Laboratory 12
POWER IN SINUSOIDAL STEADY-STATE CIRCUITS
In this experiment we shall investigate the phasor relationships between voltage and
current in reactive circuits operating in sinusoidal steady-state, and power, power factor,
and power factor correction.
Prelab
Figure 12.1
1. Referring to the RL circuit of Figure 12.1, assume that the circuit is in sinusoidal
steady-state with, Vs at 5V amplitude, zero phase, frequency f = 50KHz.
Calculate the phasors Vs, I , V2, VL , and VR . Show that the voltage across the
load, ( VL + VR ) is approximately equal to Vs.
2. We shall therefore consider VL + VR = Vs for the rest of this experiment. Draw a
phasor diagram showing Vs, I , VL , and VR .
3. Calculate the average powers PR , PL , P and the complex powers S R , S L , and S .
Draw an accurate diagram of the three complex power vectors, showing that
S = S R + S L . Calculate the power factor of the load; be sure to specify leading or
lagging.
4. Suppose that the owner of the voltage source charges us cold hard cash for the
average power that our RL load absorbs. (A wattmeter measures average power.)
From the point of view of the supplier of power, a small pf (i.e., large angle
between current and voltage) is bad because it means that the average power is
small relative to the reactive power, and so the supplier does not get full payment
for the power that he delivers. For this reason, power suppliers (namely electric
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utility companies) charge higher rates to users who have larger phase angles (such
as large industrial plants that have big inductive loads). Such consumers will
often adjust their pf (by changing their load impedance) in order to obtain lower
rates. Let us try to do this in our circuit. Obviously we need to add a capacitive
reactance if we want to increase the pf, but where to add it is the question. We
shall add a capacitor in parallel with the RL load impedance (see Fig. 12-2) so
that the voltage across the RL load will be the same.
Figure 12-2
Let θ old be the phase angle in the original circuit of Fig. 12-1 and let θ new be the
target phase angle (of the circuit of Fig. 12-2), where 0 ≤ θ new < θ old (or
1 ≥ pf new > pf old ). We want to determine the value of C that will result in a given
value of θ new , subject to the constraint that the average power delivered by the
source be the same in both circuits.
(a) Let S = P + jQ be the complex power delivered by the source in the original
circuit (Fig. 12-1); you calculated this power in part 3.
Then
θ old = arctan(Q / P) . Let S′ = P + jQ′ be the power delivered by the source in
the new circuit (Fig. 12-2) so that θ new = arctan(Q′ / P) . Finally, let SC be the
power absorbed by the capacitor in Fig. 12-2. Show that
SC = jP(tan θ new − tan θ old )
(b) Using SC = I*C Vs (or I*C = SC / Vs ), ZC = Vs / I C , and ZC = 1/ jω C , show that
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P(tan θ old − tan θ new )
ω Vm2
This is our design equation for power factor correction. Calculate the value of
C required if we want θnew = 0 deg.
C=
5. Finally, considering R2 as the parasitic line resistance between generator
and load, calculate how much power is lost in R2
In Lab
1. Build the circuit of Fig. 12-1. You may use the resistance substitution box to set
R2. Set the signal generator to provide a sinusoid at frequency f = 50KHz and
amplitude 5V. Simultaneously display on the oscilloscope the voltage waveforms
Vs(t) and VR(t). To do this, use CH1 to measure across “positive” and “gnd” of
the function generator (ie CH1 ~ Vs) and use CH2 to measure across node “A”
and “gnd” (ie CH2 ~ VR). Measure the time difference between these voltages,
and from this measurement determine the phase difference between them.
Specify whether the resistor voltage, VR(t) is leading or lagging the source
voltage. Why are you unable to directly measure the voltage across the inductor
with the oscilloscope?
2. Using the source Vs as reference phasor, determine the phasors VL and VR from
your measurement in part 1. Compare with your prelab calculations.
3. Determine the pf of the circuit from your measurements.
4. Choose a capacitor close to the value you calculated in Prelab part 5. Build the
circuit of Fig. 12-2 using this capacitor.
5. Verify that the pf of this circuit meets the design specification. Note: In order to
determine the current supplied by the source, you may use R2, as a “current
sense” resistor, measure the voltage across it and hence determine the phasor I.
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