Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
Authors:
Denard Lynch
Date:
Aug 30 - Sep 28, 2012
Sep 23, 2013: revisions-djl
Description:
This laboratory explores the behaviour of resistive, capacitive and inductive elements in
alternating current (AC) circuits. The student will observe and measure the phase
difference between voltage and current in AC circuits with combinations of resistance,
capacitance or inductance.
Learning Objectives:
In this laboratory, the student will:
• Measure voltage and current in AC circuits,
• Understand the difference between RMS, peak-to-peak and zero-to-peak values,
• Measure phase difference between voltage and current in AC circuits,
• Calculate reactance and impedance based on AC circuit measurements,
• Verify phasor addition of voltages in an AC circuit
ReportingUse your lab notebook (logbook) to document
• the key objectives of this laboratory,
• your theoretical calculations/expectations,
• Parts List: your equipment and circuit components used
• any measured values of components
• your measurements verifying your theoretical expectations (you can paste in
screen shots from your ADM where appropriate),
• use Power Triangles and Phasor Diagrams to help illustrate results (e.g. the
relationship between voltages or currents or powers).
• your observations and comments about how closely your observations matched
your expectations,
• related comments on practical limitations for your observations and comments on
possible sources of error
Safety Considerations:
In addition to general electrical safety considerations, the student should also be aware of
the following considerations specific to this laboratory exercise:
Denard Lynch
Page 1 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
•
•
•
Resistors carrying current will generate heat energy and can be overheated in AC
circuits. Power is based on RMS voltages and currents. In all other respects, the
same considerations as in DC circuits apply.
Unlike DC circuits, capacitors and inductors under AC excitation will conduct
current and can have potentials across them even in steady-state (not only
transient) conditions. The reactance (opposition to flow), current and voltage
present are also a function of the frequency of the excitation source. Always
consider these parameters when anticipating potential safety hazards and the
required ratings of components.
Measurement of AC circuit parameters requires suitable test gear or the selection
of the appropriate scale and range. Use of DC instruments or incorrect scales can
result in equipment damage or safety risks.
Background and Preparation:
Continue to familiarize yourself with the capabilities and operation of your Analog
Discovery Module, particularly the “Measurements” feature, which will be very useful in
this lab. You will also make use of the differential oscilloscope inputs and “Math”
features to make several observations and measurements in this lab. Review the
instructive material given in Lab #1, and refer to any included “Help” documentation.
Please refer to the class notes on AC circuits. A summary of key points is provided in
Appendix A at the end of this procedure.
Terms:
AC
Alternating Current. Usually used generally to refer to
sinusoidally alternating voltage and current
Reactance, X
“Opposition to flow” in the AC world. This is a scalar
value, measured in Ωs.
Impedance, Z
“Opposition to flow” in the phasor world. Z is a complexvalued number whose components are the scalar values of
resistance (real part) and reactance (imaginary part)
RMS
Root-Mean-Square. This is the effective value of an AC
waveform, or the equivalent DC voltage or current that
would provide the same heating effect.
φ
Phase angle of a waveform. This is with reference to
another waveform, or an arbitrary phase or reference point
Phasor
A representation of an AC waveform that can be used to
easily add and subtract AC signals of the same frequency.
It is a complex-valued number representing the magnitude
and phase angle of the waveform.
Resonant frequency
The frequency at which XL = XC in an AC circuit
Denard Lynch
Page 2 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
Theoretical calculations of various circuit parameters should be performed as part of the
lab preparation (i.e. prior to your lab period). Calculating the expected currents and
voltages will also allow you to determine the required ratings for your components (i.e.
how much power they must dissipate, how much voltage they must withstand etc.)
Please refer to the Class Notes (4) for background theory on impedance, reactance and
complex power.
`
Procedure:
The procedure will involve constructing several different circuits with combinations of
resistance, capacitance and inductance, and using your Analog Discovery Module (ADM)
to verify the AC voltage and current in each element as well as the phase difference
between the voltage and current. You will also measure parameters required to calculate
real, reactive and apparent power and compare those to theoretical values.
Modeling- (determining what you would expect to see)
This will involve using AC circuit theory to predict the circuit parameters for each circuit.
The required parameters are described in the detailed procedure below. Please read the
entire procedure over carefully before the lab and calculate the expected values of the
various parameters using AC circuit theory.
Note that you are again using a “practical inductor”, which has some internal resistance.
You should account for this fact in your theoretical calculations, and adjust your
observable expectations accordingly.
MeasurementsI. R-C Circuit Impedance
In this part of the procedure, you will observe and measure various parameters of
an AC waveform. You also want to verify that a capacitor (unlike in a DC circuit)
offers an opposition to the flow of AC current somewhat like a resistor, and that
this “opposition” (Reactance, X) is inversely proportional to the capacitance and
the frequency. Also, you will want to verify that the phase difference between the
voltage and current in this element is as expected (i.e. which leads which, and by
how much?).
Use your solderless breadboard and set up the circuit shown in Figure 1 below. A
good first step is to examine the circuit, make a list of the parts you will need,
obtain the necessary parts and construct the circuit. (The parts for this lab are the
same as for the previous lab, except that the resistors are a different value). Refer
You may also need to measure the actual value of your components (e.g. using the
RLC tester or equivalent) versus their nominal value. For example:
Denard Lynch
Page 3 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
Component
1/4W resistor
Nominal Value
Measured Value
220Ω
216.4Ω
Capacitor
0.1µF
0.096µF
Etc.
Your lab instructor will indicate where to obtain the necessary parts if they are not
already in your parts kit, and how to measure their actual value. You can use your
Analog Discovery Module (ADM) as the AC source for these procedures. (Note:
there is a very limited amount of current available from the ADM; ~10mA
maximum from the WaveGen. Check your theoretical calculation to make sure this
supply isn’t overloaded whenever you use it. Also, check the actual waveform
across your circuit to detect any variations from you nominal setting.)
Digilent
ADM
WaveGen (W1)
Amplitude: 5V
Offset: 0V
Symmetry: 50%
Figure 1
Check your breadboarded circuit for correctness. Start a Waveform Generator
window in your ADM and set the output to a 7.20kHz sine wave, amplitude of 5
VAC, zero offset, and connect it to the ‘W1(AC)’ input shown in Figure 1
(remember to make good ground connections from the ADM to your circuit). In
this circuit, you can conveniently determine the current by measuring the voltage
across the 220Ω resistor, and dividing by the actual R value; you do not need the
10Ω current sense (CS) resistor in this case. (In cases where this isn’t convenient,
for instance in a circuit with parallel elements, you can add a small sense resistor,
like R1, and connect the differential oscilloscope inputs, 1+ and 1-, to the test
points CS+ and CS- respectively. Note: the “differential ± inputs” are not
connected to ‘ground’, which allows you to measure voltages across any point in
the circuit without inadvertently grounding the wrong point in the circuit!). Using
Ohm’s Law, the series current in the circuit is simply the voltage measured divided
by the value of the sense resistor. You can display this directly on your
oscilloscope display by selecting “Add a Mathematical Channel” from the “+C Add
Channel” button above the traces pane. Select “Custom” from the pull-down menu
and enter “C1/10” in the “Enter Function” dialog box. The settings window for the
additional trace will appear in the parameters pane, and you can use the icon in the
settings window for that channel (immediately left of the ‘X’) and select “A”
(amperes) from the “Units:” menu and the display will show mA for units for that
channel. You can then add a “measurement” to display the RMS current from that
channel (In the Measurements pane, “+Add/ M1/ +Vertical/ ACRMS” then “+Add
Denard Lynch
Page 4 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
Selected Measurement”). Alternatively, you can use a Digital Multimeter (DMM)
on the mA current scale in series with the circuit to measure AC current separately.
(Note: DMMs usually read in RMS values on the AC scales.)
Next connect your other oscilloscope channel, 2+ and 2-, across the R-C circuit of
interest (use the points V+ and V- shown in Figure 1) to display the AC voltage
across the elements of interest (“load”). Again, you can add a measurement to
display the RMS voltage impressed across the circuit.
Now compare the phase of the two traces, C1 representing the current and C2
representing the voltage across the R-C impedance. Pick similar points on both
traces and use the “X” cursors (X1, X2) to determine the time differential between
the voltage and current in this R-C impedance. (You may need to select a smaller
horizontal time base to “spread” the trace and make it easier to adjust the cursors
accurately.) Convert this time difference to an angle, which is the phase angle of
the complex valued impedance of the R-C element under test.) Remember, the sign
of the angle depends on the whether the voltage is ahead of the current (+ve) or
behind (-ve). The magnitude of the impedance, |Z|, is simply the ratio of V/I.
Finally, move your C2 ‘scope probes and measure the voltage across the resistor,
R2, and then the capacitor, C1, again noting the phase difference relative to the
current in the circuit. (Remember: in a series circuit, the current is common to all
elements and is a good reference.) you can draw a Phasor Diagram to represent the
current and all the component voltage in your circuit.
You may want to use a table similar to Table 1 to record your measurements and
results in your laboratory notebook (logbook).
Table 1
Circuit
Parameter
IRMS
VP-P
V0-P
VRMS
Expected
Result*
Measured
Value
Δt
period
Comments
Between voltage and current
T=f-1 (can add a measurement
for this on ADM too)
Δt/T*3600
Δφ
|Z|
φZ
VR2
φVR2
VC1
φVC1
* where applicable
Denard Lynch
Page 5 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
II. R–L Circuit Impedance
In this part of the procedure, you will verify that an inductor (unlike in a DC
circuit) offers an opposition to the flow of AC current somewhat like a resistor, and
that this Reactance, X, is proportional to the frequency and the value of the
inductor. Also there the phase difference between the voltage and current in this
element is as expected, and opposite to that of the capacitor.
Use your solderless breadboard and set up a similar circuit using an inductor as
shown in Figure 2 below. Again, gather and check your components and construct
the circuit.
Continue to use your Analog Discovery Module (ADM) as the AC source for these
procedures. (Be careful not to draw too much current.)
Digilent
ADM
WaveGen (W1)
Amplitude: 5V
Offset: 0V
Symmetry: 50%
Figure 2
Check your breadboard circuit for correctness. Adjust the Waveform Generator
output from your ADM to a 4.50kHz sine wave output of 5 VAC, zero offset and
connect it to the ‘W1(AC)’ input shown in Figure 2 (remember to make good
ground connections from the ADM to your circuit). You can use the same
connections and setup as in Part I using both input channels to observe and measure
the desired circuit parameters.
Compare the phase of the two traces, C1 representing the current and C2
representing the voltage across the R-L circuit (or the individual elements as
required). Remember, with an inductive circuit you will expect the sign of the
impedance’s angular argument to be opposite that for the capacitive circuit you
measured in Part I. (Also remember, the is phase difference is generally considered
to be the smaller of the two choices; i.e. <1800.)
Finally, move your C2 ‘scope probes and measure the voltage across the resistor,
R2, and then the Inductor, L1, again noting the phase difference relative to the
current in the circuit.
You may again want to use a table similar to Table 1 to record your measurements
and results in your laboratory notebook (logbook). (The individual voltage and
phase measurements will be across the inductor this time.)
III. Verifying Phasor summation in an R – L – C Circuit
In this part of the procedure, you will again verify Ohm’s Law for AC circuits as
well as KVL for AC circuits. This will also allow you to verify phasor addition in a
series AC circuit. You will also be able to observe a phenomenon in AC circuits
known as resonance.
Denard Lynch
Page 6 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
Adding the capacitor to the circuit from Part II, you can determine the phasor
current and voltages in this circuit (Figure 3). This time, set the frequency of the
WaveGen to 6.0 kHz. (Note: If you try this circuit with the 220Ω resistor first, you
will see the effect of the current limits of the WaveGen output.)
Digilent
ADM
WaveGen (W1)
Amplitude: 5V
Offset: 0V
Symmetry: 50%
Figure 3: Phasor Addition R-L-C
Again using the current as reference, measure the RMS magnitude and phase of
each of the voltages in the circuit, and the overall voltage and phase. Compare this
to your theoretical calculations for the individual voltages and the sum.
Record relevant values, such as those shown in Table 2 and any others you may
need in your lab notebook. Show that the values you observed are substantially
what you would expect form your theoretical predictions. Remember that the
inductor has some internal resistance which cannot be separated from the true
inductance. How does this affect your magnitude and phase measurements?
Table 2
Circuit
Parameter
IRMS
VR
Expected
Result*
Measured
Value
Comments
Common to all series elements
φR
VC
φC
VL
φL
VT
φZ
Again, a Phasor Diagram makes a good graphical representation of your
measurements.
Denard Lynch
Page 7 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
Optional: Vary the WaveGen frequency while observing the overall voltage and
current phase difference, and observe where the total impedance phase angle is 00.
This is know as the resonant frequency of the circuit; a frequency where VL and VC
are equal and opposite directions (i.e. cancel each other out completely) on the
phasor diagram! You will also note that at this point, the voltage across the resistor
is ~ E, the source voltage. This phenomenon also has some relevance for power, as
we’ll see in the next lab.
Denard Lynch
Page 8 of 9
Sep 2013
Laboratory #2: AC Circuits, Impedance and Phasors
Electrical and Computer Engineering
EE 204.3
University of Saskatchewan
APPENDIX A: Background Theory
“Opposition to flow” of one of the basic elements in AC circuits is a scalar value
measured in Ohms (Ω) and called reactance, X, for capacitors and inductors. Resistance
is generally the same value in AC circuits (RAC = RDC), but the reactance of capacitors
and inductors is given by:
1
XC =
Ω , (C in Farads)
XL = ω LΩ , (L in Henrys),
ωC
where ω = 2π ⋅ frequency( Radians sec ond )
Use of these scalar values will provide some useful information about the individual
elements, but for an AC circuit with a combination of R, L and C, we find is very useful
to represent the AC voltage and current as phasor quantities (which are represented as
complex-valued numbers). Phasors are a transformation of AC voltages and currents
form the time domain into a “phasor” domain: a two-dimensional complex plane, where
the operations of adding and subtracting sinusoidal waveforms of the same frequency is
greatly simplified. Phasors are generated from the time domain expression by using only
the magnitude and phase angle of the wave form:
Time Domain to Phasor Domain Conversion
Time: v(t) = Asin (ω t + ϕ )V
Phasor: V =
A
2
∠ϕ V (polar form), =
A
2
cos( ϕ ) + j
A
2
sin( ϕ ) (rectangular form)
The reverse conversion is also straight forward, except that the actual frequency cannot
be determined from the phasor quantity alone.
Phasor Domain to Time Domain Conversion
Phasor: V = B∠θV (polar form) or V = (X +jY)V (rectangular form)
⎛
⎛ Y ⎞⎞
Time: v( t ) = 2B sin( ω t + θ )V , or v( t ) = 2 X 2 + Y 2 sin ⎜ ω t + tan−1 ⎜ ⎟ ⎟ V
⎝ X ⎠⎠
⎝
The “opposition to flow” in the phasor domain is called the impedance, Z, and is also
measured in Ohms and is defined as:
phasor _ voltage
V
, or Z = , where we use the standard units of measure (Ohms,
phasor _ current
I
Volts, Amperes), but each quantity is a complex valued number (usually represented as a
magnitude and angle, if the polar form is used, or an ‘x’ and ‘y’ component if the
rectangular form is used, in a two-dimensional complex plane). In rectangular form, the
complex impedance can be written in terms of the scalar resistance and reactances:
Z = R + j ( XL − XC ) , and can be shown graphically on a Impedance Diagram
(again, on a two-dimensional complex plane).
Denard Lynch
Page 9 of 9
Sep 2013