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Appendix G
MAXIMUM POWER TRANSFER CONDITIONS
Derivation of maximum power transfer conditions for the situation
where the resistive component of the load is adjustable but the load
reactance is set in magnitude.*
For the circuit of Fig. G.1, the power delivered to the load is determined
by
E Th
+
–
I
R Th
Z T
P � V 2 R L
�RL
Applying the voltage divider rule:
VRL �
The magnitude of VRL is determined by
RLETh VRL �����
� �� X� �L) 2 RLETh ����
RL � RTh � XTh �90° � XL �90°
�
and V 2 R L �
Z Th
X Th
FIG. G.1
�(R� L��� R� Th �) 2 � �� (�X� Th
with P � �
Using differentiation (calculus), maximum power will be transferred
when dP/dRL � 0. The result of the preceding operation is that
RL � �R� 2 Th� �� (�X� �Th �� X� L) �2� [Eq. (18.21)]
The magnitude of the total impedance of the circuit is
ZT � �(R� �Th �� R� L) �2��� (�X� Th � �� X� L) �2 RL E
�
2 Th
���
(RL � RTh) 2 � (XTh � XL) 2
V 2 RL �
RL
Substituting this equation for RL and applying a few algebraic
maneuvers will result in
ZT � 2RL(RL � RTh) X L
I
R L
R 2 L E 2 Th
���
(R L � R Th) 2 � (X Th � X L) 2
*With sincerest thanks for the input of Professor Harry J. Franz of the Beaver Campus of
Pennsylvania State University.
Z L
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