Laboratory Report
COURSE:
EE222 (Electrical Engineering Laboratory I) Section 2.
EXPERIMENT:
N ◦ 5.
TITLE:
Complex Power and Power Factor.
AUTHOR:
Ramprasad Potluri1 .
INSTRUCTOR:
Dr. Keith W. Whites.
Executive Summary
Complex power was computed in R−C, R−L and R−L−C circuits from the measurement
of current, voltage and the phase difference between current and voltage. Complex power
diagrams were constructed using the measured values of time-average power and reactive
power in the circuit. The load power factor for an R − L − C circuit was varied by varying
the value of C. Time-average power dissipated in the load and in the rest of the circuit
was measured at various values of the load power factor. It was observed that the losses
in the R − L transmission line supplying an R − L − C load are minimum when the power
factor of this load is 1.
1
Department of Electrical Engineering, University of Kentucky, Spring 2001. E-mail: potluri@engr.uky.edu
1.
Objective
This experiment had the following goals:
1. To learn to compute the complex power in R-C, R-L and R-L-C circuits from the measurement
of current, voltage and the phase difference between current and voltage.
2. To graphically represent the complex power through complex power diagrams.
3. To observe the time-average power dissipated (i) in the load, and (ii) in the rest of a series
circuit at various values of the load power factor.
2.
Theory
The theoretical tools relevant to this experiment are presented in this section. The notation
has been adapted from [J.R96].
A sinusoid x(t) = Xmax cos(ωt + θ), with ω = 2πf , can also be written as follows:
x(t) = Re{Xmax ejθ ejωt } = Re{Xejωt }.
Here, X is a phasor. It encodes the amplitude and phase of the sinusoid in the form X = Xmax ejθ .
Let V be the phasor voltage across a load, and I be the phasor current through the load. Then,
impedance Z of this load is defined as
V
Z=
[Ω].
I
The impedance of an inductor and of a capacitor are as follows:
ZL = jωL,
−j
1
=
.
ZC =
jωC
ωC
The phasor currents and the phasor voltages in the circuit of Figure 4, are related thus:
Vs = ZIo = (R1 + R2 + jωL)Io
= R1 Io + R2 Io + jωLIo = VR1 + VR2 + VL .
The relationships between phasor currents and phasor voltages for the other circuits can be written
similarly. The phasor diagrams for the circuits in Figure 4, 5, and 6 are shown in Figures 1, 2 and
3. In these diagrams, a circled number adjacent to the label of each phasor denotes the sequence in
which that phasor was added to the corresponding phasor diagram. The phasor numbered 1 serves
as the phase reference. The length of each phasor in the phasor diagram represents the magnitude
of that phasor (which is the same as the amplitude of the corresponding signal).
If the sinusoidal voltage across a load is v(t) and the current i(t) through this load lags behind
v(t) by angle ϕ, then the instantaneous power into the load is
p(t) = v(t)i(t)
[W]
(1)
= Vmax cos(ωt)Imax cos(ωt − ϕ)
Vmax Imax
=
[cos ϕ + cos(2ωt − ϕ)]
2
Vmax Imax
=
[cos ϕ + cos(2ωt) cos ϕ + sin(2ωt) sin ϕ]
2
= PAV [1 + cos(2ωt)] + Q sin(2ωt)
2
VS 10
Im
Im
VR1+ VR2 2
VS 4
VL
3
V1
Re
O
1
I1
5
O
VR1+ VR2
2
VC
Io Re
Im
9
VR1 7
3
IC
I1 6
VL2
VO 4
VS 4
3
1
Figure 1: Phasor diagram
for the circuit of Figure 4.
VL1
Figure 2: Phasor diagram
O
8
for the circuit of Figure 5.
VR2
2
1
IR2 Re
Figure 3: Phasor diagram for the circuit
of Figure 6.
Here, the time-average power PAV and the reactive power Q to the load have the following expressions:
Vmax Imax
PAV =
cos ϕ = Vrms Irms cos ϕ [W]
(2)
2
Vmax Imax
Q =
sin ϕ = Vrms Irms sin ϕ [VAR]
(3)
2
The power factor of the load is cos ϕ. The complex power S delivered to the load is as follows:
V I∗
= PAV + jQ.
2
The apparent power delivered to the load is
(4)
S≡
|S| = Vrms Irms
(5)
[W].
A complex power diagram for S is an Argand diagram (from high school mathematics) for S.
The characteristic frequency for the circuit of Figure 4 is ωc =
R1 +R2
L
τ is the time constant of this circuit. For the circuit of Figure 5, ωc =
ωc
characteristic frequency can also be expressed in Hz as fc = 2π
.
=
1
(R1 +R2 )C
The following math is for the circuit of Figure 4:
Vo (ω) =
R2
R2 + jωL
Vs and VR2 (ω) =
Vs
R1 + R2 + jωL
R1 + R2 + jωL
|Vo (ω)|
|VR2 (ω)|
=
|Vo (ωc )|
|VR2 (ωc )|
=
=
≈
s
s
R22 + ω 2 L2
=
R22
s
1+
q
’
R1 + R2
L
’
R1
1+ 1+
R2
“2
“2
1 + (1 + 2)2 =
3
s
1+
ω 2 L2
R22
L2
R22
√
10 = 3.16.
1
τ
[ rad
s ]. Here,
[ rad
s ]. The
V1
R1
A
R1
VS
VS
VO
R2
R1
C
L
R2
VR2
L1
L2
VS
VO
B
C
VR2
VO
R2
IO
C
D
IO
I1
Figure 5: Circuit for Part
2. vs = 5 cos(2πf t) V,
R1 = 1 kΩ, R2 = 510 Ω
and C = 0.1 µF.
Figure 4: Circuit for Part
1. vs = 5 cos(2πf t) V, R1 =
1 kΩ, R2 = 510 Ω and L =
39 mH.
Figure 6: Circuit for Part 3.
vs =
5 cos(2πf t) V, f = 2 kHz, R1 = 1 kΩ,
R2 = 510 Ω, L1 = 39 mH, and L2 =
70 mH (Used IS for L2 and CS for C).
The fact that R1 ≈ 2R2 has been used in the last line above. Thus, for the circuit of Figure 4:
|Vo (ωc )| ≈ 3.16 × |VR2 (ωc )|
(6)
The following math is for the circuit of Figure 5:
Vo (ω) =
j
ωC
R2 −
R1 + R2 −
|Vo (ω)|
|VR2 (ω)|
=
|Vo (ωc )|
|VR2 (ωc )|
=
=
≈
j
ωC
Vs and VR2 (ω) =
v
u
u R2 + 21 2
t 2
ω C
s
s
q
=
R22
1+
s
1+
R2
R1 + R2 −
j
ωC
Vs .
1
ω 2 C 2 R22
((R1 + R2 )C)2
C 2 R22
’
1+ 1+
R1
R2
“2
1 + (1 + 2)2 =
This is amazing!
√
10 = 3.16.
The fact that R1 ≈ 2R2 has been used in the last line above. Thus, for the circuit of Figure 5:
|Vo (ωc )| ≈ 3.16 × |VR2 (ωc )|
(7)
We can write the following for the branch between nodes A and B of the circuit of Figure 6:
2
2
2
I1(rms)
(R1 + jωL) = I1(rms)
R1 + jI1(rms)
ωL
= PAV 1 + jQ1
3.
(8)
Results
The following instruments were used in this experiment: Tektronix Digital Phosphor Oscilloscope (DPO), Simpson Volt-Ohm-Milliamperemeter (VOM), Tektronix Digital Multimeter
(DMM), Function Generator (FG), Capacitance Substituter (CS), Inductance Substituter (IS)
and Resistance Substituter (RS) from IET Labs, Inc., and various resistors. In this report, the
superscript m denotes a value measured in the experiment (or calculated from such measurements)
and the superscript p denotes a value predicted in the pre-laboratory calculations.
4
3.1.
Part 1 of the Experiment
In Part 1a, the circuit of Figure 4 was assembled. The frequency of the source voltage was
set at f = 1.2 kHz using the DMM. The voltage waveforms corresponding to VRm2 and Vom were
m
m
observed on the screen of the DPO. The current Io(max)
was computed as Io(max)
= VRm2 (max) /R2
p
m
and was equal to 3.04 mA. The difference between Vo(max)
(which was equal to 1.92 V) and Vo(max)
p
m
was 0.52%. The difference between Io(max)
(3.04 mA) and Io(max)
was ≈ 10% in absolute value.
In Part 1b, the waveforms observed in Part 1a were sketched in the lab notebook. Iop lagged
behind Vop by 30◦ . The measured value of this phase difference was 31.15◦ . The time intervals
during which the energy was being returned from the load were identified as the intervals over
which the instantaneous power (see Equation (1)) was negative.
In Part 1c, f was varied from 200 Hz to 20 kHz in steps of integer multiples of 1 kHz, and
Vo(max) and VR2 (max) corresponding to each value of f were recorded. Graphs of VR2 (max) and
Vo(max) vs. f were drawn on a semilog paper, with the amplitudes along the linear axis and the
frequencies along the logarithmic axis (Figure 7).
m and the reactive power Qm delivered to the load were
In Part 1d, the time-average power PAV
calculated from the values recorded in parts 1a and 1b according to Equations (2) and (3). The
m (2.5 mW) and P p was 7%, and that between Qm (1.51 mVAR) and Qp
difference between PAV
AV
m and Qm .
was 2.4%. A complex power diagram was drawn using the values of PAV
In Part 1e, the frequency at which Vo(max) ≈ 3.16 × VR2 (max) was identified on the graph drawn
in Part 1c (see Equation 6). The difference between fcm (5.88 kHz) and fcp was found to be ≈ 5%.
3.2.
Part 2 of the Experiment
The sequence of steps performed in Part 1 was repeated here for the circuit of Figure 5.
In Part 2a, the circuit of Figure 5 was assembled. The frequency of the source voltage was
set at f = 1.2 kHz using the DMM. The voltage waveforms corresponding to VRm2 and Vom were
m
m
was computed as Io(max)
observed on the screen of the DPO. The current Io(max)
= VRm2 (max) /R2 .
p
m
m
The difference between Vo(max) (3.482 V) and Vo(max) was 1.5%. The difference between Io(max)
p
(2.44 mA) and Io(max)
was 2%.
In Part 2b, the waveforms observed in Part 2a were sketched in the lab notebook. Iop lead
Vop by 69◦ . The measured value of this phase difference was 72.7◦ . The time intervals during
which the energy was being returned from the load were identified as the intervals over which the
instantaneous power (see Equation (1)) was negative.
In Part 2c, f was varied from 200 Hz to 20 kHz in steps of integer multiples of 1 kHz, and
Vo(max) and VR2 (max) corresponding to each value of f were recorded. Graphs of VR2 (max) and
Vo(max) vs. f were drawn on a semilog paper, with the amplitudes along the linear axis and the
frequencies along the logarithmic axis (Figure 8).
m and the reactive power Qm delivered to the load were
In Part 2d, the time-average power PAV
calculated from the values recorded in parts 2a and 2b according to Equations (2) and (3). The
m (1.263 mW) and P p was 22%, and that between Qm (−4.055 mVAR) and
difference between PAV
AV
m and Qm .
p
Q was 1.3%. A complex power diagram was drawn using the values of PAV
5
Voltages vs. frequency for R−C circuit
0
1
1
Vo_max and Vr2_max [V]
Vo_max and Vr2_max [V]
Voltages vs. frequency for R−L circuit
0
2
3
4
5
2
3
4
0
5
1
10
10
0
1
10
f [kz]
10
f [kHz]
Figure 7: Semilog plot of Vo(max) vs. f (solid
curve) and VR2 (max) vs. f (dotted curve) for Part
1c.
Figure 8: Semilog plot of Vo(max) vs. f (solid
curve) and VR2 (max) vs. f (dotted curve) for Part
2c.
In Part 2e, the frequency at which Vo(max) ≈ 3.16 × VR2 (max) was identified on the graph drawn
in Part 2c (see Equation 7). The difference between fcm (1.1 kHz) and fcp was found to be 4.3%.
3.3.
Part 3 of the Experiment
The circuit of Figure 6 was used for this part.
In Part 3a, C was varied from 0 to 0.3 µF in steps of 20 nF. Vs was observed on Channel 1 of
the DPO and Vo on Channel 2. The phase shift θV between these two signals along with the value
of current I1(rms) were recorded for each value of C. The DMM was used to measure the current.
In Part 3b, for each value of C in Part 3a, the phase shift θI between Vs and I1 was recorded.
This was done by introducing a RS set to 50 Ω resistance in series between the nodes C and D in
the circuit of Figure 6 and observing the voltage across this resistor on Channel 2 of the DPO.
In Part 3c, for each value of C in Part 3a and 3b, the power factor (P F ) for the load L2 −R2 −C
was computed using the relation P F = cos(θV − θI ). A graph of P F vs. C was plotted (Figure 9).
P F ≈ 1 where C = 60 nF.
In Part 3d, the time-average power PAV 1 delivered to the R1 − L1 combination was calculated
for each value of C in Part 3a using Equation 8, the values of I1(rms) recorded in Part 3a, and the
value of R1 measured with the DMM. A graph of PAV 1 vs. C was also plotted (Figure 10).
4.
Conclusions
1. In Part 2d, the difference between the measured and predicted values of PAV was 22%. This
needs to be investigated.
2. In Part 3b, a more accurate method to conduct the measurement would have been to exchange
the places of branches A-B and C-D and observe the voltage across R1 using the DPO.
3. In Part 3d, and in figures 9 and 10 it can be seen that the time-average power expended in
the R1 − L1 combination is minimum where the power factor of the R2 − L2 − C load is the
closest to 1. If the R1 − L1 combination of Figure 2 models the trasmission line supplying a
6
Pav in R1−L1 vs. C
−3
11
PF of load R2−L2−C vs. C
x 10
1
10
0.9
9
0.8
8
0.7
Pav
7
PF
0.6
6
0.5
5
0.4
4
3
0.3
2
0.2
0.1
0
1
0
0.5
1
1.5
2
C
2.5
3
0.5
1
1.5
2
C
3.5
−7
2.5
3
3.5
−7
x 10
x 10
Figure 9: Power factor of the R2 − L2 − C load
vs. C for circuit of Figure 6. Dotted curve –
measured plot. Solid curve - predicted plot.
Figure 10: Time-average power expended in the
R1 − L1 combination vs. C for the circuit of
Figure 6. Dotted curve – measured plot. Solid
curve – predicted plot.
R2 − L2 − C load, then transmission losses can be reduced by bringing the PF of the load as
close to 1 as possible.
4. The predicted and measured plots for PAV shown in Figure 10 diverge as the capicitance is
increased. This needs to be investigated.
5. Whereas in earlier experiments, only those features of the DPO that can be found in any
analog oscilloscope were used by me, in conducting this experiment, many of the neat features
of the DPO were well exploited. Especially, using the vertical cursors along with displaying
the average shapes of waveforms helped to determine more accurately than before the phase
difference between two signals.
References
[J.R96] J.R.Cogdell. Foundations of Electrical Engineering. Prentice Hall, Upper Saddle River,
NJ, 2nd edition, 1996.
7