Experiment 28: Alternating Currents Circuits
Purpose
(1) To observe properties of simplest AC circuits.
(2) To observe resonance in an LRC series circuit.
Apparatus
(a) an AC generator (Function generator), an AC voltmeter
(b) a sample with an inductor
(c) a capacitor, and a resistor
(d) decade resistance box
Theory
A) Alternating Current
It is understood that such current changes in time “harmonically”,
as shown in Fig. 1.
and is described by formula
i = ÎMAX · sin2πft = ÎMAX · sin  t (  = 2πf)
(1)
where ÎMAX = amplitude of the current; f = frequency;  = angular frequency.
B) Phase Relationships
When a current given by (1) flows through a resistor, the instantaneous voltage
between the resistor terminals is
VR = R · i = R · ÎMAX · sin  t

(2)
On these pages, small case script letters will denote instantaneous values of currents
and voltages.We say that “the voltage is in phase with the current” (because VR and i are
zero at the same instant of time, and they also reach their maxima at the same instant of
time).
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Experiment 28
When a capacitor is inserted in the path of current given by (1) then the current can
still exist, because the capacitor can be alternately and continuously charged and
discharged. Theory shows that the instantaneous voltage across the capacitor will
still change in time “harmonically”, but out of phase with the current. Its graph
will be like that shown in Fig. 2.
Such (instantaneous) voltage may be described by formula
VC = - 1 · ÎMAX · cos  t = 1 · ÎMAX · sin(  t +  )
2
C
(3)
C
and we say that this voltage “lags the current by 90º” (because 
means the same
2
as 90º). Alternatively, we may say “the current leads the voltage by 90º” (see Fig. 2).
C) RMS values
In AC circuits, it is customary to quote voltages and currents by their “Effective
values” (or RMS values). They are related to amplitudes by formulae
V = 1 VMAX
I = 1 · ÎMAX
(4)
2
2
Note: On these pages, capital letters (for currents and voltages) will always imply
RMS values.
D) Phasor diagrams (Vector diagrams)
Harmonically varying quantities are routinely represented by so-called PHASORS.
They are also referred to as vectors because they are added and subtracted by the
same rules as two dimensional vectors.

Caution: Phasors also have some additional properties which vectors do not have!
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Experiment 28
To draw a phasor diagram of voltages, follow these rules:
1) A voltage is represented by an arrow ( = vector). Its RMS value is the
magnitude.
2) Voltages which are in phase with the current (as in example (2) above) are
plotted in the positive x direction, quoting a specified scale.
3) Voltages lagging the current by 90º are plotted in the negative y direction.
4) All other voltages are plotted according to their phase (=phase angle) relative
to the current.
5) When voltages are to be added, as is the case of series connections, the result is a
vector sum of individual voltages.
For instance, a resistor and a capacitor in series produce a phasor diagram as
shown in Fig. 3.
Note: In AC circuits, phases are always quoted in the range -180º < Ø  + 180º,
that is, they could be positive or negative.
E) Reactance and Impedance
The quantity
XC = 1
C
(CAPACITIVE REACTANCE)
(6)
which appears in formula (3), is measured in ohms (provided that  is in Hertzes,
and C is in Farads).
When a capacitor is in series with a resistor, the quantity
ZRC =
R2 + XC2
(6a)
is known as the impedance of this combination. Its SI unit is also the ohm(Ω).
See below for similar quantities involving inductors.
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Experiment 28
F) Inductances and Their Properties
When an inductor is inserted in the path of the current given by (1) then, by
Faraday’s Law, and induced EMF appears. Theory shows that the instantaneous
voltage across the inductor will be:
VL =  L · ÎMAX · cos  t =  L· ÎMAX · sin(  t - 
2
)
(7)
and we say that this voltage “leads the current by 90º” (or: “the current lags the
voltage”. The graph of VL is shown in Fig 4.
Compare this situation with that of a capacitor, as in formula (3) and Fig.2.
The parameter L is known as inductance, and the quantity
XL =  · L
(8)
is the inductive reactance. If  is in hertzes and L is in henries then
XL is in ohms.
In a phasor diagram the voltage VL is plotted in the positive y direction. The
phasor diagram for an LR series combination is shown in Fig. 5.
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Experiment 28
G) The LCR Series Circuit. Resonance
When an inductor, a capacitor, and a resistor are connected in series then
Fig. 6 applies:
Also, the following definitions apply:
XTOT = XL - XC
(TOTAL REACTANCE)
ZTOT = R2 + XTOT2 (TOTAL IMPEDANCE)
and it follows that:
I = VLCR
tanØLCR = XTOT
ZTOT
R
(11)
(12)
The total reactane will be zero if XL = XC, and this will happen at the frequency
fRES =
1
2π
(13)
LC
In such situation we say that the circuit and the AC generator are “in resonance”.
Preliminary Procedure
a) On your data sheet, record the code number of your sample, and the values of
R, L, and C.
b) Your instructor will explain to you the operation of your AC generator, and your
AC multimeter. In this experiment the AC multimeter will be used only as an
AC voltmeter and must be set to read the 10 V scale. You should be able to read this
scale within 0.05 volt accuracy.
Make sure you can properly operate properly these devices.
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Experiment 28
Procedure Part I. RC Series Circuit
c) Assemble your circuit as shown,
and set the frequency at 2,000 Hz.
Engage the AC voltmeter across the
terminals of your AC generator.
Before turning the power “ON”
obtain CLEARANCE from
YOUR INSTRUCTOR.
d) On your data sheet, prepare
a table as shown. Adjust the
output voltage anywhere
between 9.00 and 9.20 volts.
Then measure and record
voltages VR, VC, and VRC (as
shown in Fig. 3).
f
(Hertz)
VR
VC
VRC
. . . . (volts) . . .
..
2,000
600
400
e) Repeat (d) for new frequencies 600 Hz and 400 Hz.
Note: Your output voltage, as well as, VRC , need not be the same as in (d),
but should be close.
Procedure Part II. LR Series Circuit
f) Assemble your circuit as shown,
with f = 2,000 Hz. Proceed in a
similar way as in (d) and (e), but
using 4,000 Hz and 6,000 Hz as new
frequencies. Measure VR, VL, and
VLR and record them in a (new)
table similar to that in (d).
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Experiment 28
Procedure Part III. LCR Series Circuit
g) Assemble your circuit as shown,
with f = 2,000 Hz. Adjust the
output between 9.0 and 9.2 v.
Measure and record VL, VC,
VLC, VR, and VLCR on your data
sheet. (Note: no other frequencies
need to be used).
Procedure Part IV. Series Resonance
h) Keeping your circuit as in (g), engage the voltmeter across both L and C – so that it
will read VLC.
Now, slowly reduce the frequency (from 2,000 Hz down) and watch carefully the
voltmeter needle. The voltage should decrease until (at some frequency f ΄) it will
reach a minimum, before increasing again. Go back and forth a few times until you
can determine its value to the best accuracy of your skills. Record f ΄, and record
your estimated accuracy on the data sheet.
BEFORE LEAVING THE LAB, personally return your AC multimeter to your
lab instructor, disassemble all connections, and unplug the AC generator.
Lab Report
Part I. RC Circuit
1) Construct a table as shown. Transfer the measured values from your data sheet. Then
complete the table by using formulae (5). Display the % differences between the
measured and calculated values of VRC ; use their average as the basis.
FREQUENCY
f
(HERTZ)
TABLE ONE: RC CIRCUIT. BASIC RESULTS
MEASURED VALUES
CALCULATED VALUES
%
VR
VC
VRC
DISCREPANCY
Ø
V
RC
RC
. . . . (VOLTS). . . . .
IN VRC
. . . .
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Experiment 28
2) Use a sheet of unruled paper. With help of a (metric) ruler and a (drafting) triangle,
draw (separately) phasor diagrams for each of the three frequencies. Use the measured
data and the scale: 1 cm ↔ 1 volt. Display ØRC , as measured by a protractor.
Note: At least two diagrams should fit on a single page.
3) Construct another table,
as shown. Using R and
the measured values
of VR, VC, and VRC
from your data sheet,
fill out the table,
column by column.
TABLE TWO: CALCULATIONS OF CAPACITANCE
f
(HERTZ)
I =VR
R
XC = V C
I
ZRC
VRC
ZRC
AVERAGE <C>
Display the value of <C> and compare it with the value given on your sample.
Part II. LR Circuit
4) Construct a table for your LR circuit similar to Table One (for the RC circuit,
as above). Complete this table following a similar procedure as in (1) above.
5) Draw phasor diagrams similar to those for the RC circuit.
Note: No calculations like those in pt. (3) are required.
Part III. LCR Circuit
6) Using your measured values,
(6a) Calculate the % discrepancy between (VL – VC) and VLC.
Use VLC as basis.
(6b) Calculate the % discrepancy between VR2 + VLC2 and VLCR.
Use VLCR as basis.
Part IV. Resonance
7) Calculate fRES from formula (13), using L and C from your sample. Calculate
the % discrepancy between fRES and your measured value of f ´
151
C = 1
X
C