EE 242 EXPERIMENT 3: AC STEADY STATE POWER
1
PURPOSE:
•
•
•
•
•
•
LAB EQUIPMENT:
1 Agilent 54621A Oscilloscope
2 Decade Resistance Boxes
2 Decade Capacitor (1
µ
F range with steps of 0.1 and 0.01
µ
F)
1 5 mH High-Q inductor
1 Agilent 34410A Digital Multimeter
1 Agilent 33120A Function Generator (FG)
STUDENT PROVIDED EQUIPMENT:
3 BNC-double banana
1 5 mH high-Q inductor (obtain from checkout counter)
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) = V m
cos (
ω
t +
θ v
) and i(t) = I m
cos (
ω
t +
θ i
) represent the voltage and current, then it can be shown: p(t)
= P + P cos 2
ω
t – Q sin 2
ω
t where P = ½ V m
I m
cos (
θ v
-
θ i
) and Q = ½ V m
I m
sin (
θ v
-
θ i
)
The term P is a constant and represents the average of the instantaneous power p(t) (since the averages of cos
2
ω
t and Q sin 2
ω
t are both zero)
. The term P + P cos 2
ω
t represents the instantaneous real power
(instantaneous active power) . Q is called the reactive power and the term Q sin 2
ω
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.
1
New experiment developed by John Saghri, last revised 03/02/06

Complex power S is defined as the complex sum of the average and the reactive power
S = P + j Q
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 (
θ v
-
θ i
) and is referred to as the power factor angle.
The power factor pf is defined: pf = cos (
θ v
-
θ i
) = P /
⎢
S
⎢
For ideal reactive elements
(no internal resistance) P is equal to zero and hence apparent power equals the reactive power. For resistive
S
Complex power
θ
Q
Reactive power elements, Q is zero and the pf = 1
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 Z
L matches the complex conjugate of the source Thevenin impedance Z
Th
, i.e.,
Z
L
= Z
Th
*
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 R
S will be equal to R
L
, the maximum power transferred to the load in this condition will be:
P
Max
= (RMS value of V
S
)
2
/ ( 4 R
S
)
It can be shown that, if the load consists of only a resistive component R
L
, then the maximum power will be delivered to the load when:
R
L
⎢
Z
Th
⎢

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 .
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 e) If the load is purely resistive, what will be the resistance of the load for maximum power transfer?
V
S
Frequency
= 2.5 kHz sinusoid
5 V rms
DC offset =
0 V
+
-
Function
Generator
Load
Section 1 : Maximum Power Transfer and Function Generator Internal impedance
Set up the circuit shown in Figure 1. R
S represents the finite output resistance of the function generator ( about 50 Ω ) . Use the 5
Scope Ch. 1
5 mH
High-Q induc tor
L mH high-Quality inductor for L shown in the circuit. Assume that the inductor is ideal.
Use a resistor and capacitor decade boxes for
R
L and C
L
, 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. a) Set R
L
equal to 50
Ω
. b)
Frequency
= 2.5 kHz sinusoid
5 V rms
DC offset =
0 V
Scope Ch. 1 & 2 common ground
For C
L
, use a decade capacitor box with a maximum range of 1
µ
F and steps of 0.1
µ
F and 0.01
µ
F. c)
Scope Ch. 2
Display the source voltage (function generator output) and the voltage across R
L
, simultaneously on the oscilloscope in the x-deflection mode (use Autoscale , then Main/Delayed->XY softkey)

d) Vary the value of C
L
in steps of 0.1
µ
F or 0.01
µ
F and observe the resulting lissajous patterns on the scope e) From the displayed lissajous patterns, determine the value of C
L voltage across R
L
for which the source voltage and the are in phase. Note that when the two voltages are in phase the resulting lissajous pattern narrows to a single slanted line. f) With C
L
set at the value in step e above, vary R
L
in the range of 20
Ω
to 150
Ω
in steps of 10
Ω
. At each
R
L
setting, determine the voltage drop across R
L
and the average power dissipated by it. g) Plot the average power dissipated by R
L
versus the value of R
L
. Note that you may have to readjust (i.e., fine tune) the value R
L
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) R
S
. Record this value in your report. i) Using the information obtained in steps e
and h
above, determine the load impedance Z
L
for maximum power transfer j) Compare the theoretical value of Z
L
for maximum power transfer experimentally obtained value in step i above
(refer to your prelab work)
with its k) l)
Compare the theoretical and experimental values of the maximum power delivered to the load
Explain the differences between the measured and calculated values
Section 2 : Maximum Power Transfer for Purely Resistive Load
Set up the circuit shown of Figure 2 which is the same as Figure 1 but with the capacitor
C
L
removed. The load R
L
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
Scope Ch. 1
5 mH
High-Q induc tor
Scope Ch. 2
Resistive
Load
DC offset.
Frequency
= 2.5 kHz sinusoid
5 V rms
DC offset =
Scope Ch. 1 & 2 common ground
0 V a) Set R
L
equal to 50
Ω
. b) Vary R
L
in the range of 20
Ω
to 150
Ω
in steps of 10
Ω
. At each R
L
setting, determine the voltage drop across R
L
and the average power dissipated by it.

c) Plot the average power dissipated by R
L
versus the value of R
L
. Note that you may have to readjust (i.e., fine tune) the value R
L
in order to capture a peak power reading. d) From your plot above, record the value of R
L
for maximum power transfer. e) Compare the value of R
L
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 R
L
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) 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
R
1
in series with the capacitor. C
L
is now labeled
C
1
in the new circuit. Use the
5 mH high-Q inductor for L .
Set C
1
200
Ω
.
to 0.6
µ
F and R
L
to
Scope Ch. 1
Scope Ch. 2
Scope Ch. 1 & 2 common ground
L b) Consider the combination of L + R
L
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 R
L
. 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) .
You can assume the phase of load current is 0 o
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 R
L
(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.

d) a)
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
Scope Ch. 2
Scope Ch. 1 & 2 common ground
Modify your load by placing a variable capacitor across the original load (in parallel with the combination of L + R
L and adjust it to achieve a B A power factor as close to unity as possible. Note that the best power factor corresponds to the minimum phase difference
L 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 R
1
, you can adjust the added load capacitor until the phase difference between the voltage across R
1
and the voltage across the load is a minimum.
C
Scope Ch. 1
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 R
1
, 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 R
1 polarity. Hence the phase as measured by the scope will be 180 o is measured with inverse
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 through and voltage across the load) you should try to reach a phase difference as close to 180 o
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 R
1
. Compare the value of the capacitance in step ( a)
above with the theoretical value
(computed from the circuit specifications)