a) For the circuit shown in Fig. 10.23, what value of $Z_{L}$ results in maximum average power transfer to $Z_{L}$ ? What is the maximum power in milliwatts?
b) Assume that the load resistance can be varied between 0 and 4000Ω and that the capacitive reactance of the load can be varied between 0 and -2000 Ω. What settings of $R_{L}$ and $X_{L}$ transfer the most average power to the load? What is the maximum average power that can be transferred under these restrictions?

Question

a) For the circuit shown in Fig. 10.23, what value of [latex]Z_{L}[/latex] results in maximum average power transfer to [latex]Z_{L}[/latex] ? What is the maximum power in milliwatts?
b) Assume that the load resistance can be varied between 0 and 4000Ω and that the capacitive reactance of the load can be varied between 0 and -2000 Ω. What settings of [latex]R_{L}[/latex] and [latex]X_{L}[/latex] transfer the most average power to the load? What is the maximum average power that can be transferred under these restrictions?

Accepted Answer

a) If there are no restrictions on [latex]R_{L}[/latex] and [latex]X_{L}[/latex],the load impedance is set equal to the conjugate of the output or the Thévenin impedance. Therefore we set

[latex]R_{L}=3000Ω [/latex] and [latex]X_{L}=-4000Ω[/latex],

or

[latex]Z_{L} = 3000 - j4000 Ω[/latex].

Because the source voltage is given in terms of its rms value, the average power delivered to [latex]Z_{L}[/latex] is

[latex]P =\frac{1}{4}\frac{10^{2}}{3000} =\frac{25}{3} mW = 8.33 mW[/latex].

b) Because [latex]R_{L}[/latex] and [latex]X_{L}[/latex] are restricted, we first set [latex]X_{L}[/latex] as close to -4000Ω as possible; thus [latex]X_{L}=-2000Ω[/latex]. Next, we set [latex]R_{L}[/latex] as close to [latex]\sqrt{R^{2}_{Th}+\left(X_{L}+X_{Th} ight)^{2}}[/latex] as possible. Thus

[latex]R_{L} = \sqrt{3000^{2} + (-2000 + 4000)^{2}} = 3605.55 Ω[/latex].

Now, because[latex]R_{L} [/latex] can be varied from 0 to 4000Ω, we can set [latex]R_{L} [/latex] to 3605.55 Ω Therefore, the load impedance is adjusted to a value of

[latex]Z_{L}= 3605.55 - j2000 Ω.[/latex]

With [latex]Z_{L}[/latex] set at this value, the value of the load current is

[latex]I_{eff} =\frac{10 \angle 0°}{6605.55 + j2000} = 1.4489 \angle -16.85° mA[/latex].

The average power delivered to the load is

[latex]P = \left(1.4489 imes 10^{-3} ight)^{2}\left(3605.55 ight) = 7.57 mW[/latex].

This quantity is the maximum power that we can deliver to a load, given the restrictions on [latex]R_{L}[/latex] and [latex]X_{L}[/latex]. Note that this is less than the power that can be delivered if there are no restrictions; in (a) we found that we can deliver 8.33 mW.

Question 10.9: a) For the circuit shown in Fig. 10.23, what value of ZL res...

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