ELECTRICAL ENGINEERING DEPARTMENT
California Polytechnic State University
EE 251
AC Steady-State Power – The Power Triangle & Power Factor Correction Pre-Lab 8
IMPORTANT – SHOW ALL WORK!
1. Given the series RL load below, source impedance Zs = 0Ω, determine real, reactive, and
apparent power. Find the load power factor and construct the power triangle including
angle θ.
2. Given the circuit of problem #1 with Zs = 50Ω, determine real, reactive, and complex
power. Find the load power factor and construct the power triangle including angle θ.
What is the source power factor?
3. Determine the capacitance in parallel with inductor resistor combination required to
correct the power factor for the circuit of question #2.
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ELECTRICAL ENGINEERING DEPARTMENT
California Polytechnic State University
AC Steady-State Power – The Power Triangle & Power Factor Correction
EE 251
Lab 8
Objective
To construct a power triangle diagram of an inductive load using AC steady-state analysis. In
addition, correct the power factor of an inductive load with a parallel capacitance.
Workbench Equipment
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Function Generator, Agilent 33120A or Agilent 33220A or Rigol DG1062Z
Digital Multimeter, Agilent 34401A
Resistor Box IV, 100/250/500
Inductor Decade Box, 10mH step: 0 – 100mH
Capacitor Decade Box, 1F step: 0 – 10F
Capacitor Decade Box, 0.01F step: 0 – 1.1F
RLC Bridge, Gen Rad 1659
Check-out Equipment, 20-111 window
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Banana to banana, 3 pairs, red/black
BNC to banana
Short leads, quantity 6, 1 bag
Student Supplied Materials
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Protractor
Straight Edge
Graph paper / Engineering paper
Background
Impedance Triangle
The components of impedance (Z), resistance (R) and reactance (X), are often represented as
sides of an impedance right triangle.
|Z|
X
θz
R
Fig. 8-1 Impedance Triangle
where
Z  R2  X 2
(8-1)
X
θz  arctan 
R
(8-2)
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Power Triangle
Note: The equations below use RMS currents and voltages since AC ammeters and voltmeters
display RMS values. These equations are also valid for peak currents and voltages. For
sinusoidal currents and voltages centered about zero, the RMS value equals the peak value
divided by sqrt (2).
For an ideal source (Zs = 0Ω) connected to an impedance (Z) as in Fig. 8-3,
For a non-ideal source,
Irms =
Vrms
Z
(8-3)
Irms =
Vrms
Zs  Z
(8-4)
Average power, also called “real” power (units = W = Watts), is the power absorbed by the
resistance of impedance and equals:
Pav = Irms2R
(8-5)
where Irms is the current through R.
Using Ohm’s law for AC circuits:
Pav =
Vrms 2
R
(8-6)
Where Vrms is the voltage across R.
Reactive power (units = VAR = Volt-Ampere-Reactive) is the power exchanged periodically
between the source and all storage elements and equals:
Q = Irms2X
(8-7)
Vrms 2
X
(8-8)
where Irms is the current through X.
Using Ohm’s law for AC circuits:
Q=
where Vrms is the voltage across X.
Complex power (S) is a fictitious power (units = VA = Volt-Ampere) defined as the product of
voltage and the complex conjugate of current:
S = Vrms Irms*
(8-9)
where * denotes complex conjugate.
Complex (post-lab problem) is:
S = |Irms|2R + j |Irms|2X =
|Irms|2Z (8-10)
where Irms is the current through Z.
Using Ohm’s law for AC circuits:
S=
where Vrms is the voltage across Z.
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Vrms 2
Z
(8-11)
Since S = |Irms|^2 R + j |Irms|^2 X can be written as S = Pav + jQ, complex power (S) and
components real power (Pav) and reactive power (Q) are the sides of a power triangle, Fig 8-2.
|S|
Q
θ
Pav
Fig. 8-2 Power Triangle
Power and impedance triangle angles θ and θz are identical since the triangles are scaled versions
of each other. For example, each side of the impedance triangle multiplied by |Irms|2 (S= |Irms|2 Z,
Pav = |Irms|2 R, Q = |Irms|2 X) produces the power triangle. In addition:
Pav = s cos θ
(8-12)
Q = s sin θ
(8-13)
where complex power magnitude (S) is apparent power and cos θ is defined as the power factor
(pF).
pF = cos θ
(8-14)
Power Factor Correction
Power factor correction yields a load impedance power factor of 1. In other words, making θ
(and therefore θz since θ = θz) tend to zero, since pF = cos (0) = 1. For an inductive load, this is
accomplished by connecting a capacitor across the inductive load, see Fig. 8-4. Correcting the
power factor is desirable since line losses (|I|2RLine losses) between source and load are
minimized since the current flowing through the lines is minimized.
Power factor correction capacitance is:
QL = QC =
Vcrms 2
= ωC Vcrms 2
Xc
therefore,
C
QC
VCrms
2
where the voltage across the correction capacitor and inductive load are identical (parallel
connection).
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(8-15)
Procedure 1: Construct a Power Triangle Diagram for an Inductive Load
Inductive
load inside
dashed box
Fig. 8-3 Power Triangle Circuit
NOTE: Fig. 8-3: Exclude 100 ohm resistor if 'new' inductors are used (includes 'relatively
large' series resistance).
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Measure load inductance with impedance bridge (set to 1 kHz) and record.
Measure load resistance with ohmmeter and record. Also, measure and record the DC
resistance of the inductor.
o
Note: the function generator's internal impedance Zs is 50Ω.
Construct circuit of Fig. 8-3 and set function generator to Hi Z mode.
Adjust function generator unloaded output (display value) amplitude to 7Vrms at a
frequency of 600Hz.
o Note: The loaded FG output is equal to the voltage across the inductive load.
Insert an AC ammeter between the function generator and load. Record the current.
Calculate real power (Pav) and reactive power (Q) using measured values.
Construct a power triangle diagram using Pav and Q values from previous step. Refer to
the background section of this experiment as needed.
Determine θ and load power factor.
Procedure 2: Power Factor Correction
Fig. 8-4 Power Factor Correction Circuit
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Measure and record the voltage across the inductive load (20mH + 100 ohms).
Determine the parallel capacitance required to correct the power factor (i.e. yield θ = 0°).
Insert capacitance decade boxes (see note below) set for calculated correction capacitance
in parallel with inductive load as shown in Fig. 8-4.
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o
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Note: connect capacitor decade boxes (1µF step box with 0.1µF / 0.01µF steps
box) in parallel to obtain fine resolution.
Monitor current between source and load with AC ammeter as capacitance is varied
around calculated value.
Record the capacitor value, measured on impedance bridge) that yields a minimum
ammeter reading (i.e. minimum line loss). Compare (% error) experimental capacitance
with your calculated capacitance.
Discussion
1. Prove the definition of S = Vrms Irms* leads to S consisting of two components Irms2R and
Irms2X, where Irms2R is purely real and Irms2X is purely imaginary.
Hint: Let Vrms = Vrmscosθv + j Vrmssinθv and Irms = IrmscosθI + j IrmssinθI
Use trigonometry IDs: cos(-ß) = coscosß + sinsinß and sin(-ß) = sincosß - sinßcos
2. Correcting the power factor maximizes / minimizes / does not change line loss between the
source and load. Justify your answer from experimental results.
3. For the circuit of procedure 2, determine the value of a series power factor correction
capacitor. Use experimental values in calculations.
4. If the power factor correction capacitor of question #3 was connected in series with the load,
line losses between the source and load would decrease / increase / remain the same.
Justify your answer with analysis.
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