Maximum Power Transfer Theorem
Consider a source with voltage V and internal resistance Rs connected to a load resistance RL.
1. Circuit Current
I = V / (Rs + RL)
2. Load Power
Power delivered to the load:
PL = I²RL
Substituting the value of current:
PL = (V / (Rs + RL))² × RL
PL = V²RL / (Rs + RL)²
3. Condition for Maximum Power
Differentiate the power equation with respect to RL and set:
dPL / dRL = 0
Solving gives:
RL = Rs
Therefore, maximum power is transferred when:
RL = Rs
4. Maximum Power Value
Substituting RL = Rs:
Pmax = V²Rs / (2Rs)²
Pmax = V²Rs / 4Rs²
Pmax = V² / 4Rs
5. AC Complex Impedance Case
For AC circuits:
Zs = Rs + jXs
ZL = RL + jXL
Average load power:
PL =
|V|²RL /
[(Rs + RL)² +
(Xs + XL)²]
Maximum power occurs when:
XL = -Xs
RL = Rs
Therefore:
ZL = Rs - jXs
This is called complex conjugate matching.
6. Summary Table
| Case | Maximum Power Condition |
|---|---|
| Resistive Circuit | RL = Rs |
| AC Complex Circuit | ZL = Zs* |
| Maximum Power | Pmax = V² / 4Rs |
Conclusion: Maximum power transfer occurs when the load resistance
equals the source resistance, or in AC circuits when the load
impedance is the complex conjugate of the source impedance.