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NON
v(a) � V 0 � V m1 sin a � ⋅ ⋅ ⋅ � V mn sin na � V′ m1 cos a � ⋅ ⋅ ⋅ � V′ mn cos na
then
V rms � � V �2 0�
However, since
then
V 2 m 1
� 2
����� (25.11)
V 2 m � ⋅ ⋅ ⋅ � V 1 2 m � V′ n 2 m � ⋅ ⋅ ⋅ � V′ 1 2 mn ����
2
V m1
� �2�
V m1
� �2�
� � �� � � (V 1rms )(V 1rms ) � V 2 1 rms
Vrms � �V� 2 0��� V� 2 �1 ��� �⋅ �⋅ �⋅ �� V� rms
2 n� � �� V�′ rms
2 1� ���� �⋅ �⋅ �⋅ �� V�′ rms 2 n� �� (25.12)
rms i(a) � I0 � Im1 sin a � ⋅ ⋅ ⋅ � Imn sin na � I′ m1 cos a � ⋅ ⋅ ⋅ � I′ mn cos na
we have
and
Similarly, for
I rms � � I 2 0
Irms � �I 2 0� �� I� 2 1� rms
� � ���� (25.13)
I 2 m � ⋅ ⋅ ⋅ � I 1 2 m � I′ n 2 m � ⋅ ⋅ ⋅ � I′ 1 2 mn ����
2
��� �⋅ �⋅ �⋅ �� I� 2 �n ��� I�′ rms
2 1� ���� �⋅ �⋅ �⋅ �� I�′ rms 2 n� �� (25.14)
rms The total power delivered is the sum of that delivered by the corresponding
terms of the voltage and current. In the following equations,
all voltages and currents are rms values:
P T � V 0 I 0 � V 1 I 1 cos v 1 � ⋅ ⋅ ⋅ � V n I n cos v n � ⋅ ⋅ ⋅
P T � I 2 0 R � I 2 1 R � ⋅ ⋅ ⋅ � I 2 n R � ⋅ ⋅ ⋅
CIRCUIT RESPONSE TO A NONSINUSOIDAL INPUT ⏐⏐⏐ 1135
(25.15)
(25.16)
or PT � I (25.17)
2 rmsR
with I rms as defined by Eq. (25.13), and, similarly,
PT � �
with Vrms as defined by Eq. (25.11).
V
�
R
(25.18)
EXAMPLE 25.5
a. Sketch the input resulting from the combination of sources in Fig.
25.22.
2 rms
+
6 sin �t �
–
4 V
FIG. 25.22
Example 25.5.
+
v
–