EE 242 EXPERIMENT 3: AC STEADY STATE POWER 1
PURPOSE:
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To become familiar with AC steady state power measurements
To determine the load characteristic for maximum power transfer.
To measure the internal impedance of an output device (function generator)
To measure and compare average, apparent, and reactive powers in an AC circuit
To measure and learn about power factor
To be able to make power factor correction
LAB EQUIPMENT:
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2
2
1
1
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Agilent 54621A Oscilloscope
Decade Resistance Boxes
Decade Capacitor (1 µF range with steps of 0.1 and 0.01 µF)
5 mH High-Q inductor
Agilent 34410A Digital Multimeter
Agilent 33120A Function Generator (FG)
STUDENT PROVIDED EQUIPMENT:
2 Bags of short banana to banana leads
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1
2
BNC-double banana
5 mH high-Q inductor (obtain from checkout counter)
Banana-Banana leads
Background
Instantaneous power p(t) is defined as the product of instantaneous voltage v(t) and instantaneous current i(t).
Assuming the sinusoids waveforms v(t) = Vm cos (ω t + θv ) and i(t) = Im cos (ω t + θi ) represent the
voltage and current, then it can be shown:
p(t) = P + P cos 2ω t – Q sin 2ω t
where P = ½ Vm Im cos (θv- θi ) and Q = ½ Vm Im sin (θv - θi )
The term P is a constant and represents the average of the instantaneous power p(t) (since the averages of cos
2ω t and Q sin 2ω t are both zero). The term P + P cos 2ω t represents the instantaneous real power
(instantaneous active power). Q is called the reactive power and the term Q sin 2ω t represents the
instantaneous imaginary power (instantaneous reactive power).
The average real power P indicates the transformation of energy from electric to non-electric form (such as heat
in a resistor). The term Q is called the reactive power. It causes the interchange of energy between the source
and the load without any net transfer of energy.
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New experiment developed by John Saghri, last revised 03/02/06
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Complex power S is defined as the complex sum of the average and the reactive power
S=P+jQ
To distinguish the three powers S, P and Q, different units are used for each. Real power P is in watts (W),
reactive power is in volt-amp reactive (VAR) , and complex power is volt-amps (VA).
The magnitude of the complex power ⎢S ⎢ is called the apparent power.
Power Triangle
The power triangle on the right shows the relationship between the P,
Q, and S. The angle θ is equal to (θv - θi ) and is referred to as the
S
power factor angle. The power factor pf is defined:
Complex power
Q
pf = cos (θv - θi ) = P / ⎢S ⎢
θ
For ideal reactive elements (no internal resistance) P is equal to zero
and hence apparent power equals the reactive power. For resistive
elements, Q is zero and the pf = 1
Reactive power
P
Real power
The apparent power represents the volt-amp capacity required to supply the average power to a device.
Therefore, it is recommended to operate the devices at a power factor close to unity. Many appliances (such as
refrigerator, washing machines) operate at a lagging power factor. We can make correction to power factor to
bring it close to unity. This can be done via adding a capacitor to (or connecting across) the load.
Maximum Power Transfer in AC Steady State
For an AC source, the maximum power will be transferred (delivered to) from the source to the load when the
impedance of the load ZL matches the complex conjugate of the source Thevenin impedance ZTh, i.e.,
*
ZL = ZTh
Therefore if the source impedance has a resistive and reactive parts, the maximum power is transferred to the
load (actually to the resistive part of the load) when the load resistance equals the source resistance and the load
reactance is the opposite of the source reactance. So for maximum power transfer condition, if the source is
inductive the load impedance must be capacitive and vice versa. This means that the net reactive component of
the circuit will be zero, i.e. the circuit becomes purely resistive. Therefore, the current through the load (or the
voltage across load resistance) becomes in phase with the voltage source. Since the RS will be equal to RL, the
maximum power transferred to the load in this condition will be:
PMax = (RMS value of VS )
2
/ ( 4 RS )
It can be shown that, if the load consists of only a resistive component RL, then the maximum power will be
delivered to the load when:
RL = ⎢ZTh ⎢
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Prelab:
5 mH
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a) For the circuit shown, design a load circuit
(with the minimum number of R , L , C )
that will draw the maximum average power
P. Assume that the output impedance of
the function generator is 50 Ω.
VS
Function
Generator
+
Load
b) Calculate the maximum average power
transferred to the load.
c) Determine the apparent power ⎢S ⎢, reactive
power Q, and the power factor pf associated
with the load
Frequency
= 2.5 kHz
sinusoid
5 Vrms
DC offset =
0V
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e) If the load is purely resistive, what will be the
resistance of the load for maximum power
transfer?
Procedure:
Section 1: Maximum Power Transfer and Function Generator Internal impedance
Set up the circuit shown in Figure 1. RS
represents the finite output resistance of the
function generator (about 50 Ω). Use the 5
mH high-Quality inductor for L shown in the
circuit. Assume that the inductor is ideal.
Use a resistor and capacitor decade boxes for
RLand CL, respectively. Place the FG output
termination to “HIGH Z” and set it to
produce a 5V rms sine wave at a frequency of
2.5 kHz with no DC offset.
Scope Ch. 1
5 mH
High-Q inductor
L
Frequency
= 2.5 kHz
sinusoid
5 Vrms
DC offset =
0V
Scope Ch. 1 & 2
common ground
Scope Ch. 2
Figure 1
a)
Set RL equal to 50 Ω.
b)
For CL, use a decade capacitor box with a maximum range of 1 µF and steps of 0.1 µF and 0.01 µF.
c)
Display the source voltage (function generator output) and the voltage across RL, simultaneously on the
oscilloscope in the x-deflection mode (use Autoscale, then Main/Delayed->XY softkey)
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d)
Vary the value of CL in steps of 0.1 µF or 0.01 µF and observe the resulting lissajous patterns on the
scope
e)
From the displayed lissajous patterns, determine the value of CL for which the source voltage and the
voltage across RLare in phase. Note that when the two voltages are in phase the resulting lissajous
pattern narrows to a single slanted line.
f)
With CL set at the value in step e above, vary RL in the range of 20Ω to 150 Ω in steps of 10Ω. At each
RL setting, determine the voltage drop across RL and the average power dissipated by it.
g)
Plot the average power dissipated by RL versus the value of RL. Note that you may have to readjust (i.e.,
fine tune) the value RL in order to capture a peak power reading.
h)
From the information obtained in step g above, you should be able to determine the actual function
generator’s Thevenin’s resistance (output resistance) RS. Record this value in your report.
i)
Using the information obtained in steps e and h above, determine the load impedance ZL for maximum
power transfer
j)
Compare the theoretical value of ZL for maximum power transfer (refer to your prelab work) with its
experimentally obtained value in step i above
k)
Compare the theoretical and experimental values of the maximum power delivered to the load
l)
Explain the differences between the measured and calculated values
Section 2: Maximum Power Transfer for Purely Resistive Load
Scope Ch. 1
Set up the circuit shown of Figure 2 which is
the same as Figure 1 but with the capacitor
CL removed. The load RL is now purely
resistive. Place the FG output termination to
“HIGH Z” and set it to produce a 5V rms
sine wave at a frequency of 2.5 kHz with no
DC offset.
Frequency
= 2.5 kHz
sinusoid
5 Vrms
DC offset =
0V
Scope Ch. 2
5 mH
High-Q inductor
Resistive
Load
Scope Ch. 1 & 2
common ground
a)
Set RL equal to 50 Ω .
b)
Vary RL in the range of 20Ω to 150 Ω in steps of 10Ω. At each RL setting, determine the voltage drop
across RL and the average power dissipated by it.
Figure 2
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c)
Plot the average power dissipated by RL versus the value of RL. Note that you may have to readjust (i.e.,
fine tune) the value RL in order to capture a peak power reading.
d)
From your plot above, record the value of RL for maximum power transfer.
e)
Compare the value of RL for maximum power transfer to its expected value (refer to your prelab work).
Use the nominal value of the inductor and the actual function generator’s internal resistance (obtained in
step h of section 1) to calculate the theoretical value of RL for maximum power transfer.
f)
Compare the theoretical and experimental values of the maximum power delivered to the load
g)
Explain the differences between the measured and calculated values
Section 3: (OPTIONAL) Power
Measurements
a)
b)
Scope Ch. 1
Set up the circuit of Figure 3
which is the same as the
circuit of Figure 1 but with
the positions of the capacitor
and the inductor switched, and
insertion of a 200 Ω resistor
R1 in series with the
capacitor. CL is now labeled
C1 in the new circuit. Use the
5 mH high-Q inductor for L.
Set C1 to 0.6 µF and RL to
200 Ω.
L
Scope Ch. 2
Scope Ch. 1 & 2
common ground
Figure 3
Consider the combination of L + RL as your load. Measure and record the current through and voltage
across the load (in rms phasor form).
Note that the phase difference between the load voltage and current is the same as the phase difference
between the load voltage and voltage across RL. Use Quick Measure feature of the scope to measure the
phase difference (you should exit the “xy” mode and use scope’s softkeys to select “source 1” to be
able to read the phase difference).
o
You can assume the phase of load current is 0 and the phase of the voltage is equal to the phase
difference measured between the two channels on the scope (phase 1-2). The magnitude of the load
voltage can be read directly from the scope (channel 1) in rms. The magnitude of the current is equal to
the rms voltage across RL (which you read directly from the scope’s channel 2) divided by its resistance
of 200 Ω.
c)
From your measurements in step b above, compute the average power P, the reactive power Q, and the
apparent power ⎢S ⎢, and the power factor at the load.
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d)
Compare these values with the theoretical values computed from your circuit using the specified
component values. Explain the reasons for the differences
Section 4: (OPTIONAL) Power Factor Correction
a)
Modify your load by placing a
variable capacitor across the
original load (in parallel with
the combination of L + RL
and adjust it to achieve a
power factor as close to unity
as possible. Note that the best
power factor corresponds to
the minimum phase difference
you can achieve between the
voltage across and current
through the modified load.
Since the current through the
load is the same as the current
through R1, you can adjust the
added load capacitor until the
phase difference between the
voltage across R1 and the
voltage across the load is a
minimum.
Scope Ch. 2
Scope Ch. 1 & 2
common ground
B
A
L
C
Scope Ch. 1
Figure 4
Again, you can use the corresponding lissajous pattern on the scope to detect the minimum phase difference (as
you vary the capacitance value). As shown, connect channels 1 and 2 of the scope to voltage across the load and
voltage across R1, respectively. To ensure a common ground, point A on the diagram is used as the ground for
both channels 1 and 2 of the scope. This means that the polarity of voltage across R1 is measured with inverse
o
polarity. Hence the phase as measured by the scope will be 180 shifted. So, for power factor correction, instead
of trying to reach a phase difference of zero between the two channels (which represent the phases of the current
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through and voltage across the load) you should try to reach a phase difference as close to 180 as possible. If
you use the xy mode of the scope, then the shape of the lissajous pattern will still be a straight line regardless of
the polarity of voltage across R1. Compare the value of the capacitance in step (a) above with the theoretical
value (computed from the circuit specifications)
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