Question

In a network maximum power transfer occurs when

• A. \( R_{Th} = -R_L \)
• B. \( \frac{R_{Th}}{R_L} = 0 \)
• C. \( R_{Th} = R_L \)
• D. \( R_{Th} + R_L = 1 \)

Hint:

The Maximum Power Transfer Theorem is a critical concept in circuit design, especially for power amplifiers and communication systems. Remember the condition: maximum power is transferred when the load resistance ($R_L$) is equal to the Thevenin equivalent resistance ($R_{Th}$) of the source network. Also, know that while power transfer is maximized, the efficiency at this point is only 50\%, as an equal amount of power is dissipated within the source's internal resistance.

Answer 1

The Correct Option is C

The Maximum Power Transfer Theorem states that to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must be equal to the resistance of the source as viewed from its output terminals. In the context of Thevenin's equivalent circuit, any linear circuit can be represented by a Thevenin equivalent voltage source $V_{Th}$ in series with a Thevenin equivalent resistance $R_{Th}$. If a load resistance $R_L$ is connected across the terminals of this Thevenin equivalent circuit, the power delivered to the load is given by: $P_L = I^2 R_L = \left(\frac{V_{Th}}{R_{Th} + R_L}\right)^2 R_L$ To find the condition for maximum power transfer, we differentiate $P_L$ with respect to $R_L$ and set the derivative to zero: $\frac{dP_L}{dR_L} = V_{Th}^2 \frac{d}{dR_L} \left( \frac{R_L}{(R_{Th} + R_L)^2} \right) = 0$ Using the quotient rule: $\frac{(R_{Th} + R_L)^2 \cdot 1 - R_L \cdot 2(R_{Th} + R_L)}{(R_{Th} + R_L)^4} = 0$ For the numerator to be zero: $(R_{Th} + R_L)^2 - 2R_L(R_{Th} + R_L) = 0$ Divide by $(R_{Th} + R_L)$ (assuming $R_{Th} + R_L \neq 0$): $(R_{Th} + R_L) - 2R_L = 0$ $R_{Th} - R_L = 0$ $R_{Th} = R_L$ This condition ensures that maximum power is transferred from the source to the load.