LABORATORY 1
Phasor Representations of AC Circuits
OBJECTIVES
1. To understand sinusoidal signals and phasors representations
2. To construct a phasor diagram of two waveforms, which are out of phase, directly from the
oscilloscope’s display.
3. To explore the steady state response of an RC circuit in order to measure the amplitude gain and
phase shift.
4. To examine the steady state response of an RL circuit in order to measure the amplitude gain and
phase shift.
5. To examine the steady state response of an RLC circuit in order to measure the amplitude gain and
phase shift.
INFORMATION
1. Phasor diagrams
A phasor diagram is a representation of two or more waveforms of the same frequency, but of
different phases. The diagram consists of vectors, each starting from the same point, but drawn at
an angle, which represents the phase of each waveform. Each vector, or phasor, has two
characteristics:
 The first is the length of the vector, which represents the magnitude or amplitude of the
waveform, usually expressed in RMS units.
 The second is the angle at which the vector is drawn, in degrees, representing the phase of
that waveform relative to others.
We look at phasor diagrams at frozen moments of time, thus they represent conditions at some
particular instant. Figure 1.1 illustrates typical phasor diagrams, the left one showing two phasors,
one having an RMS value of 2V and a phase angle of 15°, the other being 3 V at an angle of 50°. It
is important to realize that this diagram represents one moment of time. A few milliseconds later,
both phasors will have rotated counterclockwise to a new position such as that shown at the right,
where both have rotated by 30°. Since phasor rotation is counterclockwise, if an observer were
stationed along the vertical axis, that observer would see the approach of the 3V phasor before the
2V phasor, thus the 3V phasor is “leading” the 2V phasor by 35°, or the 2V phasor is “lagging” the
3V phasor by 35°.
3V
3V
50o
80o
2V
2V
45o
15o
a) Initial moment T1 phasor diagram
b) Consecutive moment T2 phasor diagram
Figure 1.1 Phasor Diagrams
1-1
Note that 3V phasor in Figure 1.1-a represents the voltage V (t )  3 2 cos(t  50 0 ),
and the 2V phasor represents the voltage V (t )  2 2 cos(t  150 ), where is the angular
frequency.
2. Oscilloscope phase measurements
An oscilloscope display would represent both waveforms for all moments of time. It is therefore
useful to be able to draw a phasor diagram by looking at an oscilloscope display. As an example of
this, consider the two sinusoidal waveforms shown in Figure 1.2.
A
B
0.5 V/div
50s/div
T1 T2
Tx
Figure 1.2 An oscilloscope phase display
On the scope display, each vertical line represents one instant of time. Consider the initial instant
of time Tx, marked by the arrow in Figure 1.2. At this instant of time, waveform A has amplitude
of 4 DIV x 0.5V/DIV = 2V (i.e., RMS value of 1.41 V) and its phase is 0°, while waveform B
has amplitude of 0.5DIV x 0.5V/DIV = 0.25V (i.e., RMS value of 0.18 V) and its phase B is
between 0° and 90°, approximately at 70°. This is sufficient information to sketch a rough phasor
diagram of the phasors A and B. This is shown in Figure 1.3. The phasor diagram clearly shows
that waveform B is leading waveform A.
B
A
Figure 1.3 Phasor diagram
To measure the actual phase difference between the two waveforms for the display shown in
Figure 1.2, notice that the time base is set at 50 s/DIV. Both waveforms must be exactly centered
vertically on the center horizontal graticule line. Unless two waveforms are exactly the same
1-2
amplitude, phase should only be measured along the zero volt levels of the two waveforms –
arrows T1 and T2 in Figure 1.2.
Two measurements are required to determine the phase difference between the two waveforms:
 The first is the time difference horizontally between the two waveforms.
 The second is the period for either one of the waveforms.
The time difference between the waveforms is 0.8 divisions x 50 s/division, which is a time
difference of 40s. The period of the waveform is 4 divisions x 50s/division, which is a time of
200s. Since the period of 200µs represents one complete cycle or 360°, it is easy to find out how
much phase shift in degrees is represented by the 40µs interval by solving Equation 1.1:
40s Phase.Shift
40s  3600


Phase
.
Shift

 720 .
0
200s
360
200s
Equation 1.1
3. The sinusoidal signal source
A sinusoidal voltage or current source produces a voltage or current that varies sinusoidally with
time. Using the cosine function as a base, we can use the following Equation 1.1 to describe the
sinusoidal source:
Equation 1.2
v(t )  Vm cos(t   )
Where:
 Vm is the maximum amplitude of the sinusoidal voltage,
  = 2f = 2/T is the angular frequency of the sinusoidal signal in rad/sec (f is the
frequency of the sinusoid in Hz, T is the period of the sinusoid in seconds),
  is the phase angle of the sinusoidal voltage.
Notes: 1- Since the cosine function is bounded by ±1, the amplitude is bounded by ±Vm.
This is clearly shown in Fig. 1.4.
2- Changing the phase angle  shifts the sinusoidal function along the time axis, but has no effect
on either the amplitude Vm or the angular frequency  .
3- If  is positive, the sinusoidal function shifts to the left.
If  is negative, the sinusoidal function shifts to the right.
The same function could be graphically drown as a phasor with amplitude Vm and angle
Fig
Figure 1.4
Figure 1.5
1-3
PRE-LABORATORY PREPARATION
1. The Phasor representation of signals in RC circuit
You will explore the steady-state response of an RC circuit to sinusoidal signals at a single
frequency and represent these signals in the time and frequency domain.
Figure1.6 RC circuit
Figure1.7. Voltage/time plot for the RC circuit
A simple RC circuit is shown in Figure 1.6 and the voltage/time plot for this circuit is shown in
Fig.1.7.
1.1. Determine the equivalent impedance of the circuit shown in Figure 1.6. For the signal voltage
of vS(t) =2 cos(t +0) and the frequency of f=2kHz, calculate the current ic(t) and voltage
across the capacitor vc(t) and resistor vR(t). Show your calculations in section 1and fill the data
in Table 1.2 of the Lab Measurement Sheets (LMS).
Note: Convert all Voltages and currents to their rms values!
1.2. Simulate the circuit on Fig.1.6 in MicroCap.
1.3. Run the MicroCap transient analysis and plot the voltage/time diagram for Vc, VR and Vs
Hint: Obtain VR as a voltage difference between voltages at node 2 and note 1. Convert all
Voltages and currents to their rms values!
1.4. From the plot diagram obtain amplitudes of Vc and VR and corresponding phase angles cand
R, using MicroCap tools for measuring time and amplitudes. Fill the data in Table 1.2 of the
Lab Measurement Sheets (LMS).
1.5. In section 1.4-a of your LMS sketch the phasor diagram of these signals showing Vs=VR + Vc
using the MicroCap data.
2. The Phasor representation of signals in RL circuit
You will explore the steady-state response of an RL circuit to sinusoidal signals at a single
frequency and represent these signals in the time and frequency domain.
A simple RL circuit is shown in Figure 1.5
 Determine the equivalent impedance of the circuit shown in Figure 1.8. For the signal voltage
of Vs(t) =2cos(t +0) and the frequency of f=30kHz, calculate the current and voltage across
the inductor and resistor. Show your calculations in section 2 and fill the data in Table 2.2 of
the Lab Measurement Sheets (LMS). Please consider the internal resistance of a real inductor
RL = 3
1-4
Figure 1.8. RL circuit: a) With Ideal inductor L
b) With Real inductor L
2.2 Simulate the circuit on Fig.1.8-b. in MicroCap
2.3 Run the MicroCap transient analysis and plot the voltage/time diagram for VL, VR and Vs
Hint: Obtain VR as a voltage difference between voltages at node 1 and node 2.
From the plot diagram obtain amplitudes of VL and VR and corresponding phase angles Land
R, using MicroCap tools for measuring time and amplitudes. Fill the data in Table 2.2 of the Lab
Measurement Sheets (LMS).
In section 2.4-a of your LMS sketch the phasor diagram of these signals showing Vs = VR + VL
using the MicroCap data.
3. The Phasor representation of signals in RLC series circuit
You will explore the steady-state response of an RLC circuit to sinusoidal signals at a single
frequency and represent these signals in the time and frequency domain.
A simple RLC circuit is shown in Figure 1.9:
Figure 1.9. RLC series circuit
3.1 Determine the equivalent impedance of the circuit shown in Figure 1.9. For the signal
voltage of Vs(t) =2cos(t +0) and the frequency of f=10kHz, calculate the current and
voltage across the capacitor, inductor and resistor. Show your calculations in section 3 and
fill the data in Table 3.2 of the Lab Measurement Sheets (LMS).
3.2 Simulate the circuit on Fig.1.9 in MicroCap
3.3 Run the MicroCap transient analysis and plot the voltage/time diagram for Vc, VL and Vs
Hint: Obtain VC as a voltage difference between voltages at node 2 and note 3.
3.4 From the plot diagram obtain amplitudes of Vc and VL and corresponding phase angles
cand L, using MicroCap tools for measuring time and amplitudes. Fill the data in
Table 3.2 of the Lab Measurement Sheets (LMS).
3.5 In section 3.4-a of your LMS sketch the phasor diagram of these signals showing Vs=
VR+Vc+VL using the MicroCap data.
1-5
1.



5.
6.
EQUIPMENT
PROTO-BOARD PB-503 (breadboard)
Digital Oscilloscope Tektronix TDS 210
Function Generator Wavetek FG3B
Resistors: 100
Capacitor 100nF
Inductor 220uH/ 3 
PROCEDURE
1.
RC circuit.
1.1. Build the circuit in Figure 1.10
1.2. Apply 2kHz sinusoidal waveform signal from the Signal Generator to the input and use
the dual trace oscilloscope to observe both input and output waveforms. Adjust the
magnitude of the input signal to be 4Vp-p as measured by the Ch1 of the Oscilloscope.
1.3. Display VC(t) on CH2 and measure the amplitude and phase.
1.4. In order to observe the voltage across the resistor you should switch the positions of the
resistor R and capacitor C and repeat the measurements for VR(t) on CH2.
1.5. Measure the voltages VS, VC and VR using the AC setting of your multimeter (note that
this will measure rms voltage) and record the measurements in Table 1.2. Explain do the
Multimeter readings satisfy KVL.
When the layout has been completed, have your TA check your breadboard for errors and
get his/her signature in the Signature section of the LMS.
Figure 1.10
1.6. Set the oscilloscope to a proper time/div scale and draw the measured waveforms on top
of your Micro-Cap simulation plots.
1.7. Calculate the phase shift between the measured signals using the Equation 1.1 and record
the results in Table 1.1 of the LMS.
1.8. In section 1.4-b of your LMS sketch the phasor diagram of these signals showing
Vs = VR + Vc using the measured data. Explain how close the experimental and simulated
results are and what could cause the discrepancy.
2.
RL circuit.
2.1. Build the circuit in Figure 1.11
1-6
2.2. Apply 30kHz sinusoidal waveform signal from the Signal Generator to the input and use
the dual trace oscilloscope to observe both input and output waveforms. Adjust the
magnitude of the input signal to be 4Vp-p as measured by the Ch1 of the Oscilloscope.
2.3. Display VL(t) on CH2 and measure the amplitude and phase.
2.4. In order to observe the voltage across the resistor you should switch the positions of the
resistor R and inductor L and repeat the measurements for VR(t) on CH2.
2.5. Measure the voltages VS, VL and VR using the AC setting of your multimeter (note that
this will measure rms voltage) and record the measurements in Table 2.2. Explain do the
Multimeter readings satisfy KVL.
When the layout has been completed, have your TA check your breadboard for errors and
get his/her signature in the Signature section of the Lab Measurements Sheet. You will be
penalized marks if your sheet is not initialed.
Figure 1.11
2.6. Set the oscilloscope to a proper time/div scale and draw the measured waveforms on top
of your Micro-Cap simulation plots.
2.7. Calculate the phase shift between the measured signals using the Equation 1.1 and record
the results in Table 2.1 of the LMS.
2.8. In section 2.4-b of your LMS sketch the phasor diagram of these signals showing
Vs = VR + VL using the measured data. Explain how close the experimental and simulated
results are and what could cause the discrepancy.
3.
RLC circuit.
3.1. Build the circuit in Figure 1.12
3.2. Apply 10kHz sinusoidal waveform signal from the Signal Generator to the input and use
the dual trace oscilloscope to observe both input and output waveforms. Adjust the
magnitude of the input signal to be 4Vp-p as measured by the Ch1 of the Oscilloscope.
3.3. Display VL(t) on CH2 and measure the amplitude and phase.
3.4. In order to observe the voltage across the capacitor C you should switch the positions of
the capacitor C and inductor L repeat the measurements for VC(t) on CH2.
3.5. Measure the voltages VS, VR, VL and VC using the AC setting of your multimeter (note
that this will measure rms voltage) and record the measurements in Table 3.2. Explain do
the Multimeter readings satisfy KVL.
3.6. Set the oscilloscope to a proper time/div scale and draw the measured waveforms on top
of your Micro-Cap simulation plots.
3.7. Calculate the phase shift between the measured signals using the Equation 1.1 and record
the results in Table 3.1 of the LMS.
1-7
3.8. In section 3.4-b of your LMS sketch the phasor diagram of these signals showing
Vs = VR + VC +VL using the measured data. Explain how close the experimental and
simulated results are and what could cause the discrepancy.
Figure 1.12
1-8
LAB MEASUREMENTS SHEET – LAB 1
Name _________________________
Student No_____________________
Workbench No_____
NOTE: Questions are related to observations, and must be answered as a part of the procedure
of this experiment.
Sections marked * are pre-lab preparation and must be completed BEFORE coming to the lab.
1. Phasor diagram of RC circuit
1.1. *Calculations
1.2. Table 1.1 RC circuit
* Calculations
Vc
* MicroCap
Measurements
c
VR
R
Vs
S
1.3. How close the experimental and simulated results are and what could cause the discrepancy.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
1-9
_______________________________________________________________________________
_______________________________________________________________________________
1.4-a. *MicroCap Phasor diagram
1.4-b. Measured Phasor diagram
1.5. Compare measured to the estimated phasors and comment on differences.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
Table 1.2
Voltage
VS
Multimeter (rms)
Amplitude on Oscilloscope
VC
VR
Do the Multimeter readings satisfy KVL? Give the explanation.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
2. Phasor diagram of RL circuit
2.1. *Calculations
1-10
2.2. Table 2.1 RL circuit
* Calculations
VL
* MicroCap
Measurements
L
VR
R
Vs
S
2.3. How close the experimental and simulated results are and what could cause the discrepancy.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
2.4-a. *MicroCap Phasor diagram
2.4-b. Measured Phasor diagram
1-11
2.5. Compare measured to the estimated phasors and comment on differences.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
Table 2.2
Voltage
VS
Multimeter (rms)
Amplitude on Oscilloscope
VL
VR
2.6. Do the Multimeter readings satisfy KVL? Give the explanation. .
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
3. Phasor diagram of RLC circuit
3.1. *Calculations
1-12
3.2. Table 2.1 RL circuit
* Calculations
Vs
* MicroCap
Measurements
S
VR
R
VC
C
VL
L
3.3. How close the experimental and simulated results are and what could cause the discrepancy.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
3.4-a. *MicroCap Phasor diagram
3.4-b. Measured Phasor diagram
3.5. Compare measured to the estimated phasors and comment on differences.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
Table 3.2
Voltage
Multimeter (rms)
Amplitude on Oscilloscope
VS
VL
VR
VC
1-13
2.7. Do the Multimeter readings satisfy KVL? Give the explanation. .
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
SIGNATURES
TA name:________________________
To be completed by TA during the lab session.
Check
boxes
Task
Max.
Marks
30
Pre-lab completed.
Circuit of Figure 1.7 connected and
equipment used correctly
Circuit of Figure 1.8 connected and
equipment used correctly
Circuit of Figure 1.9 connected and
equipment used correctly
Data collected and observations made
TOTAL MARKS
20
20
20
10
100
Final marking to be completed by TA after the lab session.
1-14
Granted
Marks
TA
Signature