Foundations of Analog and Digital Circuits Mas - Wordpress ...

v i
V i
0
π/ω 2π/ω
t
13.7 Power and Energy in an Impedance CHAPTER THIRTEEN 763
p(t)
0
π/ω 2π/ω
For the general case when the network contains resistors, capacitors, and inductors,
the power flow will have some intermediate form between Figure 13.52
and Figure 13.53. Assuming that the circuit is net inductive at the frequency of
interest, then θ is positive but less than π/2, and Equation 13.149 with φ = 0
becomes
V 2 i
p(t) = 1
[cos(2ωt − θ) + cos θ].
2 R2 + X2 The power waveform is as depicted in Figure 13.54.
13.7.4 EXAMPLE: POWER IN AN RC CIRCUIT
Let us examine a specific circuit with both resistive and reactive components,
the RC circuit of Figure 13.55. To calculate the average power from either
Equation 13.151 or 13.152, we must find the complex amplitude of the current.
By inspection of Figure 13.55:
Ii = Vi
Z =
Vi
R + 1/jωC
=
Vi
R 2 + (1/ωC) 2 e−jθ
p
t
(13.168)
(13.169)
where
θ = tan −1
1
ωRC
(13.170)
Now the average power dissipated in the circuit is, from Equation 13.151:
p = 1
cos(θ) (13.171)
2 R2 + (1/ωC) 2
V 2 i
V 2 i
= 1
cos(θ). (13.172)
2 |Z|
FIGURE 13.54 Power flow in an
inductive circuit. The average
power is given by
p = 1/2V 2
i / R 2 + X 2 cos θ. The
maximum value of p(t) is
1/2V 2
i / R 2 + X 2 (cos θ + 1), and
the minimum value is
1/2V 2
i / R 2 + X 2 (cos θ − 1).
v i = V i cos (ωt)
V i
+
-
+
-
i(t) C
I
R
1/jωC
FIGURE 13.55 Series RC
circuit.
R