Power Computations
2.9 POWER COMPUTATIONS USING PSPICE
52 C H A P T E R 2 Power Computations
Instantaneous Power, Energy, and Average Power Using PSpice
PSpice can be used to display instantaneous power and to compute energy. A simple example is a sinusoidal voltage across a resistor. The voltage source has amplitude Vm10 V and frequency 60 Hz, and the resistor is 5 . Use VSIN for the source, and select Time Domain (Transient) in the Simulation Setup. Enter a Run Time (Time to Stop) of 16.67 ms for one period of the source.
The circuit is shown in Fig. 2-12a. The top node is labeled as 1. When placing the resistor, rotate it 3 times so that the first node is upward. After running the simulation, the Netlist should look like this:
*source EXAMPLE 2-11
V_V1 1 0
SIN 0 10 60 0 0 0
R_R1 1 0 5
When the simulation is completed, the Probe screen appears. The waveforms of volt- age and current for the resistor are obtained by entering V(1) and I(R1). Instantaneous EXAMPLE 2-11
(a) VOFF = 0
VAMPL = 10 FREQ = 60
0
V1 R1 5
1
p(t)
(b) Time 20
10
0
v(t)
i(t)
0 s -10
5 ms 10 ms 15 ms 20 ms
I(R1) W(R1) V(1)
+ –
Figure 2.12 (a) PSpice circuit for Example 2-11; (b) voltage, current, and instantaneous power for the resistor; (c) energy absorbed by the resistor;
(d) average power absorbed by the resistor.
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2.9 Power Computations Using PSpice 53
power p(t) v(t)i(t) absorbed by the resistor is obtained from Probe by entering the expression V(1)*I(R1) or by selecting W(R1). The resulting display showing V(1), I(R1), and p(t) is in Fig. 2-12b.
Energy can be computed using the definition of Eq. (2-2). When in Probe, enter the expression S(V(1)*I(R1)) or S(W(R1)), which computes the integral of instantaneous power. The result is a trace that shows that the energy absorbed increases with time. The energy absorbed by the resistor after one period of the source is determined by placing the cursor at the end of the trace, revealing WR166.66 mJ (Fig. 2-12c).
Figure 2.12 (continued)
0 s 5 ms 10 ms 15 ms 20 ms
0 s 5 ms 10 ms 15 ms 20 ms
200 m
100 m
0
Time
ENERGY AFTER ONE PERIOD 166.664 mJ
(16.670 m, 166.664 m)
(c) S (W(R1))
15 W
10 W
5 W
0 W
Time
AVERAGE POWER 9.998 W (16.670 m, 9.998)
(d) AVG (W(R1))
54 C H A P T E R 2 Power Computations
The Probe feature of PSpice can also be used to determine the average value of power directly. For the circuit in the above example, average power is obtained by enter- ing the expression AVG(V(1)*I(R1)) or AVG(W(R1)). The result is a “running” value of average power as computed in Eq. (2-3). Therefore, the average value of the power wave- form must be obtained at the endof one or more periods of the waveform. Figure. 2-12d shows the output from Probe. The cursor option is used to obtain a precise value of aver- age power. This output shows 9.998 W, very slightly different from the theoretical value of 10 W. Keep in mind that the integration is done numerically from discrete data points.
PSpice can also be used to determine power in an ac circuit containing an inductor or capacitor, but the simulation must represent steady-state response to be valid for steady-state operation of the circuit.
(b)
V1 = 0 V2 = 20 TD = 0 TR = 1 n TF = 1 n PW = 8 m PER = 20 m
R1 2 1 L1 V1
1
10 mH 0
v(t)
t 20 V
0 8 ms
(a) 20 ms
60 ms 10 A
5 A
0 A
80 ms
70 ms 90 ms 100 ms 110 ms
(80.000 m, 4.6389) RMS CURRENT
(100.000 m, 4.6389)
I(R1) RMS(I(R1)) Time
+ -
Figure 2-13 (a) A pulse waveform voltage source is applied to a series R-Lcircuit; (b) Probe output showing the steady-state current and the rms value.
RMS and Fourier Analysis Using PSpice
Fig. 2-13ashows a periodic pulse voltage that is connected to a series R-L circuit with R10 and L10 mH. PSpice is used to determine the steady-state rms current and the Fourier components of the current.
EXAMPLE 2-12
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2.9 Power Computations Using PSpice 55
In PSpice power calculations, it is extremely important that the output being analyzed represent steady-state voltages and currents. Steady-state cur- rent is reached after several periods of the pulse waveform. Therefore, the Simulation Settings have the Run Time (Time to Stop) at 100 ms and the Start Saving Data set at 60 ms. The 60-ms delay allows for the current to reach steady state. A maximum step size is set at 10 s to produce a smooth waveform.
Current is displayed in Probe by entering I(R1), and steady state is verified by noting that the starting and ending values are the same for each period. The rms current is obtained by entering the expression RMS(I(R1)). The value of rms current, 4.6389 A, is obtained at the end of any period of the current waveform.
Fig. 2-13bshows the Probe output.
The Fourier series of a waveform can be determined using PSpice. Fourier analysis is entered under Output File Options in the Transient Analysis menu.
The Fast Fourier Transform (FFT) on the waveforms of the source voltage and
(a) 3.75 A (0.000, 4.0010)
(50.000, 3.2523)
(100.000, 567.635 m) 2.50 A
1.25 A
0 A
0 Hz 50 Hz 100 Hz 150 Hz 200 Hz 250 Hz
FOURIER ANALYSIS (FFT)
Frequency (b) I(R1)
Figure 2-14 (a) Fourier analysis setup; (b) Fourier Series Spectrum from Probe using FFT.
56 C H A P T E R 2 Power Computations
the load current will appear in the output file. The fundamental frequency (Center Frequency) of the Fourier series is 50 Hz (1/20 mS). In this example, five periods of the waveform are simulated to ensure steady-state current for this L/Rtime constant.
A portion of the output file showing the Fourier components of source volt- age and resistor current is as follows:
FOURIER COMPONENTS OF TRANSIENT RESPONSE I(R_R1) DC COMPONENT4.000000E00
HARMONIC FREQUENCY FOURIER NORMALIZED PHASE NORMALIZED NO (HZ) COMPONENT COMPONENT (DEG) PHASE (DEG)
1 5.000E01 3.252E00 1.000E00 3.952E01 0.000E00 2 1.000E02 5.675E01 1.745E01 1.263E024.731E01 3 1.500E02 2.589E01 7.963E02 2.402E01 9.454E01 4 2.000E02 2.379E01 7.317E02 9.896E01 5.912E01 5 2.500E02 1.391E07 4.277E08 5.269E00 2.029E02 6 3.000E02 1.065E01 3.275E02 6.594E01 1.712E02 7 3.500E02 4.842E02 1.489E02 1.388E02 1.378E02 8 4.000E02 3.711E02 1.141E02 3.145E01 2.847E02 9 4.500E02 4.747E02 1.460E02 1.040E02 2.517E02 TOTAL HARMONIC DISTORTION 2.092715E01 PERCENT
When you use PSpice output for the Fourier series, remember that the values are listed as amplitudes (zero-to-peak), and conversion to rms by dividing by is required for power computations. The phase angles are referenced to the sine rather than the cosine. The numerically computed Fourier components in PSpice may not be exactly the same as analytically computed values. Total harmonic distortion (THD) is listed at the end of the Fourier output. [Note: The THD com- puted in PSpice uses Eq. (2-69) and assumes that the dc component of the wave- form is zero, which is not true in this case.]
The rms value of the load current can be computed from the Fourier series in the output file from Eq. (2-43).
A graphical representation of the Fourier series can be produced in Probe. To dis- play the Fourier series of a waveform, click the FFT button on the toolbar. Upon entering the variable to be displayed, the spectrum of the Fourier series will appear. It will be desirable to adjust the range of frequencies to obtain a useful graph. Fig. 2-14bshows the result for this example. Fourier component magni- tudes are represented by the peaks of the graph, which can be determined pre- cisely by using the cursor option.
IrmsC(4.0)2a 3.252
12 b2a0.5675 12 b2p
L 4.63 A
12
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2.9 Power Computations Using PSpice 57
PSpice Solution of Example 2-3
Use PSpice to simulate the inductor circuit of Fig. 2-4a with the parameters of Example 2-3.
■ Solution
Fig. 2-15 shows the circuit used in the PSpice simulation. The transistor is used as a switch, so a voltage-controlled switch (Sbreak) can be used in the PSpice circuit. The switch is idealized by setting the on resistance to Ron ⫽ 0.001 ⍀. The control for the switch is a pulse voltage source which has a pulse width of 10 ms and period of 100 ms.
The diode Dbreak is used.
Some of the possible results that can be obtained from the Probe output are listed below. All traces except the maximum inductor current and the stored inductor energy are read at the end of the Probe trace, which is after one complete period. Note the agreement between the results of Example 2-3 and the PSpice results.
Desired Quantity Probe Entry Result
Inductor current I(L1) max ⫽4.5 A
Energy stored in inductor 0.5*0.2*I(L1)*I(L1) max ⫽2.025 J
Average switch power AVG(W(S1)) 0.010 W
Average source power (absorbed) AVG(W(VCC)) ⫺20.3 W
Average diode power AVG(W(D1)) 0.464 W
Average inductor power AVG(W(L1)) ⬇0
Average inductor voltage AVG(V(1,2)) ⬇0
Average resistor power AVG(W(R1)) 19.9 W
Energy absorbed by resistor S(W(R1)) 1.99 J
Energy absorbed by diode S(W(D1)) 0.046 J
Energy absorbed by inductor S(W(L1)) ⬇0
RMS resistor current RMS(I(R1)) 0.998 A
Figure 2-15 Circuit for Example 2-13, a PSpice simulation of the circuit in Example 2-4.
EXAMPLE 2-13
+-
V1 = -10 V2 = 10 TD = 0 TR = 10 n TF = 10 n PW = 10 m PER = 100 m
VCONTROL
4 S1
Ron = 0.001 L1 1
2 2
R1 1
20
30V D1 VCC
Dbreak 200 mH
0 Sbreak +
-
58 C H A P T E R 2 Power Computations