Question

The Maximum power transfer in an AC network occurs when:

• A. $RL = RS$
• B. $ZL = ZS^*$
• C. $ZL = ZS$
• D. $RL = 0$

Hint:

If the load is purely resistive (\(X_L = 0\)) but the source is complex, the condition for maximum power transfer changes to \(R_L = |Z_S| = \sqrt{R_S^2 + X_S^2}\).
Always verify if the load is restricted to being resistive or if it can be complex.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical condition required to achieve maximum power transfer in an AC network when both the source impedance and the load impedance are complex quantities.

Step 2: Key Formula or Approach: Let the source impedance be represented as:
\[ Z_S = R_S + jX_S \] And the load impedance be represented as:
\[ Z_L = R_L + jX_L \] To maximize the active power transferred to the load, the load impedance must be the complex conjugate of the internal source impedance:
\[ Z_L = Z_S^* = R_S - jX_S \]

Step 3: Detailed Explanation:

• The average power delivered to the load in an AC circuit is given by:
\[ P = I^2 R_L \]
• The current \(I\) in the loop is:
\[ I = \frac{V_S}{|Z_S + Z_L|} = \frac{V_S}{\sqrt{(R_S + R_L)^2 + (X_S + X_L)^2}} \]
• Substituting the current into the power formula:
\[ P = \frac{V_S^2 R_L}{(R_S + R_L)^2 + (X_S + X_L)^2} \]
• To maximize \(P\) with respect to the load reactance \(X_L\), we choose \(X_L\) such that it cancels the source reactance \(X_S\):
\[ X_S + X_L = 0 \implies X_L = -X_S \]
• With the reactances canceled, the expression for power simplifies to:
\[ P = \frac{V_S^2 R_L}{(R_S + R_L)^2} \]
• Differentiating this power with respect to \(R_L\) and setting the derivative to zero yields the standard resistive matching condition:
\[ R_L = R_S \]
• Combining both conditions (\(R_L = R_S\) and \(X_L = -X_S\)), we get:
\[ Z_L = R_S - jX_S = Z_S^* \]
• This means the load impedance must be equal to the complex conjugate of the source impedance.

Step 4: Final Answer:
Maximum power transfer in an AC network occurs when \(Z_L = Z_S^*\).