6.2.1 Low‐pass Filter (LPF)

6.2.1.1 Series LR Circuit

Figure 6.4(a) shows a series LR circuit where the voltage v_R(t) across the resistor R is taken as the output to the input voltage source v_i(t). With V_R(s) = {v_R(t)} and V_i(s)={v_i(t)}, its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.1a)

(6.2.1b)

Since the magnitude |G(jω)| of this frequency response becomes smaller as the frequency ω gets higher as depicted in Figure 6.4(b), the circuit is a lowpass filter (LPF) that prefers to have low‐frequency components in its output. Noting that |G(jω)| achieves the maximum G_max = G(j0) = 1 at ω = 0, we can find the cutoff frequency ω_c at which it equals as

(6.2.2)

The cutoff frequency has a physical meaning as the boundary frequency between the passband and the stopband. Why do we use for its definition? In most cases, the transfer function and the frequency response is the ratio of output and input voltages/currents in s‐frequency domain. Since an electric power is proportional to the squared voltage and current, the ratio of between voltages/currents corresponds to the ratio of 1/2 between powers. For this reason, the cutoff frequency is also referred to as a half‐power frequency or a 3dB frequency on account of that 10log₁₀(1/2) = −3 [dB]. In the case of LPF with cutoff frequency ω_c like this circuit, the passband is the range of frequency [0, ω_c] and its width is the bandwidth, where the bandwidth means the width of the range of frequency components that pass the filter better than others.

6.2.1.2 Series RC Circuit

Figure 6.4(c) shows a series RC circuit where the voltage v_C(t) across the capacitor C is taken as the output to the input voltage source v_i(t). With V_C(s)={v_C(t)} and V_i(s)={v_i(t)}, its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.3a)

(6.2.3b)

Everything is the same with the series LR circuit in Figure 6.4(a) except for that the cutoff frequency is

(6.2.4)

6.2.2 High‐pass Filter (HPF)

6.2.2.1 Series CR Circuit

Figure 6.5(a) shows a series CR circuit where the voltage v_R(t) across the resistor R is taken as the output to the input voltage source v_i(t). Its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.5a)

(6.2.5b)

Since the magnitude |G(jω)| of this frequency response becomes larger as the frequency ω gets higher as depicted in Figure 6.5(b), the circuit is a highpass filter (HPF) that prefers to have high‐frequency components in its output. Noting that |G(jω)| achieves the maximum G_max=G(j∞)=1 at ω=∞, we can find the cutoff frequency ω_c at which it equals as

(6.2.6)

6.2.2.2 Series RL Circuit

Figure 6.5(c) shows a series RL circuit where the voltage v_L(t) across the inductor L is taken as the output to the input voltage source v_i(t). Its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.7a)

(6.2.7b)

Everything is the same with the series CR circuit in Figure 6.5(a) except for that the cutoff frequency is

(6.2.8)

6.2.3 Band‐pass Filter (BPF)

6.2.3.1 Series Resistor, an Inductor, and a Capacitor (RLC) Circuit and Series Resonance

Let us consider the series resistor, an inductor, and a capacitor (RLC) circuit of Figure 6.6(a) in which the voltage v_R(t) across the resistor R is taken as the output to the input voltage source v_i(t). Its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.9a)

(6.2.9b)

Referring to the typical magnitude curve of the frequency response of a bandpass filter (BPF) shown in Figure 6.6(b), we can find the maximum of the magnitude, |G(jω)|, of the frequency response by setting the derivative of its square |G(jω)|² w.r.t. frequency ω to zero as

This yields the peak or center frequency

(6.2.10)

at which the frequency response achieves the maximum magnitude as

This frequency happens to coincide with the resonant frequency ω_r at which the frequency response is real so that the input and the output are in phase at that frequency. We can also find the lower and upper 3dB frequencies at which the magnitude |G(jω)| of this frequency response is as

(6.2.11)

Multiplying these two 3 dB frequencies and taking the square root yields

(6.2.12)

This implies that the peak frequency ω_p is the geometrical mean of two 3 dB frequencies ω_l and ω_u and also the arithmetical mean of their logarithmic values that is at the midpoint between the two 3 dB frequency points on the log‐frequency (log ω) axis. That is why ω_p is also called the center frequency. The frequency band between the two 3 dB frequencies ω_l and ω_u is the passband and its width is the bandwidth:

(6.2.13)

Note that the bandwidth for a BPF is the width of the range of frequency components that pass the filter better than others.

For a BPF, we define the quality factor or selectivity (sharpness) as

(6.2.14)

This quality factor is referred to as the selectivity since it is a measure of how selectively the filter responds to the peak frequency and frequencies near it within the passband, discriminating against frequencies outside the passband. It is also referred to as the sharpness in the sense that it describes how sharp the magnitude curve of the frequency response is around the peak frequency. It coincides with the voltage magnification ratio, that is, the ratio of the magnitude of the voltage across the inductor or the capacitor to that of the input voltage at peak frequency as

(6.2.15)

which may be greater than unity. Isn't it interesting that the voltages across some components of a circuit can be higher than the applied voltage?

Now, let us see the following example showing how to determine the parameters of a series RLC circuit so that it can function as a BPF satisfying a design specification on the passband.

6.2.3.2 Parallel RLC Circuit and Parallel Resonance

Let us consider the parallel RLC circuit of Figure 6.7(a) in which the current i_R(t) through the resistor R is taken as the output to the input current source i_i(t). Its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.16a)

(6.2.16b)

Since this frequency response is the same as that, Eq. (6.2.9b), of the series RLC circuit in Figure 6.6(a) except for ω_b, we can obtain the peak or center frequency, the upper/lower 3dB frequencies, the bandwidth, and the quality factor as follows:

(6.2.17a)

(6.2.17b)

(6.2.17c)

(6.2.17d)

The peak frequency happens to be coinciding with the resonant frequency ω_r in the sense that the frequency response is real so that the input and the output are in phase at that frequency. The quality factor coincides with the current magnification ratio, that is, the ratio of the amplitude of the current through the inductor or the capacitor to that of the input current at resonant frequency as

(6.2.18)

which may be greater than unity. How strange it is that the currents through some components of a circuit can be larger than the applied current!

Let us now consider RLC circuit of Figure 6.7(b) in which the voltage across L‖C (the parallel combination of L and C) is taken as the output to the input voltage source RI_i. We can use the voltage divider rule to obtain the transfer function as

This is the same as Eq. (6.2.16a), which is the transfer function of the parallel RLC circuit of Figure 6.7(a). In fact, the circuit of Figure 6.7(b) is obtained by transforming the current source I_i in parallel with R into a voltage source RI_i in series with R in the circuit of Figure 6.7(a).

6.2.4 Band‐stop Filter (BSF)

6.2.4.1 Series RLC Circuit

Let us consider the series RLC circuit of Figure 6.9(a) in which the voltage v_LC(t) across the series combination of L and C is taken as the output to the input voltage source v_i(t). Its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.19a)

(6.2.19b)

Referring to the typical magnitude curve of the frequency response of a BSF shown in Figure 6.9(b), we can find the minimum of the magnitude of the frequency response by setting the derivative of its square w.r.t. the frequency ω to zero as

This yields the notch, rejection, or center frequency

(6.2.20)

at which the frequency response achieves the minimum magnitude as

(Note) It is hoped that the readers will enjoy the dual sense of ω_r, as the resonant frequency of BPF or the rejection (notch) frequency of BSF (depending on the context), rather than being confused.

Noting that the magnitude of the frequency response achieves its maximum of unity at another extremum frequency ω = 0 or ω = ∞ as

we can find the lower and upper 3 dB frequencies at which the magnitude |G(jω)| of this frequency response is as

Multiplying these two 3 dB frequencies and taking the square root yields

(6.2.21)

This implies that the notch frequency ω_r is the geometrical mean of two 3 dB frequencies ω_l and ω_u and also the arithmetical mean of their logarithmic values that is at the midpoint between the two 3 dB frequency points on the log‐frequency (log ω) axis. The band of frequencies between the two 3 dB frequencies ω_l and ω_u is the stopband and its width is the bandwidth:

(6.2.22)

Note that the bandwidth for a BSF is the width of the range of frequency components that are relatively more attenuated or rejected by the filter than other frequencies.

For a BSF, we define the quality factor or selectivity (sharpness) as

(6.2.23)

This quality factor is referred to as the selectivity since it is a measure of how selectively the filter rejects the notch frequency and frequencies near it within the stopband, favoring frequencies outside the stopband. It is also referred to as the sharpness in the sense that it describes how sharp the magnitude curve of the frequency response is around the rejection or notch frequency.

6.2.4.2 Parallel RLC Circuit

Let us consider the parallel RLC circuit of Figure 6.10(a) in which the current i_L(t) + i_C(t) through the parallel combination of L and C is taken as the output to the input current source i_i(t). Its input‐output relationship can be described by the transfer function and the frequency response as

(6.2.24a)

(6.2.24b)

Since this frequency response is the same as that, Eq. (6.2.19b), of the series RLC circuit in Figure 6.9(a) except for ω_b, we have the same results for the notch or rejection or center frequency, the upper/lower 3 dB frequencies, the bandwidth, and the quality factor.

Let us now consider the RLC circuit of Figure 6.10(b) in which the voltage across R is taken as the output to the input voltage source RI_i. We can use the voltage divider rule to obtain the transfer function as

This is exactly the same as Eq. (6.2.24a), which is the transfer function of the parallel RLC circuit of Figure 6.10(a). In fact, this circuit is obtained by transforming the current source I_i in parallel with R into a voltage source RI_i in series with R in the circuit of Figure 6.10(a).

6.2.5 Quality Factor

In the previous sections, the quality factors for series and parallel RLC circuits were defined as the ratio of the center frequency ω_c to the bandwidth B= ω_u ‐ ω_l where ω_u and ω_l are the upper and lower 3 dB frequencies, respectively:

(6.2.25a)

(6.2.25b)

Note that the quality factor Q of a circuit can be viewed as the ratio of the reactive power (i.e. the maximum power stored in L or C) to the active power of the circuit.

Now, let us consider the quality factor of an individual reactive element L or C at the center frequency ω_c of the circuit containing the element. It depends on whether the element is modeled as a series or parallel combination with its loss (parasitic) resistance:

(6.2.26a,b)

where

(6.2.27)

(6.2.28a,b)

(6.2.29a,b)

Note that these equivalence conditions between the series and parallel combination models of a reactive element are valid only at a certain frequency since the reactances are frequency dependent. The series‐to‐parallel and parallel‐to‐series conversion processes of L‐R and C‐R circuits have been cast into the following MATLAB functions:

Now, let us consider the (loaded) quality factor [L-1] Q_LD of a series RLC circuit with the source resistance R_s and the load resistance R_L as shown in Figure 6.12(a). Noting that the circuit is a series connection of (R_s+R+R_L)‐L‐C, we can use Eq. (6.2.25a) to write the loaded quality factor Q_LD as

(6.2.30)

where the filter quality factor Q_F and the external quality factor Q_E are as follows:

(6.2.31)

Likewise, the loaded quality factor Q_LD of a parallel RLC circuit with the source resistance R_s and the load resistance R_L as shown in Figure 6.12(b) can be written as

(6.2.32)

where the filter quality factor Q_F and the external quality factor Q_E are as follows:

(6.2.33)

6.2.6 Insertion Loss

When designing a filter to be inserted in a telecommunication system as can be modeled by, say, Figure 6.12(a) or (b), it is important to consider the insertion loss (IL) of the filter, which is defined as the ratio of the power (delivered to the load without the filter) to the power (delivered to the load with the filter inserted):

(6.2.34)

6.2.7 Frequency Scaling and Transformation

Note the following remarks on frequency scaling and frequency transformation:

Frequency transformations	MATLAB function
LPF with cutoff frequency ωc: s → s/ωc(6.2.36a) This corresponds to substituting and .(6.2.36b)	lp2lp()
HPF with cutoff frequency ωc: s → ωc/s(6.2.37a) This corresponds to substituting and .(6.2.37b)	lp2hp()
BPF with upper/lower cutoff frequencies ωu/ωl: (6.2.38a) This corresponds to substituting () (6.2.38b) (6.2.38c)	lp2bp()
BSF with upper/lower cutoff frequencies ωu/ωl: (6.2.39a) This corresponds to substituting () (6.2.39b) (6.2.39c)	lp2bs()

6.3.1 LC Ladder

Figure 6.14.1(a) and (b) shows the П‐type and T‐type of LC ladder that can be used to realize a prototype LPF where the values of L's, C's, and R_L are listed as {g₁, g₂, … } in Tables 5.2 (Butterworth), 5.3 (linear‐phase), 5.4 (Chebyshev 3 dB), and 5.5 (Chebyshev 0.5 dB) of [L-1] and can also be determined by using the following MATLAB function ‘LPF_ladder_coef()’.

The formulas listed in Table 6.1 can be used to magnitude/frequency scale the LC filter prototypes to make them realize LPF/HPF/BPF/BSF with unnormalized cutoff frequency as depicted in Figures 6.14.1‐6.14.4.

6.3.2 L‐Type Impedance Matcher

Figure 6.18 shows two L‐types of LC filters (with Z_k = jX_k = jωL_k or Z_k = jX_k = 1/jωC_k) that can be used to accomplish the impedance matching condition [Y-1], (6.38) (^*: complex conjugate) between the source impedance Z_s = R_s + jX_s and load impedance Z_L = R_L + jX_L. The impedance matching condition for maximum power transfer of L1‐type filter (Figure 6.18(a)) can be written as

(6.3.1)

Likewise, noting that the structure of L2‐type filter (Figure 6.18(b)) is just like the reverse of L1‐type filter with Z_s and Z_L switched, the impedance matching condition of L2‐type filter can be written as

(6.3.2)

The above MATLAB function ‘imp_matcher_L()’ can be used to find possible L‐type filter(s) for impedance matching between given source/load impedances. Note that the function without the fourth input argument ‘type’ returns all possible L‐type LC filters while with type = 1/2, it returns just L1‐/L2‐type LC filter(s).

6.3.3 T‐ and П‐Type Impedance Matchers

Figure 6.21 shows T‐ and П‐type LC ladders (with Z_k = jX_k = jωL_k or Z_k = jX_k = 1/jωC_k) that can be used to accomplish impedance matching between the source impedance Z_s = R_s + jX_s and load impedance Z_L = R_L + jX_L. The conditions for impedance matching of T‐type filter (Figure 6.21(a)) with quality factor Q can be formulated into the formulas for computing Z₃ = jX₃, Z₂ = jX₂, and Z₁ = jX₁ as

(6.3.3a)

(6.3.3b)

(6.3.3c)

The Y‐Δ conversion of a T‐type impedance matcher yields its equivalent П‐type impedance matcher (Figure 6.21(b)).

The above MATLAB functions ‘imp_matcher_T()’/‘imp_matcher_pi()’ can be used to find possible T/П‐type LC filter(s) (with given quality factor Q) tuned for impedance matching between given source and load impedances, respectively.

6.3.4 Tapped‐C Impedance Matchers

Figure 6.24(a) shows a tapped‐C(capacitor) impedance matching circuit for a desired quality factor of Q/2 between R₁ and R₂. Figure 6.24(b) shows the intermediate equivalent circuit where the parallel R₁|C₂ part (with a target quality factor Q_p to be determined) has been transformed into its series equivalent consisting of R_1s‐C_2s and the parallel L|R₂ part (with a quality factor Q given) has been transformed into its series equivalent consisting of L_s‐R_2s by applying Eq. (6.2.28a). Figure 6.24(c) shows the final equivalent circuit with the series C₁‐C_2s part replaced by its series combination where

(6.3.4a,b,c,d)

(6.3.5a,b)

Noting that if the quality factors of the source and load parts from node 2 are equally Q, their composite quality factor will be 1/(1/Q + 1/Q) = Q/2, we write the condition for the quality factor of the load (L|R₂) part (i.e. the parallel combination of L and R₂) to be Q as

(6.3.6)

Also, the impedance matching conditions to make the source and load impedances, Z₁ and Z₂, a complex conjugate pair can be written as

(6.3.7)


(6.3.8)

Once the series combination of C₁‐C_2s part, that is C, is determined from Eq. (6.3.8), we can use the determined value of C_2s (Eq. (6.3.4c)) together with Eq. (6.3.8) to find C₁:

(6.3.9)

This process of designing a tapped‐C impedance matcher has been cast into the above MATLAB function ‘imp_matcher_tapped_C()’, in which the resulting value of quality factor Q, in the variable name of ‘Qr’, is computed as

(6.3.10a)

(6.3.10b)

since if the series R_1s‐C part (in Figure 6.24(c)) is converted to its parallel equivalent R_p|C_p while the series R_2s‐L_s part (in Figure 6.24(c)) is converted back to its parallel equivalent (as shown in Figure 6.24(d)), the resistance in parallel with L and C_p is not just R₂ but R_p‖R₂.

Problems

1. 6.1 Design of Passive Filter in the form of LC Ladder Circuit
   1. Design a Butterworth LPF with cutoff frequency f_c = 2 GHz in the form of N = 5‐stage П‐type LC ladder circuit (depicted in Figure 6.14.1(a)) where the source impedance is assumed to be 50 Ω. Plot the frequency response magnitude [dB] (relative to that without the filter) to check the validity of the designed filter.
   2. Design a Chebyshev BPF with passband ripple R_p = 0.5 dB, peak frequency f_p = 1 GHz, and bandwidth 0.1 GHz in the form of N = 3‐stage T‐type LC ladder circuit (depicted in Figure 6.14.1(b)) where the source impedance is assumed to be 50 Ω. Plot the frequency response magnitude or its reciprocal (called the insertion loss) [dB] to check the validity of the designed filter.
2. 6.2 Design of Passive Filters in the form of LC Ladder Circuit
   1. Design a Butterworth LPF with cutoff frequency f_c = 10 MHz, passband ripple R_p = 3 dB, and stopband attenuation A_s = 30 dB at f_s = 31.62 MHz where the source and load impedances are equally 50 Ω. Note that the order of the Butterworth LPF can be determined by using the MATLAB function ‘buttord()’ as

              >>wp=2*pi*7e6; ws=2*pi*31.62e6; Rp=3; As=30;
               [N,wc]=buttord(wp,ws,Rp,As,'s'), fc=wc/2/pi
      

   2. Design a Chebyshev LPF with cutoff frequency f_c = 100 MHz, passband ripple R_p = 0.01 dB, and stopband attenuation A_s = 5 dB at f_s = 40 MHz where the source and load impedances are equally 75 Ω. Note that the minimum order of the Chebyshev LPF can be determined by using the MATLAB function ‘cheb1ord()’ as

              >>wp=2*pi*1e8; ws=2*pi*4e8; Rp=0.01; As=5;
                [N,wc]=cheb1ord(wp,ws,Rp,As,'s')
      

   3. Design a Chebyshev HPF with cutoff frequency f_c = 100 MHz, passband ripple R_p = 0.01 dB, and stopband attenuation A_s = 5 dB at f_s = 25 MHz where the source and load impedances are equally 75 Ω. Note that the minimum order of the Chebyshev HPF can be determined by using the MATLAB function ‘cheb1ord()’ as

              >>wp=2*pi*1e8; ws=2*pi*25e6; Rp=0.01; As=5;
                [N,wc]=cheb1ord(wp,ws,Rp,As,'s')
      

   4. Design a sixth‐order Chebyshev BPF with cutoff frequencies {f_c1 = 10 MHz, f_c2 = 40 MHz} and passband ripple R_p = 0.01 dB, where the source and load impedances are equally 75 Ω.
   5. Design a sixth‐order Butterworth BSF with cutoff frequencies {f_c1 = 10 MHz, f_c2 = 40 MHz} and passband ripple R_p = 0.1 dB, where the source and load impedances are equally 75 Ω.
3. 6.3 Design of L‐Type Impedance Matcher for Maximum Power Transfer

   Design as many L‐type impedance matchers as possible for maximum power transfer from a transmitter with output impedance Z_s = 50 Ω to an antenna with input impedance Z_L = R_L + 1/jωC_L (R_L = 80 Ω, C_L = 2.653 pF) at the center frequency f_c = 1 GHz. Plot the power delivered to the load R_L as shown in Figure 6.20 twice, once by using the MATLAB function ‘imp_matcher_L_and_freq_resp()’ and once by using PSpice (with AC Sweep analysis type and a power marker on R_L in the schematic). Which one of the L‐type filters are you going to use for impedance matching after viewing the power curves?

4. 6.4 Design of L‐Type Impedance Matcher
   1. Make as many L‐type impedance matchers as possible to match a load of Z_L = 600 Ω to a source of Z_s = 50 Ω at a frequency f_c = 400 MHz and plot their frequency responses for some frequency band around f_c.
   2. Make as many two‐stage impedance matchers as possible to match a load of Z_L = 600 Ω to a source of Z_s = 50 Ω at a frequency f_c = 400 MHz where the second stage matches Z_L = 600 to 173.2 Ω and the first stage matches 173.2 to Z_s = 50 Ω. Use the following MATLAB function ‘ladder_xfer_ftn()’ to find the frequency responses (for two decades around f_c) and the input impedances (Z_i(j2πf)'s) of the impedance matchers (with the load impedance Z_L = 600 Ω) at f_c. Plot the frequency responses to see if they have peaks at f_c and check if Z_i(j2πf)'s match the source impedance Z_s = 50 Ω at f_c.
   3. Make as many L‐type filters as possible to match a load of Y_L = 8 − j12 [mS] to a 50 Ω line at a frequency f_c = 1 GHz. Use the above MATLAB function ‘ladder_xfer_ftn()’ to find the input impedances of the impedance matchers (with the load impedance Z_L = 1/Y_L) and check if they match the source impedance Z_s = 50 Ω at f_c = 1 GHz.
5. 6.5 Design of T‐Type Impedance Matcher
   1. Use the MATLAB function ‘imp_matcher_T()’ to make as many T‐type filter(s) as possible matching a load of Z_L = 225 Ω to a source of Z_s = 15 + j15 Ω with quality factor Q = 5 at a frequency f_c = 30MHz. You can visit the website https://www.eeweb.com/tools/t‐match to find two of the designs, one with DC current passed and one with DC current blocked.
   2. Use the MATLAB function ‘imp_matcher_T()’ to determine just the reactances constituting T‐type filter(s) to match a load of Z_L = 50 Ω to a source of Z_s = 10 Ω with quality factor Q = 10. (See Example 4.5 of [B-2].)
6. 6.6 Design of П‐Type Impedance Matcher

   Determine just the reactances constituting П‐type filter(s) to match a load of Z_L = 1000 Ω to a source of Z_s = 100 Ω with quality factor Q = 15. (See Example 4.4 of [B-2].)

7. 6.7 Design of Tapped‐C Impedance Matcher

   Make a tapped‐C impedance matcher to match a load of Z_L = 5 Ω to a source of Z_s = 50 Ω with quality factor Q = 5 at f₀ = 150 MHz. Plot the frequency response magnitude curve as shown in Figure 6.25(b).

8. 6.8 Design, Realization, and Implementation of High‐Order Butterworth Lowpass Filter (LPF)

   Given the order, say, N = 4 and cutoff frequency, say, ω_c = 2πf_c =2π × 10⁴ [rad/s] of an LPF, we can use the following MATLAB statements:

     >>format short e; N=4; fc=1e4; wc=2*pi*fc;
      [B,A]=butter(N,wc,'s')
   

   to find the numerator (B) and denominator (A) polynomials in s of the transfer function of the desired fourth‐order Butterworth LPF as

    B =           0           0           0           0 1.5585e+019
    A = 1.0000e+000 1.6419e+005 1.3479e+010 6.4819e+014 1.5585e+019
   

   This means the following transfer function:

   (P6.8.1)
   1. Plot the frequency response magnitude 20log₁₀|G(jω)| [dB] of the LPF (for the frequency range of two decades around f_c = 10⁴ [Hz]) and show that the transfer function can be realized in cascade form, i.e. as a product of second‐order transfer functions as
      (P6.8.2)

      or in parallel form, i.e. as a sum of second‐order transfer functions as

      (P6.8.3)

      In fact, this implies that the transfer function can be realized as a cascade connection of the two SOSs (second‐order sections), G_C1(s) and G_C2(s), or a parallel connection of the two SOSs, G_P1(s) and G_P2(s). You can complete and run the above MATLAB script “elec06p08a.m” to find the cascade and parallel realizations.

   2. Noting that each SOS of the circuit in Figure P6.8(a) has the following transfer function

      

      (P6.8.4)

      let R₁₁ = R₁₂ = R₂₁ = R₂₂ = 10 kΩ and find the values of C₁₃, C₁₄, C₂₃, and C₂₄ such that the desired transfer function will be implemented. Plot the frequency response of the cascade realization (P6.8.2) to see if it conforms with that of G(s) obtained in (a).

   3. Noting that the circuit of Figure P6.8(b) has the following transfer function
      (P6.8.5)

      let L₁ = L₂ = L = 0.01 H and find the values of R₁, C₁, R₂, C₂, and R_f such that the desired transfer function will be implemented. Plot the frequency response of the parallel realization (P6.8.3) to see if it conforms with that of G(s) obtained in (a).

   4. Perform the PSpice simulation to see that the cutoff frequency of the two circuits, one realized in cascade form and the other in parallel form, is f_c = 10 kHz at which the magnitude of the frequency response is = 0.7071 = ‐3 dB. You can use the VDB marker (from PSpice > Markers > Advanced) in the Schematic window to measure the frequency response in dB.
9. 6.9 Frequency Scaling

   Referring to Remark 6.2(2), try the frequency scaling for Example 6.17 to reduce the discrepancy between the two frequency responses, one obtained from PSpice simulation and the other obtained from MATLAB analysis. Using PSpice, plot the frequency response magnitude of the modified filter to show that you worked properly.

10. 6.10 Design and Realization of High‐Order Chebyshev I Bandpass Filter (BPF)

    Given the filter order, say, N = 4 and the lower/upper 3 dB frequencies, say, ω_l = 2π × 8000 and ω_u = 2π × 12500 [rad/s] of a BPF, we can run the following MATLAB statements to find the numerator (B) and denominator (A) polynomials in s of the transfer function of the desired fourth‐order Chebyshev I BPF.

     >>format short e; N=4; Rp=3;
       fl=8000; fu=12500; wl=2*pi*fl; wu=2*pi*fu;
       [B,A]=cheby1(N/2,Rp,[wl wu],'s')
    

    This yields

     B =           0           0 4.0067e+008             0     0
     A = 1.0000e+000 1.8234e+004 8.4616e+009 7.1985e+013 1.5585e+19
    

    meaning the following transfer function:

    (P6.10.1)

    This transfer function can be expressed in the form of product of two second‐order transfer functions as

    (P6.10.2)

    In fact, this implies that the transfer function can be realized as a cascade connection of the two SOSs, G_C1(s) and G_C2(s).

    1. Noting that each SOS of the circuit in Figure P6.10.1 has the following transfer function
       (P6.10.3)

       let C₁₃ = C₁₄ = C₂₃ = C₂₄ = 10 nF and find the values of R₁₁, R₁₂, R₁₅, R₂₁, R₂₂, and R₂₅ such that the desired transfer function will be realized. You can complete and run the above script “elec06p10.m,” which uses the MATLAB function ‘BPF_MFBa_design()’.

       

    2. As depicted in Figure P6.10.2(a), plot the magnitude curve of the frequency response of the filter having the transfer function (P6.10.1) for f = 1–100 kHz. Does it have any ripple in the passband or the stopband? Then perform the PSpice simulation to get the frequency response of the BPF circuit that has been designed in part (a) (as depicted in Figure P6.10.2(b)). Check if the lower and upper 3 dB frequencies of the circuit are close to f_l,3 dB = 8000 Hz and f_u,3 dB = 12500 Hz, respectively, at which the magnitudes of the frequency response are = 0.7071 = −3 dB.

       

    3. If the PSpice simulation result tells you that the upper/lower 3 dB frequencies are considerably deviated from the expected ones, scale the capacitance values by an appropriate factor for frequency scaling (Remark 6.2(1)) and perform the PSpice simulation again to see if the upper/lower 3 dB frequencies get closer to the expected ones.
11. 6.11 Fourier Series Solution of an Active Second‐Order OP Amp Circuit Excited by a Triangular Wave

    Consider the active second‐order OP Amp circuit in Figure P6.11(a), which is excited by a triangular wave voltage source of period 0.0004π as depicted in Figure P6.11(b). Find the magnitudes of the leading three frequency components (up to three significant digits) of the input v_i(t) and output v_o(t) in two ways, one using MATLAB and one using PSpice, and fill in Table P6.11.

    

    Order of harmonics	k = 1		k = 3		k = 5
    	MATLAB	PSpice	MATLAB	PSpice	MATLAB	PSpice
    Input vi(t)	8.11			0.903	0.324
    Output vo(t)		8.16	0.0675			0.0135

    1. Complete the following MATLAB script “elec06p11.m,” which uses the MATLAB function ‘Fourier_analysis()’ introduced in Section 1.5.2, together with the MATLAB function ‘elec06p11_f()’ (to be saved in a separate M‐file named “elec06p11_f.m”) and run it to get the waveforms and Fourier spectra of v_i(t) and v_o(t) as shown in Figure P6.11(c) and (d).
    2. Find the peak (center) frequency ω_p and bandwidth ω_b of the circuit from its transfer function or frequency response. How is ω_p related with the frequency component of the input passing through the filter that can be seen in the input and output spectra shown in Figure P6.11(d)?
    3. Use the PSpice simulation where the parameters of the periodic voltage source VPULS can be set as follows:
       

       In the Simulation Settings dialog box, set the Analysis type, Run_to_time, and Maximum step as Time Domain (Transient), 20 ms, and 1 u, respectively, and click the Output File Options button to open the Transient Output File Options dialog box, in which the parameters are to be set as