Appendix 2B: Transmission Lines: Power Calculation

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2B.1Transmission Lines: Power Calculation

A transmission line of length d is shown in Figure 2B.1a. The equivalent circuit is shown in Figure 2B.1b.

FIGURE 2B.1
Transmission line: (a) length d and (b) equivalent input impedance.

The input impedance into the transmission line of length d with load Z_L is given by

(2B.1)

Z_{in} = Z_{0} \frac{Z_{L} + j Z_{0} tan (β d)}{Z_{0} + j Z_{0} tan (β d)} = Z_{0} \frac{(1 + Γ_{0} e^{- j 2 β d})}{(1 - Γ_{0} e^{- j 2 β d})} .

$Z_{in} = Z_{0} \frac{Z_{L} + j Z_{0} tan (β d)}{Z_{0} + j Z_{0} tan (β d)} = Z_{0} \frac{(1 + Γ_{0} e^{- j 2 β d})}{(1 - Γ_{0} e^{- j 2 β d})} .$

The voltage at the input of the transmission line can be found as

(2B.2)

\begin{matrix} \tilde{V} (d) & = & \frac{{\tilde{V}}_{g} Z_{in}}{Z_{g} + Z_{in}} \\ = & {\tilde{V}}_{0}^{+} e^{j β d} + {\tilde{V}}_{0}^{-} e^{- j β d} = {\tilde{V}}_{0}^{+} e^{j β d} [1 + Γ_{0} e^{- j 2 β d}], \end{matrix}

$\begin{matrix} \tilde{V} (d) & = & \frac{{\tilde{V}}_{g} Z_{in}}{Z_{g} + Z_{in}} \\ = & {\tilde{V}}_{0}^{+} e^{j β d} + {\tilde{V}}_{0}^{-} e^{- j β d} = {\tilde{V}}_{0}^{+} e^{j β d} [1 + Γ_{0} e^{- j 2 β d}], \end{matrix}$

where ${\tilde{V}}_{0}^{+}$ ${\tilde{V}}_{0}^{+}$ is the voltage of the positive traveling wave and ${\tilde{V}}_{0}^{-}$ ${\tilde{V}}_{0}^{-}$ is the voltage of the reflected wave at the load end. Rearranging, we obtain

(2B.3)

{\tilde{V}}_{0}^{+} = \frac{{\tilde{V}}_{g} Z_{in}}{(Z_{g} + Z_{in}) e^{j β d} [1 + Γ_{0} e^{- j 2 β d}]},

${\tilde{V}}_{0}^{+} = \frac{{\tilde{V}}_{g} Z_{in}}{(Z_{g} + Z_{in}) e^{j β d} [1 + Γ_{0} e^{- j 2 β d}]},$

(2B.4)

{\tilde{V}}_{0}^{+} = {\tilde{V}}_{g} \frac{Z_{in}}{(Z_{g} + Z_{in}) (e^{j β d} + Γ_{0} e^{- j β d})} .

${\tilde{V}}_{0}^{+} = {\tilde{V}}_{g} \frac{Z_{in}}{(Z_{g} + Z_{in}) (e^{j β d} + Γ_{0} e^{- j β d})} .$

In the above,

(2B.5)

Γ_{0} = \frac{Z_{L} - Z_{0}}{Z_{L} + Z_{0}} .

$Γ_{0} = \frac{Z_{L} - Z_{0}}{Z_{L} + Z_{0}} .$

The average incident power is given as

P_{av}^{i} = Re [\frac{1}{2} {\tilde{V}}_{0}^{+} {({\tilde{I}}_{0}^{+})}^{*}] = \frac{1}{2} Re [{\tilde{V}}_{0}^{+} \frac{{({\tilde{V}}_{0}^{+})}^{*}}{Z_{0}}],

$P_{av}^{i} = Re [\frac{1}{2} {\tilde{V}}_{0}^{+} {({\tilde{I}}_{0}^{+})}^{*}] = \frac{1}{2} Re [{\tilde{V}}_{0}^{+} \frac{{({\tilde{V}}_{0}^{+})}^{*}}{Z_{0}}],$

(2B.6)

P_{av}^{i} = \frac{1}{2} \frac{{|{\tilde{V}}_{0}^{+}|}^{2}}{Z_{0}} .

$P_{av}^{i} = \frac{1}{2} \frac{{|{\tilde{V}}_{0}^{+}|}^{2}}{Z_{0}} .$

The average reflected power is given by

(2B.7)

\begin{matrix} P_{av}^{r} & = & Re [\frac{1}{2} {\tilde{V}}_{0}^{-} {({\tilde{I}}_{0}^{-})}^{*}] = - \frac{1}{2} Re [{\tilde{V}}_{0}^{-} \frac{{({\tilde{V}}_{0}^{-})}^{*}}{Z_{0}}], \\ P_{av}^{r} & = & - \frac{1}{2} \frac{{|Γ_{0}|}^{2} {|{\tilde{V}}_{0}^{+}|}^{2}}{Z_{0}} . \end{matrix}

$\begin{matrix} P_{av}^{r} & = & Re [\frac{1}{2} {\tilde{V}}_{0}^{-} {({\tilde{I}}_{0}^{-})}^{*}] = - \frac{1}{2} Re [{\tilde{V}}_{0}^{-} \frac{{({\tilde{V}}_{0}^{-})}^{*}}{Z_{0}}], \\ P_{av}^{r} & = & - \frac{1}{2} \frac{{|Γ_{0}|}^{2} {|{\tilde{V}}_{0}^{+}|}^{2}}{Z_{0}} . \end{matrix}$

The total power consumed by the load is

(2B.8)

\begin{matrix} P_{av}^{tot} & = & P_{av}^{i} + P_{av}^{r} = \frac{1}{2} \frac{{|{\tilde{V}}_{0}^{+}|}^{2}}{Z_{0}} [1 - {|Γ_{0}|}^{2}], \\ P_{av}^{tot} & = & P_{av}^{i} [1 - {|Γ_{0}|}^{2}] . \end{matrix}

$\begin{matrix} P_{av}^{tot} & = & P_{av}^{i} + P_{av}^{r} = \frac{1}{2} \frac{{|{\tilde{V}}_{0}^{+}|}^{2}}{Z_{0}} [1 - {|Γ_{0}|}^{2}], \\ P_{av}^{tot} & = & P_{av}^{i} [1 - {|Γ_{0}|}^{2}] . \end{matrix}$

This is shown in Figure 2B.2.

FIGURE 2B.2
Transmission line circuit showing incident and reflected power.

2B.2Transmission Lines: Special Case Z_g = Z₀

Consider the special case of the transmission line circuit in Figure 2B.1a with Z_g = Z₀, the characteristic impedance of the transmission line. It will be shown that ${\tilde{V}}_{0}^{+}$ ${\tilde{V}}_{0}^{+}$ is independent of Z_L if Z_g = Z₀. When Z_g = Z₀, from Equation 2B.1, we obtain

(2B.9)

Z_{g} + Z_{in} = Z_{0} + Z_{0} \frac{(1 + Γ_{0} e^{- j 2 β d})}{(1 - Γ_{0} e^{- j 2 β d})} .

$Z_{g} + Z_{in} = Z_{0} + Z_{0} \frac{(1 + Γ_{0} e^{- j 2 β d})}{(1 - Γ_{0} e^{- j 2 β d})} .$

Substituting Equations 2B.1 and 2B.9 in the expression for ${\tilde{V}}_{0}^{+}$ ${\tilde{V}}_{0}^{+}$ given in Equation 2B.3:

(2B.10)

{\tilde{V}}_{0}^{+} = \frac{{\tilde{V}}_{g} Z_{0} (1 + Γ_{0} e^{- j 2 β d}) / (1 - Γ_{0} e^{- j 2 β d})}{Z_{0} [1 + (1 + Γ_{0} e^{- j 2 β d}) / (1 - Γ_{0} e^{- j 2 β d})] e^{j β d} [1 + Γ_{0} e^{- j 2 β d}]} .

${\tilde{V}}_{0}^{+} = \frac{{\tilde{V}}_{g} Z_{0} (1 + Γ_{0} e^{- j 2 β d}) / (1 - Γ_{0} e^{- j 2 β d})}{Z_{0} [1 + (1 + Γ_{0} e^{- j 2 β d}) / (1 - Γ_{0} e^{- j 2 β d})] e^{j β d} [1 + Γ_{0} e^{- j 2 β d}]} .$

Rearranging,

(2B.11)

{\tilde{V}}_{0}^{+} = \frac{{\tilde{V}}_{g}}{2} e^{- j β d},

${\tilde{V}}_{0}^{+} = \frac{{\tilde{V}}_{g}}{2} e^{- j β d},$

which holds whenever Z_g = Z₀ regardless of the value of Z_L.

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Table of Contents for Appendix 2B: Transmission Lines: Power Calculation

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2B.1Transmission Lines: Power Calculation

2B.2Transmission Lines: Special Case Zg = Z0

Table of Contents for
Appendix 2B: Transmission Lines: Power Calculation

2B.2Transmission Lines: Special Case Z_g = Z₀