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Experiment 27: AC Circuits II: LR & LCR Circuits
Purpose
(1) To study the properties of an AC circuit containing a resistor and an inductor (coil).
(2) To observe the basic properties of a series LCR circuit.
(3) To observe resonance in an LCR circuit.
Apparatus
(a) an AC Power Supply; an AC multimeter
(b) a sample containing an inductor (coil)
(c) two resistors, and a capacitor.
Theory
Faraday’s Law
When an inductor (coil) is inserted in the path of an alternating current (AC), then an
induced EMF, εIND , appears across the terminals of the coil, according to Faraday’s Law of
Electromagnetic Induction. It is commonly called an induced voltage. The instantaneous value
of εIND will be here denoted by VL and is expressed by the formula:
VL =  L · ÎMAX · cos  t =  L· ÎMAX · sin(  t - 
2
)
(1)
The inductor voltage opposes the change in voltage of the power supply and peaks before the
current peaks. We say that the “induced voltage leads the current by 90˚ (or  , in radians)” or,
2
alternatively, “the current lags VL by 90˚”. See Fig. 1 and compare it with Fig. 2 in Experiment
26.
The parameter L is the inductance of the coil and the quantity

Before reading this, make sure you have read and understood the Theory Section in
Experiment 26.
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Experiment 27
XL =  · L
(2)
is the inductive reactance of the coil. If L is measured in henries (SI unit for inductance) and 
in hertz, then XL is in ohms.
Employing the RMS values (see Exp. 26), the RMS voltage across the inductor is:
VRMS = IRMS XL
(3)
(Compare with equations (6) and (7) in Exp. 26.)
The Phasor Diagram of an LR Series Circuit
The principles of the phasor diagrams were explained in Exp. 26. The phasor
diagram for a coil and a resistor connected in series is shown in Fig. 2, together with
relevant formulae (4).
Important Note: As well as its reactance, a coil also has a resistance R´ which creates
an additional voltage VR' = R´i across the coil. This voltage is in phase with the current,
so that it lags VL by 90º. What a voltmeter connected across a coil actually measures
is the phasor sum of the RMS magnitudes of VL and VR' - that is the quantity
VLR2 =  VL2 + VR'2
(4a)
rather than VL. However, if VR' is much smaller than VL then the voltmeter essentially measures
VL .
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Experiment 27
The LCR Series Circuit
When a coil, a capacitor, and a resistor are connected (in that order) in series, then
Fig. 3 applies:
Relevant formulae are
VLCR =  VR2 + (VL – VC) 2
tan ØLCR = VL – VC
VR
(5)
It is also customary to define:
XL – XC
XTOTAL =
ZTOTAL =  R2 + XTOTAL 2
= TOTAL REACTANCE
= TOTAL IMPEDANCE
(6)
and the following formulae then hold:
I = VLCR
ZTOTAL
tan ØLCR = XTOTAL
R
(7)
Resonance.
The total reactance
XTOTAL = XL – XC =  L – 1
C
depends on frequency f (recall:  = 2πf ).
(8)
When the frequency happens to be
f RES =
1
2π
LC
(9)
then XTOTAL = 0 and the circuit is in resonance with the applied frequency.
139
Experiment 27
Procedure Part I. The LR Circuit
a) With the AC power supply unplugged and OFF,
set up a circuit as in Fig. 4, using resistance
R1 ~ 4,000Ω from your sample. Record its
exact value.
You should know how to use the AC power
supply and the AC multimeter from Exp.26
but, if you do not, check with your instructor.
Set the frequency at f 1 = 2,000 hertz and
record. Prepare your multimeter to read the
10 volt AC scale.
b) Upon your instructor’s approval, turn ON the power, and adjust the output voltage
between 9.80 and 9.95 volts (or your maximum voltage if you cannot reach 9.80 V).
Record this as VOUT to the accuracy of 0.05 V.
c) Measure and record the voltages VL and VR across the coil and the resistor separately.
Measure and record the voltage VLR across both of them together to 0.05 volt accuracy.
Return the output voltage knob to MINIMUM POSITION.
d) Change the frequency to f 2 = 4,000 hertz and record. Re-adjust VOUT to be between
9.80 and 9.95 volts and record the exact value. Repeat (c) above.
e) Repeat (d) above with f 3 = 6,000 hertz.
Procedure Part II. The LCR Circuit
f) With the power OFF, assemble a
circuit as in Fig. 5, carefully
observing the sequence L-C-R. Use
the resistance R2 ~ 1,000Ω from
your sample and record its exact
value.
Set f = 2,000 hertz and record.
g) Upon your instructor’s approval turn ON the power. Engage your voltmeter (using the
same 10 V scale) across the inductor (NOT the output voltage, as always before!) and
adjust VL to be between 9.80 and 9.95 volts (or your maximum value). Record the
exact value of VL.
h) Measure and record VL , VC , VR , and VLCR (as in Fig.3) to 0.05 volt accuracy.
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Experiment 27
Procedure Part III. Resonance
i) With your circuit the same as in (f) above, start with the output voltage at the
MINIMUM. Set f = 1,000 hertz and engage the voltmeter to measure VLC (the total
voltage across the LC combination).
Slowly increase power until VLC is about 8.5 volts. Next, slowly increase the
frequency. You will notice that VLC will be decreasing until it reaches a minimum
at some frequency fVLCmin. Record the value of fVLCmin and also the minimum value
of VLC .
j) Use the same set-up as in (i) above (start with MINIMUM VOUT) but now engage
the voltmeter to measure VR.
Slowly increase power until VR is about 5 volts (NO MORE!).
Next, increase f and watch VR increasing, until a maximum is reached at some
frequency fVRmax. Record fVRmax and the maximum value of VR . Also record
VOUT in this situation.
BEFORE YOU LEAVE THE LAB:
Make sure that you have recorded the values of L, C, and R′, marked on your samples and
the values of resistances and frequencies which you personally used.
Unplug and turn OFF the AC power supply and leave your station in order.
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Experiment 27
Lab Report
Part I. LR Circuit
1) Using your measured values of VL , and VR , draw phasor diagrams (consult Fig. 6)
for each or your three runs.
2) Draw a table as shown. Quote all physical units. The graphical values are to be
measured by ruler and protractor from your phasor diagrams.
FREQUENCY
f
TABLE ONE: LR CIRCUIT
MEASURED VALUES
GRAPHICAL VALUES
VL
VR
VLR
VLR
ØLR
. . . . . . . .(quote units). . . . . . . . . .
%
DISCREPANCY
IN VLR
Part II
3) Using your measured
values of VL, VC, and
VR, draw the phasor
diagram (consult
Fig.7).
VL
TABLE TWO: LCR CIRCUIT
MEASURED:
VC
VR
VLCR CALCULATED
. . (volts). .
VLCR (VOLTS)
%
DISCR’CY
IN VLCR
4) Using your measured values of VL, VC, and VR, calculate the expected value of
VLCR, according to formula (14) and complete Table Two, as shown.
5) From the values of L, C, and R′, as well as R1 and R2 (all of which should be recorded
on your data sheet), construct and fill out Table Three; the graphical value of
ØLCR must come from your phasor diagram and the calculated value of ØLCR
must come from equation (16).
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Experiment 27
TABLE THREE: LCR CIRCUIT. FREQUENCY: 2,000 Hz
XL
CALCULATED VALUES:
XC
XTOTAL ZTOTAL
. . . (quote units). . .
. . .
ØLCR
(DEGREES)
GRAPHICAL
VALUE
OF ØLCR
(DEGREES)
ABSOLUTE
DISCREPANCY
IN ØLCR
(DEGREES)
Part III. Resonance.
6) Show the average (=arithmetical mean value) of frequencies f ′ and f ˝ from your
measurements. Compare this value with the formula (18), by displaying the
% discrepancy (use the given values of L and C as the basis).
7) Answer the questions:
Question #1: What is the RMS value of the current in your LCR circuit at the
resonance frequency?
Question #2: Should VL and VC be equal at resonance frequency (explain why “yes”
or “no”).
Question #3: In your particular experiment (with the parameters you had), can the
value of R′ be ignored (explain why “yes” or “no”).
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