In the circuit shown in Fig. 13.37, the source output resistance is Ro=100Ω and the load resistance is RL=10Ω. The source voltage is sinusoidal, with frequency f=60 Hz . Find the values of the inductance L and capacitance C in the L-section matching circuit for which maximum power is

Question

In the circuit shown in Fig. 13.37, the source output resistance is [latex] R_o=100 \Omega [/latex] and the load resistance is [latex] R_L=10 \Omega [/latex]. The source voltage is sinusoidal, with frequency f=60 Hz . Find the values of the inductance L and capacitance C in the L-section matching circuit for which maximum power is transferred to the load. For those values of L and C , find the frequency for which maximum power is transferred to the load.

Accepted Answer

Because the source resistance is larger than the load resistance, we associate the shunt capacitor with the source (to decrease the apparent source impedance) and the series inductor with the load (to increase the apparent load impedance). The modified source and load impedances are given by

[latex] \begin{aligned} &Z_L(j \omega)=R_L+j \omega L \ &Z_o(j \omega)=\frac{R_o}{1+j \omega R_o C}, \omega=2 \pi f \end{aligned} [/latex]

For maximum power transfer at the match frequency [latex]f_0=60 Hz[/latex], we require [latex]Z_L\left(j \omega_0 ight)=Z_o{ }^*\left(j \omega_0 ight)[/latex], which leads to

[latex] \frac{R_o\left(1-j \omega_0 R_o C ight)}{1+\left(\omega_0 R_o C ight)^2}=R_L-j \omega_0 L ; \omega_0=2 \pi f_0 [/latex]

It follows that

[latex] \begin{aligned} &\frac{R_o}{1+\left(\omega_0 R_o C ight)^2}=R_L \ &\frac{R_o^2 C}{1+\left(\omega_0 R_o C ight)^2}=R_L R_o C=L . \end{aligned} [/latex]

Solving the first of these relations for the capacitance and then using the second to calculate the inductance yields

[latex] \begin{gathered} C=\frac{1}{\omega_0 R_o} \sqrt{\frac{R_0}{R_L}-1}=79.6 \mu F \\ L=R_o R_L C=79.6 mH \end{gathered} [/latex] (13.63)

The power delivered to the load (as a function of frequency) is given by

[latex] \begin{aligned} P_L(f) &=\frac{1}{2}| ilde{I}(f)|^2 \operatorname{Re}\left[Z_L(j 2 \pi f) ight] \ &=\frac{1}{2}\left|\frac{ ilde{V}(f)}{Z_o(j 2 \pi f)+Z_L(j 2 \pi f)} ight|^2 R_L . \end{aligned} [/latex]

Figure 13.39 shows the power transferred to the load, normalized to the power transferred at the match frequency, as a function of frequency. The power transferred to the load is maximum at a frequency slightly higher than the match frequency [latex] f_0 [/latex] . As an exercise, you can show that the frequency for maximum power transfer is [latex] f_1=65.8 Hz [/latex] .

Question 13.22: In the circuit shown in Fig. 13.37, the source output resist...

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