 'R(v)'
  5708990770823839524233143877797980545530986496/
                                                    3
  (45566839819273848699938746290718052889107427673 s
                                                    2
  + 27212766570123614952226532668306611436833996800 s
  + 42300936562233083587497427629177459394738126848 s
  + 11417981541647679048466287755595961091061972992)

This, with ‘h’ = horizontal and ‘v’ = vertical, means the filter of Figure 7.17.1.

(Step 3) Convert the L/C values into the impedances of short‐/open‐circuited stubs by running
```
 >>[Zs,SPsos]=LC2Z(LCRs,LCRs_)
```
- where the following MATLAB function ‘LC2Z()’ is supposed to be saved on the computer.

Figure 7.17.1 Normalized prototype low‐pass filter (LPF) of order N = 3.

function [Zs,SPsos]=LC2Z(LCs,LCs_)
% To convert the L/C's into short/open-circuited stubs
for m=1:numel(LCs)
  if LCs_{m}(1)=='L'
     Zs(m)=LCs(m); % Eq. (7.3.26a)
     if LCs_{m}(3)=='h', SPsos{m}='Ss'; else SPsos{m}='Ps'; end
  elseif LCs_{m}(1)=='C'
     Zs(m)=-1/LCs(m); % Eq. (7.3.26b)
     if LCs_{m}(3)=='h', SPsos{m}='So'; else SPsos{m}='Po'; end
  else break;
 end
end

This yields the characteristic impedances of the Ss (series short‐circuited stub), Po (parallel open‐circuited stub), and Ss constituting the microstrip filter in Figure 7.17.2(a):
```
 Zs = 3.3489  -1.4052  3.3489,  SPsos = 'Ss'  'Po'  'Ss'
```
where the +/− signs stand for L/C or short/open‐circuited stubs, respectively.
(Step 4) Now, to use Kuroda identity 2 to convert the two SSs (series short‐circuited stubs) Z₁ and Z₃ into POs (parallel open‐circuited stubs), we insert UEs between Z₁/Z₂ and the TL (as depicted in Figure 7.17.2(b)) and run the following statements:
```
 >>[Z0s(1),Zs(1),SPsos{1}]=Kuroda(Zs(1),SPsos{1});
   [Z0s(3),Zs(3),SPsos{3}]=Kuroda(Zs(3),SPsos{3});
   Z0s, Zs, SPsos
```
- This yields the characteristic impedances of the UEs and stubs
```
    Z0s =  4.3489         0     4.3489
    Zs = -1.2986   -1.4052    -1.2986
    SPsos = 'Po'     'Po'       'Po'
```
- With the +/− signs of Zs values standing for L/C or short‐/open‐circuited stubs, respectively, this implies the microstrip filter in Figure 7.17.2(c) where, say, Zs(1) = −1.2986 together with SPsos(1) = Po means that the first stub of characteristic impedance 1.2986 Ω is a parallel open‐circuited one and, say, Z0s(1) = 4.3489 means that the characteristic impedance of the UE next to the first stub is 4.3489 Ω.
(Step 5) Denormalize the normalized characteristic impedances by multiplying them with R_s = 50 Ω:
```
 >>Z0s=Z0s*Rs, Zs=Zs*Rs
```
Figure 7.17.2 Microstrip filter design for Example 7.5.
- This yields
```
     Z0s = 217.4466     0  217.4466
     Zs = -64.9301  -70.2575  -64.9301
```
- This implies the microstrip filter in Figure 7.17.2(d).
(Step 6) Frequency scale the circuit by determining the line/stub lengths as l = λ/8 = v_p/f_c/8[m] where v_p[m/s] is the phase velocity of signal propagating on the line and f_c is the cutoff frequency given as a design specification.
(Step 7) To plot the frequency responses of the two filters, one (called a lumped‐element filter) consisting of lumped elements L/C (Figure 7.17.1) and the other (called a distributed‐element filter) consisting of microstrip lines/stubs (Figure 7.17.2(d)), we do the following:

We first construct a vector of the characteristic impedances of each stub/line:
```
     >>Z0strip=[Zs(1) Z0s(1) Zs(2) Z0s(3) Zs(3)];
```
- and the corresponding cell
```
>>SPsos={SPsos{1} 'Tl' SPsos{2} 'Tl' SPsos{3}};
```
- which contains one of the five strings {‘Po’, ‘Tl’, ‘Po’, ‘Tl’, ‘Po’} for each stub/line depending on whether it is a parallel‐/series‐connected stub or a section of TL and also whether it is short‐ or open‐circuited if it is a stub.

Referring to Eqs. (7.3.27) and (7.3.28), set the frequency range which to plot the frequency response for:

      >>l=1/8; fc=4e9; % Stub length and Cutoff frequency
        fN=linspace(0,pi/4,1000); % Normalized frequency range
        w=fN*2*pi*fc; % Real frequency range
        bl=2*pi*w*l/fc; % Range of bl (bl=2*pi*l at f=fc=4e9[Hz])

To get the overall frequency response considering R_s, we use the MATLAB function ‘GZ_strip()’ to get the frequency response G(ω) (Gw) and input impedance Z_in(ω) (Zinw) of the filter (excluding R_s) for the frequency range set above:
```
      >>ZL=Rs; [Gw,Zinw]=GZ_strip(Z0strip,SPsos,ZL,bl);
```

and we multiply G(ω) with the voltage divider ratio Z_in(ω)/(R_s+Z_in(ω)) on the source side:
```
       >>Gw=Zinw./(Rs+Zinw).*Gw; % the overall frequency response
```
Here, the below MATLAB function ‘GZ_strip()’ is supposed to be saved on the computer.

function [Gw,Zinw]=GZ_strip(Z0strip,SPsos,ZL,bl)
% To get the frequency response Gw and input impedance Zinw
% Input:
% Z0strip = Impedances of stub, line, stub, line, ...
% SPsos   = e.g. {'Ss','Po',..} for Series-short, Parallel-open, ..
%  ZL     = Load impedance
%  bl     = Range of electrical length
% Output:
%  Gw     = Frequency response
%  Zinw   = Input AC impedance
Zl=@(ZL,Z0,bl)(ZL+j*Z0*tan(bl))./(Z0+j*ZL.*tan(bl))*Z0;
% Impedance of a line of characteristic impedance Z0 and
% electrical length bl with load ZL
Zls=@(Z0,bl)j*Z0*tan(bl); % Impedance of a an SS
Zlo=@(Z0,bl)-j*Z0./tan(max([bl; eps*ones(1,numel(bl))]));
% Impedance of an open-circuited stub
Zlso=@(Z0,bl,so)(so=='s')*Zls(Z0,bl)+(so=='o')*Zlo(Z0,bl);
Z_par=@(Z1,Z2)Z1.*Z2./(Z1+Z2); % Parallel(shunt) combination
Z0strip=abs(Z0strip); Zinw=ZL; I=1;
for m=numel(Z0strip):-1:1
  if isempty(SPsos{m})|Z0strip(m)<eps, continue; end
  SP=upper(SPsos{m}(1)); so=lower(SPsos{m}(2));
  if SP=='P'
   Zp=Z_par(Zlso(Z0strip(m),bl,so),Zinw); % Parallel combination
   I=I.*Zinw./Zp; Zinw=Zp; % Sum of shunt currents.
  elseif SP=='S'
   Zinw=Zinw+Zlso(Z0strip(m),bl,so); % Series combination
  elseif SP=='T', Zinw=Zl(Zinw,Z0strip(m),bl); % Serial line
 end
end
Gw=ZL./(Zinw.*I); % Ratio of voltage across ZL to Input voltage

To plot the magnitudes [dB] of the overall frequency responses considering R_s (with maximum of 0 dB) for the two filters, we run the following MATLAB statements:
```
      >>Gwmag=20*log10(max(abs(Gw),1e-5));
        clf, plot(w,Gwmag-max(Gwmag)), hold on
        s=j*w/fc; Gw0=eval(Gs); Gwmag0=20*log10(max(abs(Gw0),1e-5));
        plot(w,Gwmag0-max(Gwmag0),'r') % To make the maximum 0dB
```
- where Gs is the transfer function obtained in Step 2. This yields Figure 7.17.3.
An easy alternative to implement an LC ladder using stubs is a stepped‐impedance or hi‐Z, low‐Z filter [P-1] obtained by replacing the normalized values of inductance L and capacitance C with TL sections of high/low characteristic impedances Z_h/Z_l and electrical lengths

Figure 7.17.3 Frequency responses of the lumped‐element filter and distributed‐element filter.

(7.3.35a)

(7.3.35b)
- respectively.

Example 7.6 Design of Stepped‐Impedance Filter [P-1]

For line/load impedances R₀ = 50 [Ω]/Z_L = 50 [Ω], design a sixth‐order Butterworth LPF prototype with passband ripple R_p = 3 dB and implement it as a stepped‐impedance filter with Z_h = 150 [Ω] and Z_l = 10 [Ω].

To design such an LPF, we run the following MATLAB script “elec7e06.m”

%elec07e06.m: Butterworth LPF design in Example 8.7 of [P-1]
N=6; Rp=3; % Order and Passband ripple of Butterworth LPF
type='B'; % Butterworth
gg=LPF_ladder_coef(N,type,Rp)
% To find the L & C values of the prototype Pi filter
R0=50; Rs=R0; pT='pi'; wc=1; % for a prototype filter
[LCRs,LCRs_,Gs]=LC_ladder(Rs,gg,pT,wc);
disp('Values of the lumped elements L, C, and RL')
format short e; LCRs, LCRs_
LCs=LCRs(1:N); LCs_=LCRs_(1:N);
Zh=150; Zl=10; % Highest/lowest practical line impedances
bl=@(g,LC_)(LC_=='L')*g*R0/Zh+(LC_=='C')*g*Zl/R0; % Eq. (7.3.35a,b)
Z0=@(LC_)(LC_=='L')*Zh+(LC_=='C')*Zl;
% Find the electrical lengths of the hi/low-Z transmission line sections
% to replace the series inductors and shunt capacitors.
for m=1:N
 bls(m)=bl(gg(m),LCs_{m}(1)); Z0s(m)=Z0(LCs_{m}(1));
end
disp('Electrical lengths, Characteristic impedances of microstrip lines')
Z0s, Els=rad2deg(bls) % Characteristic impedances, Electrical lengths

Figure 7.18 Sixth‐order Butterworth 3 dB LPF realized with a П‐type LC ladder and implemented as a stepped‐impedance filter.

to get the electrical lengths and characteristic impedances of each TL section as

 Z0s =   10      150     10     150      10     150
 ELs = 5.9317  2.7009  2.2137  3.6896  1.6206  9.8862

This means the stepped‐impedance LPF of Figure 7.18(b).

7.3.6 Impedance Matching with Lumped Elements

This section presents several examples illustrating how some types of impedance matchers consisting of lumped elements can be designed by using the Smith chart.

Example 7.7 Design of L‐type Impedance Transformers Using the Smith Chart

Find L‐type LC circuits transforming the impedance Z₁ = 40 + j50[Ω] or admittance Y₁ = 1/Z₁ = 0.009756 − j0.012195 [S] represented by point Γ₁ into the impedance Z₂ = 50 + j17 [Ω] or Y₂ = 1/Z₂ = 0.017928 − j0.060954 [S] (so that it can be matched to Z_s = 50 − j17 [Ω]) by using the Smith chart and compare the results with those obtained in Example 6.12.

Noting that there are four paths along resistance/conductance curves from point Γ₁ (representing Z₁ = 40 + j50 [Ω]) to Γ₂ (representing Z₂ = 50 + j17 [Ω]), each via Γ₃, Γ₅, Γ₆, and Γ₇ in Figure 7.13, let us work with ‘Smith V4.1’ taking the following steps:

From the top menu bar, select Tools > Settings to open the Settings dialog box, set the values of Z₀ (characteristic impedance) and frequency to 50 [Ω] (default) and 1 [GHz], respectively, and click OK button. In fact, as long as the design objective is not to match the input impedance with the characteristic impedance, it does not matter whatever value Z₀ is set to, because ‘Smith V4.1’ accommodates even unnormalized impedances/admittances as they are.
From the toolbar (below the menu bar), select Keyboard to open Data Point dialog box, insert the impedance Z₁ = 40 + j50 [Ω] as a (start) point (DP1), and click OK button. Also, insert Z₂ = 50 + j17 [Ω] as a (target) point (DP2). If you want to cancel (undo) any work or delete the data points starting from the last one (in LIFO order), just right‐click anywhere on the Smith chart.
To design the L1‐type C‐C impedance matcher as shown in Figure 6.19(a1) corresponding to the path from Γ₁ to Γ₂ via Γ₃, click on ‘Insert series capacitor’ button and move the mouse cursor to click on Γ₃ (the intersection of the R = 40 Ω‐circle and G = 0.0179 S‐circle) at which the value of G = Re{Y} is close to 0.0179 S = 17.9 mS as can be read in the Cursor pane. Then, TP 3 will appear at Γ₃ and its impedance value will appear in the Data Points pane at the right side as shown in Figure 7.19.

Figure 7.19 Using ‘Smith 4.1’ to design L‐type impedance matchers in Example 7.7 or 6.12.
From the tool bar, click on ‘Insert parallel capacitor’ button and move the mouse cursor to click on Γ₂ (the intersection of the R = 50 Ω‐circle and the G = 17.9 mS‐circle) representing Z₂ = 50 + j17 [Ω]. Then, TP 4 will appear at Γ₂ and its impedance value will appear in the Data Points pane.
Then, the designed network with the parameter values C₂ = 6.4 pF and C₁ = 824.9 fH will be shown in the Schematic pane at the upper‐right part of the window. It conforms with Figure 6.19(a1).
If you want to read more digits of a numeric value or change it, double‐click on it to open the Edit Element dialog box.
Right‐click anywhere on the Smith chart twice to delete the last two data points.
To design the L2‐type L‐C impedance matcher as shown in Figure 6.19(b1) corresponding to the path from Γ₁ to Γ₂ via Γ₅, click on ‘Insert parallel capacitor’ button (from the tool bar) and move the mouse cursor to click on Γ₅ (the intersection of the G=0.009756 S ≈ 0.01 S‐circle and the R=50 Ω‐circle) at which the value of R=Re{Z} is close to 50 Ω as can be read in the Cursor pane.
From the tool bar, click on ‘Insert series inductor’ button and move the mouse cursor to click on Γ₂ (the intersection of the R=50 Ω‐circle and the X=17 Ω‐circle) representing Z₂=50 + j17 [Ω].
Then, the designed network with the parameter values C₂ = 3.5 pF and L₁ = 10.9 nH will be shown in the Schematic pane at the upper‐right part of the window. It conforms with Figure 6.19(b1).

Example 7.8 Design of T‐type Impedance Transformers Using the Smith Chart

Make T‐type ladders with quality factor Q = 3 for transforming Z₁ = 60 − j30 [Ω] to Z₂ = 10 + j20 [Ω] at frequency f_c = 1 GHz.

Note that, with Q ≤ 3, there are four paths along resistance/conductance curves from point Γ₁ (representing Z₁=60 − j30[Ω]) to Γ₂ (representing Z₂=10+j20[Ω]), each via Γ₅‐Γ₃, Γ₅‐Γ₄, Γ₆‐Γ₄, and Γ₆‐Γ₃, in Figure 7.20 where Γ₃ and Γ₄ have been determined as the intersections between 10 Ω‐resistance circle and Q= 3‐circles. Let us work with ‘Smith V4.1’ taking the following steps:

From the top menu bar, select Tools > Circles to open the Circles dialog box, insert 3 as a value of Q, and click OK to close the dialog box so that two Q=3‐circles appear above/below the real axis.
From the top menu bar, select Tools > Settings to open the Settings dialog box, set the values of Z₀ and frequency to 50[Ω] (default) and 1[GHz], respectively, and click OK.
From the toolbar, select Keyboard to open Data Point dialog box, insert Z₁ = 60 − j30[Ω] as a (start) point (DP1), and click OK. Also, insert Z₂=10 + j20[Ω] as a (target) point (DP2). See Figure 7.20.

Figure 7.20 Using ‘Smith 4.1’ to design T‐type impedance matchers in Example 7.8 or 6.13.
To designate Γ₃ and Γ₄ as intermediate points just before the target point Γ₂, click on the ‘Mouse’ button, find two points with R = R₂ = 10[Ω] and Q = 3, one by one, on the two Q=3‐circles (keeping your eye on the values of Q and R = Re{Z} in the Cursor pane), and click on them to insert as data points. In fact, noting that the R‐value of (the resistance circle through) Γ₂ is 10 [Ω], we can compute the X‐value of (the reactance circle through) Γ₃ with Q = 3 as
(E7.8.1)

Noting that the number of data points is not allowed to exceed 5, designate Γ₅ as an intermediate point by clicking on a point with G₅ = G₃ = 10 [mS] (equal to the conductance value of Γ₃) and R₅ = R₁ = 60 [Ω] (equal to the resistance value of Γ₁) (keeping your eye on the values of R = Re{Y} and G = Re{Y} in the Cursor pane) to insert it. Why is that? Because we are going to connect the reactive elements (each corresponding to the Γ₁‐Γ₅ and Γ₅‐Γ₃ sections, respectively) in series and in parallel, respectively. In fact, from R₅ = 60 Ω and G₅ = 0.01 S, we can compute the X₅ for Γ₅ as
(E7.8.2)

or
```
    >>R5=60; G5=0.01; [Z5,Y5]=RG2ZY(R5,G5) % see Problem 7.9
```
To design the T‐type C‐L‐L impedance matcher as shown in Figure 6.22(a) corresponding to the path from Γ₁ to Γ₂ via Γ₅‐Γ₃, click on ‘Insert series inductor’ and move the mouse cursor to click on Γ₅ (corresponding to Z₅=60+j49[Ω]). Then, click on ‘Insert parallel inductor’ and move the mouse cursor to click on Γ₃ (corresponding to Z₃ = 10 + j30 [Ω]). Then, click on ‘Insert series capacitor’ and move the mouse cursor to click on Γ₂ (corresponding to Z₂=60− j30[Ω]). Then, the designed network appears in the Schematic pane as shown in Figure 7.20 where L₃ = 12.61 nH, L₂ = 7.306 nH, and C₁ = 15.91 pF.
To design the T‐type L‐C‐C impedance matcher as shown in Figure 6.22(d) corresponding to the path from Γ₁ to Γ₂ via Γ₆‐Γ₄, click on ‘Insert series capacitor’ and move the mouse cursor to click on Γ₆ (corresponding to Z₆=60− j49[Ω]). Then, click on ‘Insert parallel capacitor’ and move the mouse cursor to click on Γ₄ (corresponding to Z₄ = 10 − j30 [Ω]). Then, click on ‘Insert series inductor’ and move the mouse cursor to click on Γ₂. Then, the designed network appears in the Schematic pane (Figure 7.20) where C₃ = 8.383 pF, C₂ = 3.488 pF, and L₁ = 7.945 nH.

Example 7.9 Design of П‐type Impedance Transformers Using the Smith Chart

Make П‐type ladders with quality factor Q = 2 for transforming Z₁ = 10 − j10 [Ω] to Z₂=20+j40[Ω] at frequency f_c = 2.4 GHz.

Note that, with Q ≤ 2, there is a path along resistance/conductance curves from point Γ₁ (representing Z₁ = 10 − j10 [Ω] or equivalently Y₁ = 50 + j50 [mS]) to Γ₂ (representing Z₂=20+j40[Ω] or equivalently Y₂=10+j20[mS]) via Γ₃‐Γ₄ in Figure 7.21 where Γ₄ has been determined as the intersection between G = 10 mS‐circle and the lower Q = 2‐circle. Let us work with ‘Smith V4.1’ taking the following steps:

Figure 7.21 Using ‘Smith 4.1’ to design П‐type impedance matchers in Example 7.9 or 6.14.

From the top menu bar, select Tools > Circles to open the Circles dialog box, insert 2 as a value of Q, and click OK to close the dialog box so that two Q = 2‐circles appear above/below the real axis.
From the top menu bar, select Tools > Settings to open the Settings dialog box, set the values of Z₀ and frequency to 50[Ω] (default) and 2.4[GHz], respectively, and click OK.
From the toolbar, select Keyboard to open Data Point dialog box, insert Z₁=10 − j10[Ω] as a (start) point (DP1), and click OK. Also, insert Z₂ = 20 + j40 [Ω] as a (target) point (DP2). See Figure 7.21.
To set an intermediate point Γ₄ just before the target point Γ₂, click on the ‘Mouse’ button, find a point with G = G₂ = 10[mS] and Q = 2 on the lower Q = 2‐circle (keeping your eye on the values of Q and G = Re{Y} in the Cursor pane), and click on it to insert as data points. Why not on the upper Q=2‐circle? Because Γ₂ is on the upper Q=2‐circle. In fact, noting that the G‐value of (the G‐circle through) Γ₂ is 0.01[S], we can compute the B‐value of (B‐circle through) Γ₄ with Q = 2 as

This implies that the admittance and impedance for Γ₄ are

(E7.9.1)

Why have we set G₄ = G₂? Because we are going to connect the reactive element (corresponding to the Γ₄‐Γ₂ section) in parallel with Z₂, as denoted by ‘Lp’.

To determine another intermediate point Γ₃ (between Γ₁ and Γ₄), we set the conductance and resistance to 50 mS (that for Γ₁) and 20 Ω (that for Γ₄), respectively, because we are going to connect the reactive elements (each corresponding to the Γ₁‐Γ₃ and Γ₃‐Γ₄ sections, respectively,) in parallel and in series, respectively:
(E7.9.2)

or
```
  >>R3=20; G3=0.05; [Z3,Y3]=RG2ZY(R3,G3)
```
If you somehow have got the numeric value of the Z, Y, or Γ of a data point to be inserted, you can select Keyboard to open the Data Point dialog box and enter the value.
To design the П‐type L‐C‐L impedance matcher as shown in Figure 6.23(b) corresponding to the path from Γ₁ to Γ₂ via Γ₃‐Γ₄, click on ‘Insert parallel inductor’ and move the mouse cursor to click on Γ₃ (corresponding to Z₃ = 20 + j0 [Ω]). Then, click on ‘Insert series capacitor’ and move the mouse cursor to click on Γ₄ (corresponding to Z₄=20 − j40[Ω]). Then, click on ‘Insert parallel inductor’ and move the mouse cursor to click on Γ₂ (corresponding to Z₂ = 20 + j40 [Ω]). Then, the designed network appears in the Schematic pane as shown in Figure 7.22 where
(E7.9.3)

Regrettably, this design result is quite different from that of Figure 6.23(b). To figure out the difference between the design result using the Smith chart and that using the MATLAB function ‘imp_matcher_pi()’, we have drawn the latter on the Smith chart in Figure 7.22, which shows that some intermediate points such as Γ₄ and Γ₆ do not satisfy the upper bound condition for Q. It implies that the design result using the Smith chart satisfies the Q‐condition at every stage while that using MATLAB may not satisfy the Q‐condition at some stage even if both design results satisfy the condition as a whole.

A window displaying a Smith chart at the left and Schematic, Datapoints, Circles, and Cursor panes at the right. The Schematic pane displays two networks. — **Figure 7.22** The design results of Figure 6.23(b) and (d) (Example 6.14) drawn on the Smith chart.

Example 7.10 Impedance Matching with an LC Ladder

Make LC ladder‐type impedance matchers (with quality factor Q ≤ 1 at every stage) for transforming Z₁=12.5+j10[Ω] to 50 Ω at frequency f_c=100 MHz.

Let us work with ‘Smith V4.1’ taking the following steps to make an LPF LC ladder‐type impedance matcher:
1. From the top menu bar, select Tools > Circles to open the Circles dialog box, insert 1 as a value of Q and click OK to close the dialog box so that two Q=1‐circles appear above/below the real axis.
2. From the top menu bar, select Tools > Settings to open the Settings dialog box, set the values of Z₀ and frequency to 50 [Ω] (default) and 100 [MHz], respectively, and click OK.
3. From the toolbar, click on ‘Keyboard’ to open Data Point dialog box, insert the impedance Z₁ = 12.5 + j10 [Ω] or Y₁ = 48.8 + j39 [mS] as a (start) point (DP1), and click OK. The target point Γ = 0, i.e. the origin corresponding to Z₀ = 50 [Ω], is so apparent that it does not have to be designated as a data point.
  
  Figure 7.23 Using ‘Smith 4.1’ for impedance matching between Z₁ = 12.5 + j10 and Z₆ = 50 in Example 7.10(a).
4. To design an LC ladder‐type impedance matcher as an LPF, take the following steps, referring to Figure 7.23:
  - Click on ‘Insert parallel capacitor’ and move the mouse cursor to click on Γ₂ (the intersection of G = G₁ = 48.8 [mS]‐circle and the horizontal axis).
  - Click on ‘Insert series inductor’ and move the mouse cursor to click on Γ₃ (the intersection of R = R₂ = 1/G₁ = 20.5[Ω]‐circle and the upper Q = 1‐circle corresponding to Y₃ = G₃+jB₃[S]).
  - Click on ‘Insert parallel capacitor’ and move the mouse cursor to click on Γ₄ (the intersection of G=G₃‐circle and the horizontal axis).
  - Click on ‘Insert series inductor’ and move the mouse cursor to click on Γ₅ (the intersection of R=R₄=1/G₃‐circle and the G=1/R₀‐circle).
  - Click on ‘Insert parallel capacitor’ and move the mouse cursor to click on Γ₆ = 0 (i.e. the origin corresponding to Z₀ = 50[Ω]).
5. Then, the designed LC ladder appears in the Schematic pane in Figure 7.23.

function Xs=imp_matcher_LC_LPF1(R0,ZL,fc,Qm)
% Design an LC ladder LPF with Q<=Qm to make impedance matching with R0
% when RL<R0
% Copyleft: Won Y. Yang, [email protected], CAU for academic use only
EPS=1e-4;
if abs(ZL-R0)<EPS, Xs=[]; return; end
RL=real(ZL); XL=imag(ZL); YL=1/ZL; GL=real(YL); BL=imag(YL);
if abs(XL)<EPS
  M1=abs(Qm); M2=sqrt(R0/RL-1);
  % Does the R-circle cross G=1/R0-circle or Q-circle first?
  [X,imin]=min([M1 M2]); X=X*RL;
  Z=RL+j*X; Y=1/Z; B=imag(Y);
  if abs(M1-M2)<EPS|imin>1 % If it crosses G=1/R0-circle first
   Xs=[1/B X]; % Series L & Parallel C from Load
  else % If it crosses Q-circle first
  RL=1/real(Y); Xsp=imp_matcher_LC_LPF1(R0,RL,fc,Qm);
  Xs=[Xsp 1/B X]; % Series L & Parallel C
 end
elseif XL>0 % First, C in parallel with ZL for LPF
 RL=1/GL;
 Xsp=imp_matcher_LC_LPF1(R0,RL,fc,Qm);
 Xs=[Xsp 1/BL]; % Parallel C
else % if XL<0
 % Move the point from the 3rd quadrant into the 2nd one by Ls (series L).
 M1=abs(Qm); M2=sqrt(R0/RL-1);
 [X,imin]=min([M1 M2]); X=X*RL;
 Z=RL+j*X; Y=1/Z; B=imag(Y);
 if abs(M1-M2)<EPS|imin>1
  Xs=[1/B (X-XL)]; % Series L and Parallel C from Load
 else
  R=1/real(Y); Xsp=imp_matcher_LC_LPF1(R0,R,fc,Qm);
  Xs=[Xsp 1/B (X-XL)]; % Series L and Parallel C
 end
end

function [LCs,LCs_]=L_or_C_from_X(Xs,fc,vh)
% vh = 'v' or 'h' for vertical/parallel or horizontal/series first
% Find L or C from given values of reactance.
% Copyleft: Won Y. Yang, [email protected], CAU for academic use only
wc=2*pi*fc;
for n=1:length(Xs)
  X=Xs(n);
  if abs(X)<eps|abs(X)>realmax, LCs(n)=0; LCs_{n}='????';
   else
   if X>0, LCs(n)=X/wc; LCs_n='L'; unit='H';
    else LCs(n)=-1/X/wc; LCs_n='C'; unit='F';
   end
   if nargin>2
    LCs_n=[LCs_n '(' vh ')'];
    if vh=='v', vh='h';
     else vh='v';
    end % To alternate 'L' and 'C'
   else fprintf([LCs_n '(%d)=%10.3e[' unit ']
'],n,LCs(n));
   end
   LCs_{n}=LCs_n;
 end
end

This LPF LC ladder‐type impedance matcher (with quality factor Q ≤ 1 at every stage) can also be designed by using the above MATLAB function ‘imp_matcher_LC_LPF1()’ as follows:

  >>fc=1e8; Qm=1; % Frequency and Maximum quality factor
   Xs=imp_matcher_LC_LPF1(50,12.5+10i,fc,Qm)
   [LCs,LCs_]=L_or_C_from_X(Xs,fc,'v')
   LCs  = 1.4913e-11 3.0573e-08 3.8818e-11 3.2627e-08 6.2109e-11
   LCs_= 'C(v)'     'L(h)'     'C(v)'     'L(h)'     'C(v)'

where the third input argument of ‘L_or_C_from_X()’ has been given as 'v' (meaning ‘vertical’) since we know that the last section of the path from Γ₁ to Γ₂ leads to a parallel C and consequently the first element of the LC ladder (from the source) is to be connected vertically. This design result conforms with that obtained using ‘Smith 4.1’ and shown in the Schematic pane in Figure 7.23.

Now, to design an LC ladder‐type impedance matcher as a HPF, take the following steps:
1. Right‐click anywhere on the Smith chart four times to delete the existing four data points Γ₂, Γ₃, Γ₄, and Γ₅ (in LIFO order) except Γ₁.
2. From the top menu bar, select Tools > Circles to open the Circles dialog box, insert 1 as a value of Q and click OK to close the dialog box so that two Q = 1‐circles appear above/below the real axis.
3. From the top menu bar, select Tools > Settings to open the Settings dialog box, set the values of Z₀ and frequency to 50[Ω] (default) and 100 [MHz], respectively, and click OK.
4. From the toolbar, click on ‘Keyboard’ to open the Data Point dialog box, insert Z₁=12.5+j10[Ω] as a (start) point (DP1), and click OK.
5. To design an LC ladder‐type impedance matcher, do the following, referring to Figure 7.24:
  - Click on ‘Insert series capacitor’ (from the tool bar) and move the mouse cursor to click on the point Γ₂ (the intersection of R=R₁=12.5[Ω]‐circle and the lower Q=1‐circle) corresponding to Y₂=40+j40[mS].
  - Click on ‘Insert parallel inductor’ and move the mouse cursor to click on the point Γ₃ (the intersection of G=G₂=40[mS]‐circle and the horizontal axis).
  - Click on ‘Insert series capacitor’ and move the mouse cursor to click on the point Γ₄ (the intersection of R = 1/G₃ = 1/0.04=25 Ω‐circle and the G=1/R₀‐circle, which coincidentally happens to be on the lower Q = 1‐circle), corresponding to Y₄=20+j20 [mS].
    
    Figure 7.24 Using ‘Smith 4.1’ for impedance matching between Z₁ = 12.5 + j10 and Z₅ = 50 in Example 7.10(b).
  - Click on ‘Insert parallel inductor’ and move the mouse cursor to click on the point Γ₅ = 0 (i.e. the origin corresponding to Z₀ = 50 [Ω]).
6. Then, the designed LC ladder appears in the Schematic pane in Figure 7.24.
This HPF LC ladder‐type impedance matcher (with quality factor Q ≤ 1 at every stage) can also be designed by using the above MATLAB function ‘imp_matcher_LC_HPF1()’ as follows:
```
    >>fc=1e8; Qm=1; % Frequency and Maximum quality factor
      format short e; Xs=imp_matcher_LC_HPF1(50,12.5+10i,fc,Qm)
      [LCs,LCs_]=L_or_C_from_X(Xs,fc,'v')
```
where the third input argument of ‘L_or_C_from_X()’ has been given as 'v' (meaning ‘vertical’) since we know that the last section of the path from Γ₁ to Γ₂ leads to a parallel L and consequently, the first element of the LC ladder (from the source) is to be connected vertically. Running these statements yields the following result:
```
  LCs =  7.9577e-08  6.3662e-11   3.9789e-08   7.0736e-11
  LCs_= 'L(v)'      'C(h)'       'L(v)'      'C(h)'
```
This design result conforms with that obtained using ‘Smith 4.1’ and shown in the Schematic pane in Figure 7.24.

function Xs=imp_matcher_LC_HPF1(R0,ZL,fc,Qm)
% Design an LC ladder HPF with Q<=Qm to make impedance matching with R0
% when RL<R0
% Copyleft: Won Y. Yang, [email protected], CAU for academic use only
EPS=1e-4;
if abs(ZL-R0)<EPS, Xs=[]; return; end
RL=real(ZL); XL=imag(ZL); YL=1/ZL; GL=real(YL); BL=imag(YL);
if abs(XL)<EPS
 M1=abs(Qm); M2=sqrt(R0/RL-1);
 % Does the R-circle cross G=1/R0-circle or Q-circle first?
 [X,imin]=min([M1 M2]); X=X*RL;
 Z=RL+j*X; Y=1/Z; B=imag(Y);
 if abs(M1-M2)<EPS|imin>1 % If it crosses G=1/R0-circle first
  Xs=[-1/B -X]; % Series C and Parallel L from Load
 else % If it crosses Q-circle first
  RL=1/real(Y); Xsp=imp_matcher_LC_HPF1(R0,RL,fc,Qm);
  Xs=[Xsp -1/B -X]; % Series C and Parallel L from Load
 end
elseif XL<0 % First, L in parallel with ZL for HPF
 RL=1/GL; Xsp=imp_matcher_LC_HPF1(R0,RL,fc,Qm);
 Xs=[Xsp 1/BL]; % Parallel L
else % if XL>0
 % Move the point from the 2nd quadrant into the 3rd one by Ls (series C).
 M1=abs(Qm); M2=sqrt(R0/RL-1);
 [X,imin]=min([M1 M2]); X=-X*RL;
 Z=RL+j*X; Y=1/Z; B=imag(Y);
 if abs(M1-M2)<EPS|imin>1 % If it crosses G=1/R0-circle first
  Xs=[1/B (X-XL)]; % Series C and Parallel L from Load
 else % If it crosses Q-circle first
  R=1/real(Y); Xsp=imp_matcher_LC_HPF1(R0,R,fc,Qm);
  Xs=[Xsp 1/B (X-XL)]; % Series C and parallel L
 end
end
%Usage:
%fc=1e8; Qm=1; Xs=imp_matcher_LC_HPF1(50,10+5i,fc,Qm);
%[LCs,LCs_]=L_or_C_from_X(Xs,fc,'v')

Example 7.11 Impedance Matching with Serial Line

Use a serial transmission line of characteristic impedance Z₀ = 50 Ω and a parallel capacitor to make an impedance matching between Z_L= 30 − j40[Ω] and 50 Ω at frequency f_c = 430 MHz.

Having started ‘Smith V4.1’ to open the main window, take the following steps:

From the top menu bar, select Tools > Settings to open the Settings dialog box, set the values of Z₀ and frequency to 50[Ω](default) and 430[MHz], respectively, and click OK.
From the toolbar, click on ‘Keyboard’ to open Data Point dialog box, insert the impedance Z_L=30−j40[Ω] as a (start) point Γ_L (DP1), and click OK. The target point Γ₃ = 0, i.e. the origin corresponding to Z₀ = 50 [Ω], is so apparent that it does not have to be designated as a data point.
To design an impedance matcher consisting of a serial line and a reactive element C or L, you need to visualize a |Γ|‐circle centered at the origin and crossing Γ_L (which shows the effect of a lossless SL [serial line] on the reflection coefficient Γ of the load impedance as described by Eq. (7.2.5)) and the G = 1/R₀‐circle along which a Γ point can be moved by a parallel C or L to reach the target point Γ = 0, as have been shown in Figure 7.25. From Figure 7.25, notice that there are two intersections between the two circles, Γ₂ and Γ₄, each of which can be an intermediate point. Therefore, there are two paths from Γ_L to the target point Γ₃ = 0, one via Γ₂ and one via Γ₄.

Figure 7.25 Using ‘Smith 4.1’ for impedance matching between Z_L = 30 − j40 and Z₆ = 50 in Example 7.11.
- First, to take the first path Γ_L‐Γ₂‐Γ₃, do the following, referring to Figure 7.25:
  - Click on ‘Insert serial line’ (from the tool bar) and move the mouse cursor to click on the point Γ₂ (one intersection of the G = 1/R0‐circle and the |Γ|‐circle crossing Γ_L).
  - Click on ‘Insert parallel inductor’ and move the mouse cursor upward to click on the point Γ₃ = 0 (i.e. the origin corresponding to Z₀ = 50[Ω]).
  - Then, the designed impedance matcher consisting of SL (serial line) and Lp (parallel inductor) appears in the Schematic pane as shown in Figure 7.25.
- Second, to take the second path Γ_L‐Γ₄‐Γ₃, do the following:
  - Click on ‘Insert serial line’ (from the tool bar) and move the mouse cursor to click on the point Γ₄ (another intersection of the G = 1/R0‐circle and the |Γ|‐circle).
  - Click on ‘Insert parallel capacitor’ and move the mouse cursor downward to click on Γ₂ = 0.
  - Then, the designed impedance matcher consisting of SL (serial line) and Cp (parallel capacitor) appears in the Schematic pane as shown in Figure 7.25.
  As an alternative, we can compose the following MATLAB script “elec07e11.m” and run it to get
```
 >>elec07e11
   ds =  0.0417  0.2083 % Transmission length(s)
   Xs =  24.7436 -24.7436 % Reactances to be adjusted
   L(1)= 1.603e-08[H]  C(2)= 8.548e-12
```

These results conform with those obtained using ‘Smith 4.1’.

%elec07e11.m
clear, clf
Z0=50; b=2*pi; fc=430e6; ZL=30-40i; G0=1/Z0;
[c,r]=Gcircle(G0,Z0); % Center and radius of the G-circle
% To determine the length(s) of line to move from Gamma 1 to Gamma 2|4
Zd=@(d)Z0*(ZL+j*Z0*tan(b*d))./(Z0+j*ZL*tan(b*d)); % Eq. (7.1.17)
GammaZ=@(Z)(Z-Z0)./(Z+Z0); % Eq. (7.1.11)
f=@(d)abs(GammaZ(Zd(d))-c)-r; % To be solved for the line length d
ds=[fsolve(f,0) fsolve(f,0.25)];
ds=mod(ds,0.5) % Zd is periodic with period 0.5 in d[wavelength]
Xs=1./(imag(1./Zd(ds)-1/Z0)) % Reactances to be adjusted for imp matching
LCs=L_or_C_from_X(Xs,fc);
% To plot the G-circle, the Gamma-circle, and their intersections
dth=0.01; ths=-pi+dth:dth:pi; ucircle=exp(j*ths);
dd=[0:1e-4:0.5];
plot(c+r*ucircle,'b'), hold on, plot(GammaZ(Zd(dd))+j*eps,'k')
legend(['G=' num2str(G0,'%6.3e') '-circle'],'Line impedance effect')
plot(GammaZ(Zd(ds)),'o'), axis('equal')

Problems

7.1 Skinship with the Smith Chart
To get familiar with the Smith chart, do the following:
1. Compose the following MATLAB functions and save them as M‐files each named “G2Z.m” and “Z2G.m” if you don't have such similar M‐files.
function Z=G2Z(Gam,Z0)
if nargin<2, Z0=1; end
Z=Z0*(1+Gam)./(1‐Gam); %Eq. (7.2.1) function Gam=Z2G(Z,Z0) if nargin<2, Z0=1; end Gam=(Z‐Z0)./(Z+Z0); %Eq. (7.2.3)
1. For the point Γ₁ = 0.2 + j0.4 (with Z₀ = 50) on the Smith chart in Figure 7.4, find the corresponding impedance/admittance and their normalized values, hopefully, by using one of the above MATLAB functions. Then, plot the point Γ₁ and the R‐/X‐/G‐/B‐circles passing through Γ₁ by completing and running the following MATLAB script “elec07p01b.m”.
```
%elec07p01b.m
clear, clf
Z0=50; Gam1=0.2+0.4i;
Z1=???(Gam1,Z0), Y1=1/Z1 % Impedance and Admittance
Z1n=Z1/Z0, Y1n=Y1*?? % Normalized impedance and admittance
ths=pi/180*[0:360];
plot(Gam1,'ro'), hold on
R1=real(Z1); [cR1,rR1]=Rcircle(R1,Z0);
plot(cR1+rR1*exp(j*ths)); % plot the R1-circle
X1=imag(Z1); [cX1,rX1]=Xcircle(X1,Z0);
plot(cX1+rX1*exp(j*ths),':'); % plot the X1-circle
G1=real(Y1); [cG1,rG1]=Gcircle(G1,Z0);
plot(cG1+rG1*exp(j*ths),'r'); % plot the G1-circle
B1=imag(Y1); [cB1,rB1]=Bcircle(B1,Z0);
plot(cB1+rB1*exp(j*ths),'r:'); % plot the B1-circle
plot(exp(j*ths)); % plot the unit circle as background
plot([-1 1],[0 0],'k', [0 0],[-1 1],'k') % Horizontal/vertical axes
axis('equal'), axis([-1 1 -1 1])
```
1. For the point Γ₂ = −0.2 − j0.4 (with Z₀ = 50) on the Smith chart in Figure 7.4, find the corresponding impedance/admittance and their normalized values, hopefully, by using one of the above MATLAB functions. Then, plot the point Γ₂ and the R‐/X‐/G‐/B‐circles passing through Γ₂ by completing and running the following MATLAB script “elec07p01c.m”.
```
%elec07p01c.m
Z0=50; Gam2=-0.2-0.4i;
Z2=G2Z(Gam2,??), Y2=1/Z2 % Impedance and Admittance
Z2n=Z2/??, Y2n=Y2*Z0 % Normalized impedance and admittance
plot(Gam2,'ro'), hold on
R2=real(Z2); [cR2,rR2]=Rcircle(R2,Z0);
plot(cR2+rR2*exp(j*ths)); % plot the R1-circle
X2=imag(Z2); [cX2,rX2]=Xcircle(X2,Z0);
plot(cX2+rX2*exp(j*ths),':'); % plot the X1-circle
G2=real(Y2); [cG2,rG2]=Gcircle(G2,Z0);
plot(cG2+rG2*exp(j*ths),'r'); % plot the G1-circle
B2=imag(Y2); [cB2,rB2]=Bcircle(B2,Z0);
plot(cB2+rB2*exp(j*ths),'r:'); % plot the B1-circle
```
1. For the points Γ₁ = 0.2 + j0.4 and Γ₂ = −0.2 − j0.4 (with Z₀ = 50), find the corresponding impedance/ admittance and their normalized values by using the Smith Chart software ‘Smith V4.1’
2. Which one of the circles drawn in (b) can the Γ‐point move from Γ₁ to Γ=0 along? Which one of the circles drawn in (c) can the Γ‐point move from Γ = 0 to Γ₂ along?
3. Find the Γ‐points for Z₃ = 25 + j25 (with Z₀ = 50) and Y₄ = 0.01 + j0.01 (with Z₀ = 50).
7.2 Using the Smith Chart
In connection with Example 7.1, find two L‐type LC circuits transforming the normalized impedance Z₁=1+j1[Ω] or admittance Y₁=1/Z₁=0.5 − j0.5 [S] represented by point Γ₁ into the normalized impedance Z₂ = 0.5 − j0.5 [Ω] represented by point Γ₂, one via Γ₄ and the other via Γ₀ = 0 on the Smith chart in Figure 7.4.
7.3 Single‐Stub Shunt Impedance Matching
Consider the single‐stub shunt impedance matching discussed in Section 7.3.2.1 to find the distance (from the load) and length of shunt single stub so that they can match a given load impedance Z_L to the lossless transmission (or feed) line of characteristic impedance Z₀.
1. Misra [M-2] presents the following formulas for determining the distance d and length l of the shunt single stub:
  (P7.3.1a)
  
  (P7.3.1b)
- where
  (P7.3.2)
  
  (P7.3.3)
```
function [ds,ls]=imp_match_1stub_shunt_Misra(Z0,ZL,r,os)
% Input: Z0 = Characteristic impedance, ZL = Load impedance
%        r = a+bi = Propagation constant
%        os = 'o'/'s' for open/short stub
% Output: ds = Distances of the stub from load
%        ls = Lengths of the stub
if nargin<4, os='s'; end
if length(Z0)>1, Z01=Z0(2); Z0=Z0(1); else Z01=Z0; end
a=real(r); b=imag(r); %lambda=2*pi/b;
YL=1/ZL; yL=YL*Z0; gL=real(yL); bL=imag(yL);
A=gL*(gL-1)+bL^2;
ds=atan((bL+[? ??]*sqrt(bL^2-A*(1-gL)))/A)/b; % Eq. (5.1.2) of [M-2]
tbd=tan(b*ds); den=(gL*tbd).^2+(1-bL*tbd).^2;
bs=-((bL+tbd).*(1-bL*tbd)-gL^2*tbd)./den; % Eq. (5.1.3,4) of [M-2]
ds=mod(ds*b,pi)/b;
if lower(os(1))=='s', ls=acot(-bs)/b; % Eq. (5.1.5) of [M-2]
 else ls=atan(bs)/b; % Eq. (5.1.6) of [M-2]
end
ls=mod(ls*b,pi)/b;
```
- Complete the above MATLAB function ‘imp_match_1stub_shunt_ Misra()’ so that it uses the above formulas to compute the distances and lengths of two shunt single stubs for impedance matching.
1. Use the MATLAB function ‘imp_match_1stub_shunt()’ (presented in Section 7.3.2.1) and/or the above function ‘imp_match_1 stub_shunt_Misra()’ to determine the distances (from the load) and lengths of single shunt short‐circuited stubs in the unit of wavelength so that they can match a load impedance Z_L = 50 − j75[Ω] to the lossless transmission (or feed) line of characteristic impedance Z₀ = 100 Ω.
2. Use ‘Smith 4.1’ to get the solution for the problem dealt with in (b).
7.4 Single‐Stub Series Impedance Matching
Consider the single‐stub series impedance matching discussed in Section 7.3.2.2 to find the distance (from the load) and length of series single stub so that they can match a given load impedance Z_L to the lossless transmission (or feed) line of characteristic impedance Z₀.
1. Misra [M-2] presents the following formulas for determining the distance d and length l of the series single stub:
  (P7.4.1a)
  
  (P7.4.1b)
- where
  (P7.4.2,3)
- Complete the following MATLAB function ‘imp_match_1stub_ series_Misra()’ so that it uses the above formulas to compute the distances and lengths of two series single stubs for impedance matching.
```
function [ds,ls]=imp_match_1stub_series_Misra(Z0,ZL,r,os)
% Input:  Z0 = Characteristic impedance of the TL and stub
%        ZL = Load impedance
%         r = a+bi = Propagation constant
%        os = 'o'/'s' for open/short stub
% Output: ds = Distance of the stub from the load
%        ls = Length of the stub
if nargin<4, os='s'; end
a=real(r); b=imag(r); lambda=2*pi/b;
zL=ZL/Z0; rL=real(zL); xL=imag(zL);
A=rL*(rL-1)+xL^2;
ds=atan((xL+[? ??]*sqrt(xL^2-A*(1-rL)))/A)/b; % Eq. (5.1.8) of [M-1]
tbd=tan(b*ds); den=(rL*tbd).^2+(1-xL*tbd).^2;
xs=-((xL+tbd).*(1-xL*tbd)-rL^2*tbd)./den; % Eq. (5.1.9) of [M-2]
if lower(os(1))=='o', ls=acot(-xs)/b; % Eq. (5.1.13) of [M-2]
 else ls=atan(xs)/b; % Eq. (5.1.12) of [M-2]
end
ds=mod(ds*b,pi)/b; ls=mod(ls*b,pi)/b;
```
1. Use the MATLAB function ‘imp_match_1stub_series()’ (presented in Section 7.3.2.2) and/or the above function ‘imp_match_1stub_series_Misra()’ to determine the distances (from the load) and lengths of single series short‐circuited stubs in the unit of wavelength so that they can match a load impedance Z_L = 100 + j80 [Ω] to the lossless transmission (or feed) line of characteristic impedance Z₀ = 50 Ω.
2. Use ‘Smith 4.1’ to get the solution for the problem dealt with in (b).
7.5 Double‐Stub Impedance Matching
Consider the double‐stub impedance matching discussed in Section 7.3.2.3 to find the lengths of stub A (at d₀ from load) and stub B (at d from stub A) so that they can match a given load impedance Z_L to the lossless transmission (or feed) line of characteristic impedance Z₀.
1. Misra [M-2] presents the following formulas for determining the distance d and length l of the shunt double stub:
  (P7.5.1a,b)
- where
  (P7.5.2)
  
  (P7.5.3a)
  
  (P7.5.3b)
```
function [lA,lB]=imp_match_2stub_Misra(Z0,ZL,r,d)
% Input:  Z0 = Characteristic impedance
%        ZL  = Load impedance
%         r  = a+bi = Propagation constant
%         d  = Distance of the stub B from A (load)
%    where d may be given as [d d0] with d0: Distance of A from load
% Output: lA = Length of the stub A at (d0 from) load
%        lB  = Length of the stub B at d from stub A
if numel(d)>1
 d0=d(2); d=d(1);
 ZL=Z0*(ZL+Z0*tanh(r*d0))/(Z0+ZL*tanh(r*d0)); % Eq. (7.1.16)
end
if imag(r)<eps, r=r*i; end % Since a real-valued r must be beta.
RL=real(ZL); a=real(r); b=imag(r); lambda=2*pi/b;
YL=1/ZL; yL=YL*Z0; gL=real(yL);
if gL>csc(b*d)^2 % Eq. (5.2.4) of [M-2]
 error('Double stub matching is impossible since gL>csc(b*d)^2!');
end
bd=b*d; tbd=tan(bd);
sgd=sqrt(gL*sec(bd)^2-(gL*tbd)^2);
bA=cot(bd)*(1-bL*tbd+[? ??]*sgd); % Eq. (P7.5.3a), Eq. (5.2.5) of [M-2]
bb1tbd=1-(bL+bA)*tbd;
sqgt=(gL*tbd)^2+bb1tbd.^2;
bB=(gL^2*tbd-(bL+bA+tbd).*bb1tbd)./sqgt; % Eq. (5.2.6) of [M-2]
lA=acot(-bA)/b; lB=acot(-bB)/b; % Eq. (P7.5.1a,b)
lA=mod(lA*b,pi)/b; lB=mod(lB*b,pi)/b;
```
- Complete the above MATLAB function ‘imp_match_2stub_Misra()’ so that it uses the above formulas to compute the lengths of two stubs for impedance matching.
1. Use the MATLAB function ‘imp_match_2stub()’ (presented in Section 7.3.2.3) and/or the above function ‘imp_match_2stub_Misra()’ to determine the lengths of two stub A (at d₀ = λ/8 from load) and stub B (at d = λ/8 from stub A) in the unit of wavelength so that they can match a load impedance Z_L = 100 + j50 [Ω] to the lossless transmission (or feed) line of characteristic impedance Z₀ = 50 Ω.
2. Noting that with the distance d₀ = λ/8 of stub A from the load, the virtual load impedance can be regraded as
  (P7.5.4)
- modify the MATLAB script “elec07e04.m” into “elec07p05.m” so that it returns the solution to this problem.
1. Use ‘Smith 4.1’ to get the solution for the problem dealt with in (b) where you may use some transit points (that can be found in (c)) to move the Γ‐point from Γ_L to Γ₀ = 0 as in Example 7.4.
7.6 Quarter‐Wave Transformer (QWT)
Consider the MATLAB function ‘chebtr()’ presented in [O-1] to design the Chebyshev QWTs (Section 7.3.4.2) satisfying the requirements for attenuation and fractional bandwidth where the attenuation is specified indirectly through reflection coefficient or SWR (standing wave ratio) as

(P7.6.1)
- Here, |Γ₁| is the reflection coefficient from the first section of the several section constituting the QWT and |Γ_L| is the reflection coefficient from the load. Note that the usage of ‘chebtr()’ is as follows:
```
[Y,Az,Bz]=chebtr(Ys,YL,A,fb)
```
- where
(P7.6.2)
1. Design a Chebyshev QWT that matches a Z_L = 200 Ω‐load to a Z₀=50 Ω‐line with SWR_max=1.25 over the frequency band [50, 150] MHz where the center frequency f₀ = 100 MHz. You can complete and run the following script “elec07p06a.m”, which uses ‘chebtr()’ to design such a QWT. Noting that the characteristic admittance vector Y returned as the first output argument of ‘chebtr()’ contains Y_s/Y_L, as the first/last entry, tell the number of sections and characteristic impedances of each section of the designed QWT.
```
%elec07p06a.m
clear, clf
Z0=50; ZL=200; % Line characteristic impedance and Load impedance
GL = Z2G(ZL,Z0); SWRmax = 1.25;
f1 = 50; f2 = 150; % Band edge frequencies
f0 = (f1+f2)/2; % Operating frequency
fb = (f2-f1)/f0; % Required fractional bandwidth
As = 20*log10(GL*(SWRmax+1)/(SWRmax-1)); % Attenuation of
% reflectionless band (relative to unmatched case)
[Y,a,b] = chebtr(1/Z0,1/ZL,As,fb); % Chebyshev transformer design
% where b/a = Numerator/Denominator polynomials of reflection response
Z = 1./? % Characteristic impedances of each section of designed QWT
N=length(Z)-2; % Number of sections
Zr=Z(2:N+1); % Real characteristic impedance vector of designed QWT
```
1. To see if the above design Zr satisfies the following QWT conditions:
  (P7.6.3)
  
  (P7.6.4)
  
  (P7.6.5)
- complete/run the above MATLAB script “elec07p06b.m” and discuss the result (see Figure P7.6.1).
Figure P7.6.1 An N‐section impedance transformer.
```
%elec07p06b.m
Zl=@(ZL,Z0,bl)(ZL+j*Z0*tan(bl))./(Z0+j*ZL.*tan(bl))*Z0;
Zo(1)=Z0; Zi(N+1)=ZL; bl=pi/2;
for n=1:N, Zo(n+1)=Zl(Zo(n),Z?(n),bl); end % Eq. (P7.6.3b)
for n=N:-1:1
  Zi(n)=Zl(Z?(n+1),Zr(n),bl); % Eq. (P7.6.3b)
  Zr1(n)=sqrt(Zo(n)*Zi(n+1)); % Eq. (7.3.19) at each stage
end
Mismatching=norm([Zi(1)-Z0 Zr-Zr1 Zo-Zi Zo(N+1)-ZL])
```
Figure P7.6.2 Reflection response and SWR.
1. Noting that the fifth input argument of ‘multi_line()’ (presented in Section 7.3.4) should have the normalized phase constant β = 2π as the first entry, use the MATLAB function to get the reflection response of the QWT designed in (a) and plot it together with SWR (Eq. (7.1.13a)) for the frequency range 0~2f_c. To this end, complete and run the following script “elec07p06c.m” to get the graphs as shown in Figure P7.6.2.
```
%elec07p06c.m
f = linspace(0,2*f0,401); % Frequency range [0,200]MHz
ls = ones(1,N)/4; % Quarter-wave lengths
bb = 2*pi*[1 f/f0]; % Normalized frequency vector
%  with phase constant beta as its 1st entry
[Zin,Gam,bb]=multi_line(Z?,ls,Z0,ZL,bb);
SWR = (1+abs(Gam))./(1-abs(Gam)); % SWR by Eq. (7.1.13a)
subplot(221), plot(f,abs(Gam)), hold on
  plot(f([1 end]),max(Gam)*10^(-As/20)*[1 1],'r:')
subplot(222), plot(f,SWR, [f1 f2],SWRmax*[1 1],'r:')
```
1. To realize the meanings of the second/third output arguments of ‘chebtr()’ as the denominator/numerator polynomials of the reflection response in z = exp(jπf/f₀), use them to compute the reflection response
  (P7.6.5)
- and see if it conforms with that obtained as the second output argument of ‘multi_line()’ in (c) by running the following MATLAB script “elec07p06d.m”.
```
%elec07p06d.m
w=pi*f/fc; Gam1=freqz(b,a,w); % Alternative way of computing Gam
Gam1_=polyval(b,exp(j*w))./polyval(a,exp(j*w)); % Eq. (P7.6.5)
discrepancy=norm([Gam-Gam1 Gam-Gam1_])
```
7.7 Design of LPF Using Stubs [L-1]
With source/load impedances R_s = 50 [Ω]/Z_L = 50 [Ω], design a Chebyshev I LPF with passband ripple R_p = 0.5 dB for f ≤ f_p (passband edge frequency) = 3 GHz and stopband attenuation A_s = 40 dB for f ≥ f_s (stopband edge frequency) = 6 GHz for fabrication using microstrip lines with shunt stubs only.

Referring to Example 7.5, take the following steps to design such an LPF:
- (Step 0) Determine the order N and cutoff frequency ω_c [rad/s] of the LPF to satisfy the given frequency response specifications by running the following MATLAB statements:
```
 >>wp=2*pi*3e9; ws=2*pi*6e9; Rp=0.5; As=40
   [N,wc]=cheb1ord(wp,ws,Rp,As,'s')
```
- (Step 1) Determine the normalized LPF prototype element values by running
```
 >>Rs=50; % Source (line) characteristic impedance
   type='C'; % for the Nth order Chebyshev with Rp[dB]
   gg=LPF_ladder_coef(N,type,Rp)
```
- (Step 2) Determine the normalized LC values of the prototype LPF by running
```
 >>Rs0=1; pT='T'; wc=1; % for a T-type prototype filter
   [LCRs,LCRs_,Gs]=LC_ladder(Rs0,gg,pT,wc);
   LCRs, LCRs_
```
  - With the results, fill in the parameter values of the circuit in Figure P7.7.1.
- (Step 3) Convert the L/C values into the impedances of short‐/open‐circuited stubs by running the following statement:
```
   >>[Zs,SPsos]=LC2Z(LCRs,LCRs_); Zs_0=Zs, SPsos_0=SPsos
```
  - With the results, fill in the parameter values in Figure P7.7.2(a).
- (Step 4) Noting that even if we are going to convert every stub into parallel (shunt) stubs, we provisionally should convert the two Po's (parallel open‐circuited stubs) Z₁ and Z₅ (at either ends) into Ss's (series short‐circuited stubs) so that the Ss's Z₂ and Z₄ can have UEs of Z₀ = 1 (needed for conversion), let us use Kuroda identity 1 to convert Z₁ and Z₅ into Ss's. Thus, we insert UEs between Z₁/Z₅ and the transmission line (as Figure P7.7.2(b)) and run the following statements:
  
  Figure P7.7.1 Normalized prototype LPF of order N = 3.
```
 >> [Z0s(2),Zs(1),SPsos{1}]=Kuroda(Zs(1),SPsos{1});
   [Z0s(4),Zs(5),SPsos{5}]=Kuroda(Zs(5),SPsos{5});
   Z0s, Zs, SPsos
```
  - With the results, fill in the parameter values in Figure P7.7.2(c).
- (Step 5) Noting that in Figure P7.7.2(c) the two Ss's Z₁ and Z₅ (at either ends) have no UE needed for conversion (while Z₂ and Z₄ have UEs), let us insert UEs of Z₀=1 between them and TL as shown in Figure P7.7.2(d).
```
 >>Z0s(1)=1; Z0s(5)=1;
```
- (Step 6) Now that every Ss has got its UEs for conversion, let us use Kuroda identity 2 to convert the Ss's Z₁, Z₂, Z₄, and Z₅ into Po's by running
```
 >>for m=[1 2 4 5]
       [Z0s(m),Zs(m),SPsos{m}]=Kuroda(Zs(m),SPsos{m},Z0s(m));
   end
   Z0s, Zs, SPsos
```
  - With the results, fill in the parameter values in Figure P7.7.2(e).
- Alternatively, the process from Step 4 to Step 6 can be performed by using the MATLAB function ‘S2P_stubs()’ (listed below) as follows:
```
 >>Zs=Zs_0; Z0s=zeros(size(Zs)); SPsos=SPsos_0;
   [Z0sf,Zsf,SPsosf]=S2P_stubs(Z0s(1:2),Zs(1:2),SPsos(1:2));
   [Z0sb,Zsb,SPsosb]= ...
        S2P_stubs(Z0s(end:-1:4),Zs(end:-1:4),SPsos(end:-1:4));
   Z0s_=[Z0sf Z0s(3) Z0sb(end:-1:1)]; Zs_=[Zsf Zs(3)
   Zsb(end:-1:1)]
   SPsos_=[SPsosf SPsos(3) SPsosb(end:-1:1)]
```
  Figure P7.7.2 Microstrip filter design for Problem 7.7.
- (Step 7) Denormalize the normalized characteristic impedances by multiplying them with R_s = 50 Ω:
```
 >>Zs=Zs*Rs, Z0s=Z0s*Rs
```
  - With the results, fill in the parameter values in Figure P7.7.2(f).
```
function [Z0s,Zs,SPsos]=S2P_stubs(Z0s,Zs,SPsos)
% To convert all series stubs (SS) into parallel stubs (PS)
% Input:
%  Z0s = Characteristic impedances of UEs with 0 for no UE
%  Zs  = Stub impedances
%  SPsos = {'Po','Ss', .. } for parallel-open, series-short stubs
% Output:
%  Z0s, Zs, SPsos = Results with all stubs parallel
% where any -ve sign of Zs implies that it is an open-circuited stub.
N=numel(Zs); nS=N+1;
for n=1:N % To find the position of the 1st SS
   if upper(SPsos{n}(1))=='S', nS=n; break; end
end
if nS<=N % If any stub is a SS
 for n=1:nS-1 % All PSs into SSs before the 1st SS
   [Z0s(n+1),Zs(n),SPsos{n}]=Kuroda(Zs(n),SPsos{n},Z0s(n));
 end
 for n=1:nS % All the SSs into PSs up to the 1^st SS
   [Z0s(n),Zs(n),SPsos{n}]=Kuroda(Zs(n),SPsos{n},Z0s(n));
 end
[Z0s,Zs,SPsos]=S2P_stubs(Z0s,Zs,SPsos); % Self-calling recursively
end
```
- (Step 8) Frequency scale the circuit by determining the line/stub lengths as l = λ/8 = v_p/f_c/8 [m] where v_p [m/s] is the phase velocity of signal propagating on the line and f_c is the cutoff frequency given as a design specification.
- (Step 9) To plot the frequency responses of the two filters, one (called a lumped‐element filter) consisting of lumped elements L/C (Figure P7.7.1) and the other (called a distributed‐element filter) consisting of microstrip lines/stubs (Figure P7.7.2(f)), complete and run the following MATLAB script “elec07p07b.m” to get the frequency responses as shown in Figure P7.7.3:
Figure P7.7.3 Frequency responses of the lumped‐element filter and distributed‐element filter.
```
%elec07p07b.m
Z0strip=[Zs(1) Z0?(1) Zs(2) Z0s(?) Zs(3) Z0s(4) ??(4) Z0s(5) Zs(5)];
SPsos={SPsos{1} 'Tl' SPsos{2} '??' SPsos{3} 'Tl' SPsos{?} 'Tl' SPsos{5}};
l=1/8; fc=3e9; % Stub length and Cutoff frequency
fN=linspace(0,pi/4,1000); % Normalized frequency range
w=fN*2*pi*fc; % Real frequency range
bl=2*pi*w*l/fc; % Range of bl (bl=2*pi*l at f=fc=4e9[Hz])
% To get the frequency response and input impedance
ZL=Rs; [Gw,Zinw]=GZ_strip(Z0?????,??sos,ZL,bl);
Gw=Zinw./(Rs+Zinw).*Gw; % Overall frequency response
Gwmag=20*log10(max(abs(Gw),1e-5)); % Frequency response magnitude in dB
plot(w,Gwmag-max(Gwmag),'b'), hold on % with the maximum 0dB
s=j*w/fc; Gw0=eval(Gs); % Frequency response from the transfer function
Gwmag0=20*log10(max(abs(Gw0),1e-5));
plot(w,Gwmag0-max(Gwmag0),'r:') % To make the maximum 0dB
legend('Microstrip','Lumped L/C'), axis([w([1 end]) -80 5])
```
- Note that the whole process from Step 1 to Step 7 can be performed by using the MATLAB function ‘ustrip_LPF()’ (listed below) as follows:
```
 >>Rs=50; type='C'; wp=2*pi*3e9; ws=2*pi*6e9; Rp=0.5; As=40
   [Z0strip,SPsos]=ustrip_LPF(Rs,type,[Rp As],[wp ws],pT)
```

7.8 Design of T‐type Impedance Transformers Using the Smith Chart

Consider making T‐type ladders with quality factor Q = 2 for transforming Z_L = 20 − j15[Ω] to Z_s = 50[Ω] at frequency f_c = 2 GHz.

function [Z0strip,SPsos,LCRs,LCRs_]= ...
                 ustrip_LPF(Rs,type,RpAs,wps,pT)
% Input:
%  Rs = Line characteristic impedance
%  type = 'B'/'C1'/'C2'/'E' for B/Chebyshev I/II/Elliptic filter
%  RpAs = [Rp As] Passband ripple/Stopband attenuation[dB]
%  wps = [wp ws] = Passband/Stopband edge frequencies[rad]
%  pT = 'pi'/'T' for pi-/T-form
% This function can be called as ustrip_LPF(Rs,type,Rp,N,pT).
% Copyleft: Won Y. Yang, [email protected], CAU for academic use only
if nargin<5, pT='pi'; end % Pi-form by default
if numel(wps)<2 % When called as ustrip_LPF(Rs,type,Rp,N,pT)
  N=wps; Rp=RpAs;
 else
 Rp=RpAs(1); As=RpAs(2); wp=wps(1); ws=wps(2);
 if lower(type(1))=='b', [N,wc]=buttord(wp,ws,Rp,As,'s');
  elseif lower(type(1))=='c'
    if numel(type)<2|type(2)=='1', [N,wc]=cheb1ord(wp,ws,Rp,As,'s');
     else [N,wc]=cheb2ord(wp,ws,Rp,As,'s');
    end
  elseif lower(type(1))=='e', [N,wc]=ellipord(wp,ws,Rp,As,'s');
  else error('What type of filter do you want? (inside ustrip_LPF)');
 end
end
gg=LPF_ladder_coef(N,type,Rp);
% To find the L & C values of the prototype pi filter
Rs0=1;
[LCRs,LCRs_,Gs]=LC_ladder(Rs0,gg,pT,1);
format short e;
LCs=LCRs(1:end-1);
LCs_=LCRs_(1:end-1);
% To convert the L/C values into the characteristic impedance
% of short/open-circuited stub
[Zs,SPsos]=LC2Z(LCRs,LCRs_);
% To convert all SSs (series stubs) into PSs (parallel stubs)
M=ceil(N/2);
if SPsos{M}(1)~='P', M=M+1; end
M1=M+1; Z0s=zeros(1,N);
Z0s_f=Z0s(1:M-1); Zs_f=Zs(1:M-1); SPsos_f=SPsos(1:M-1);
Z0s_b=Z0s(end:-1:M1); Zs_b=Zs(end:-1:M1); SPsos_b=SPsos(end:-1:M1);
[Z0s_f,Zs_f,SPsos_f]=S2P_stubs(Z0s_f,Zs_f,SPsos_f);
[Z0s_b,Zs_b,SPsos_b]=S2P_stubs(Z0s_b,Zs_b,SPsos_b);
Z0s=[Z0s_f Z0s_b(end:-1:1)];
Zs=[Zs_f Zs(M) Zs_b(end:-1:1)];
SPsos=[SPsos_f SPsos(M) SPsos_b(end:-1:1)];
% Denormalize the normalized characteristic impedances
% by multiplying them with Rs.
Z0s=Z0s*Rs, Zs=Zs*Rs
Z0strip=reshape([Zs;[Z0s 0]],1,2*N); Z0strip=Z0strip(1:end-1);
SPsos=[SPsos; repmat({'Tl'},1,N)];
SPsos=reshape(SPsos,1,2*N); SPsos=SPsos(1:end-1);

Complete and run the following MATLAB script “elec07p08.m” to make such T‐type ladders.

Referring to Figure P7.8, note that, with Q ≤ 2, there are four paths along resistance/conductance curves from point Γ_L (representing Z_L = 20 − j15 [Ω]) to Γ_s (representing Z_s = 50 [Ω]), each via Γ₂‐Γ₃, Γ₄‐Γ₃, Γ₂‐Γ₅, and Γ₄‐Γ₅ where Γ₂ and Γ₄ are on the lower and upper Q = 2‐circles, respectively. Use ‘Smith V4.1’ to solve the same design problem as (b).

A window displaying a Smith chart at the left and a Schematic pane with four networks at the right. — **Figure P7.8** Using ‘Smith 4.1’ to design T‐type impedance matchers in Problem 7.8

7.9 Impedance/Admittance from Resistance/Conductance
Complete the following MATLAB function ‘RG2ZY()’ so that it can return the impedance Z and admittance Y with given resistance R and conductance G. Use it for R₅ = 60 Ω and G₅ = 0.01 S (Eq. (E7.8.2)) to check if it works properly.
```
function [Z,Y]=RG2ZY(R,G)
X=sqrt(R/G-?^2); Z=R+j*X; % Reactance and Impedance
B=sqrt(?/R-G^2); Y=G+j*B; % Susceptance and Admittance
```

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Table of Contents for 7 Smith Chart and Impedance Matching

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7.1 Transmission Line

7.2 Smith Chart

7.3 Impedance Matching Using Smith Chart

7.3.1 Reactance Effect of a Lossless Line

7.3.2 Single‐Stub Impedance Matching

7.3.2.1 Shunt‐Connected Single Stub

7.3.2.2 Series‐Connected Single Stub

7.3.3 Double‐Stub Impedance Matching

7.3.4 The Quarter‐Wave Transformer

7.3.4.1 Binomial Multisection QWT

7.3.4.2 Chebyshev Multisection QWT

7.3.5 Filter Implementation Using Stubs[P-1]

<Richard's Transformation>

<Kuroda's Identities>

7.3.6 Impedance Matching with Lumped Elements

Problems

Table of Contents for
7 Smith Chart and Impedance Matching

7.3.5 Filter Implementation Using Stubs^[P-1]