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EXPERIMENT 5
SINUSOIDAL STEADY STATE ANALYSIS – SERIES RLC CIRCUIT
OBJECTIVE
1. Measure the current in series RLC circuit using oscilloscope.
2. Apply the dual-trace method to determine the phase angle associated with each
voltage of the circuit and determine which signal leads or lags which.
3. Determine the input impedance using experimental methods.
4. Relate the phase angle between the current and voltage of a resistor.
5. Relate the phase angle between the current and voltage of an inductor.
6. Relate the phase angle between the current and voltage of a capacitor.
INTRODUCTION
PART A: DUAL-TRACE METHOD OF MEASUREMENT
The phase angle between two signals of the same frequency can be determined using the
oscilloscope using dual-trace comparison with the calibrated time base. Despite of giving
better accuracy compared to the Lissajous pattern method, it also can compare two
signals of different amplitudes and different shapes.
The procedure essentially consists of displaying both traces on the screen simultaneously
(select vertical mode button [31] to DUAL) and measuring the distance (in scale division)
between two identical points on the two traces as in Figure 5.1. The input signal is chosen
as a reference, that is, zero-phase angle. In the comparison we assume that the signal
being compared is leading (+) if it is to the left of the reference and lagging (-) if it is
to the right of the reference.
V2
reference
V1
D2
D1
Figure 5.1: Measuring phase angle using dual-trace method
Procedures:
1. Connect the two signals to the two vertical channels, [10] and [12], make sure that
proper grounding are observed. For clarity, adjust the vertical sensitivity of each
waveform until both signals have the same relative size.
2. Engage (depress) GND button [9] to set both traces in the vertical center of the
screen.
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3. Measure the number of horizontal divisions, D1 occupied by one complete cycle
of the either waveforms. Measure the number of horizontal divisions in the phase
shift, D2.
Since D1 is associated with a full cycle of 360o while D2 is associated with phase angle or
phase shift,  then the relationship is,
D
  2 x 360 o
(5.1)
D1

For the case in Figure 5.1:
1div.
x 360 o  120 o (V2 lags V1 )
3 div.
PART B: SINUSOIDAL STEADY STATE RESPONSE OF RLC CIRCUIT
All the basic circuits, Kirchoff’s laws and methods of analysis which previously applied
to dc circuits are equally applicable to ac circuit. The only major distinguish is that in ac
we will be working in phasors in analyzing ac circuits.
Let us revise the response of the individual basic R, L and C to a sinusoidal voltage or
current. For all practical purposes, assume resistor is unaffected by the frequency of the
applied voltage. The voltage source takes up this mathematical expression,
V= Vm sin t
(5.2)
Using Ohm’s law to have the current through the resistor,
IR= V/R =( Vm sin t)/R = Im sin t
(5.3)
From equations (5.2) and (5.3), it is apparent that for purely resistive circuit, voltage and
current are in phase with the peak values related by Ohm’s law.
VR
Vm
IR
Im

Imaginary


VR
t

(a)
IR
Real
(b)
Figure 5.2 (a) time domain waveform (b) phasor diagram of voltage and current of a
resistive element.
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The current through the capacitor can be derived by,
dVC
 C (Vm cos t )  CVm cos t
dt
 I m sin (t  90 o )
IC  C
(5.4)
From equations (5.2) and (5.4), it is apparent that for purely capacitive circuit, current
leads voltage by 90o with the peak values related by Ohm’s law.
VR
Vm
Im


Imaginary
IR



t
VC
IC
90o

Real
Figure 5.3 (a) time domain waveform (b) phasor diagram of voltage and current of a
capacitive element.
The voltage across the inductor can be derived by,
dI
VL  L L  L(I m cos t )  LI m cost
dt
 Vm sin (t  90 o )
(5.5)
From equations (5.3) and (5.5), it is apparent that for purely inductive circuit, current lags
voltage by 90o with the peak values related by Ohm’s law.
VR
Vm
Im


Imaginary
IR



t
VL
IL
90o

Real
Figure 5.4 (a) time domain waveform (b) phasor diagram of voltage and current of an
inductive element.
A complete phasor diagram and schematic diagram for a series RLC circuit can be
viewed as in Figure 5.5.
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Imaginary
VL
+ VC< - + VL< -
VR
IR= IL = IC

 
90o

+
Vs
-
Vs
Real
IR = IL = IC
+ VR< -
90o
VC
(a)
(b)
Figure 5.5 (a) Phasor diagram showing the relationship of voltages and currents in series
RLC Circuit. (b) Series RLC schematic diagram.
The magnitude of the voltage source is in a series RLC circuit is given by,
Vs  VR  (VL  VC ) 2  I
2
R 2  ( X L  X C ) 2  IZ T
while the phase angle of the circuit is given by,
V  VC X L  X C
  tan 1 L

VR
R
(5.6)
(5.7)
and the impedance can be determined by,
ZT  R  j( X L  X C )
(5.8)
In a series RLC circuit, the voltage across the reactive component may be greater than the
input voltage.
EQUIPMENT/COMPONENT
Dual-trace oscilloscope (1)
Function Generator (1)
Resistor (1/4 W) – 1 k
Capacitor – 0.022 F
Inductor – 10 mH
Breadboard (1)
**For all theoretical calculation results students are strictly required to show their work
in progress (WIP in formula form/complete figures) in the PRE-LAB space provided
before the lab session. Otherwise they will be forbidden from participating the session.
There will be certain marks allocated for this part.
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PROCEDURE
1. Construct the circuit as depicted in Figure 5.6 below. Insert the measured value of R
in Table 1(a). The internal dc resistance of the inductor and capacitor will be ignored
comparable to R.
C = 0.022 uF L = 10 mH
CH1
+ VL< R = 1 k
Ip-p
+ VR< -
+
Vs
-
+ VC< -
CH2
Figure 5.6: Circuit diagram of a series RLC circuit
2. Set the function generator to produce a sine wave input signal of amplitude 8Vp-p and
frequency 5 kHz. Use this input voltage as the reference signal.
3. Obtain the Vs and VR traces on the scope. Make sure you have done the correct
settings as instructed in Introduction (Part A). Draw the waveforms in Figure R1 and
label them completely with the details. Since the resistor voltage VR is in phase with
the current I, thus the phase difference between Vs and VR equals to the phase angle
between Vs and I waveforms.
4. Determine the number of horizontal divisions for one complete cycle of either
waveform denoted as D1 and the horizontal divisions for the phase shift between the
Vs and VR denoted as D2. Now determine the phase shift, 1 in degrees using Eq.
(5.1). Insert all the measured values in Table 1. From the results of Figure R1 and
Table 1(a) represent the signals VR and I in both polar and rectangular forms and
insert the answers in Table 2. Determine Ip-p from ohm’s law, Ip-p = VR(p-p) / Rmeasured.
Also determine the total impedance, ZT = Vs/Ip-p.
5. Secondly, obtain the Vs and VC traces on the scope by interchanging the position of
resistor and capacitor as depicted in Figure 5.7. The elements need to exchange
positions to avoid “shorting-out” of the resistor and inductor if we simply place the
probe of the scope across the capacitor of Figure 5.6. In other words we must ensure
the common ground between the generator and scope. Draw the waveforms in Figure
R2 and label them completely with the details.
CH1 + VR< -
Ip-p
CH2
+ VL< C
+ VC< -
R
+
Vs
-
L
Figure 5.7: Measuring VC using the oscilloscope and ensuring common ground between
the source and output
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6. Determine the number of horizontal divisions for one complete cycle of either
waveform denoted as D1 and the horizontal divisions for the phase shift between the
Vs and VC denoted as D2. Now determine the phase shift, 2 in degrees using Eq.
(5.1). Insert all the measured values in Table 1. From the results of Figure R2 and
Table 1(a) represent the signal VC in both polar and rectangular forms and insert the
answer in Table 2.
7. Thirdly, obtain the Vs and VL traces on the scope by interchanging the position of
resistor and inductor as depicted in Figure 5.8. Draw the waveforms in Figure R3 and
label them completely with the details.
CH1
+ VC< Ip-p
+ VR< -
CH2
R
L
+ VL< -
+
Vs
-
C
Figure 5.8: Measuring VL using the oscilloscope and ensuring common ground between
the source and output
8. Determine the number of horizontal divisions for one complete cycle of either
waveform denoted as D1 and the horizontal divisions for the phase shift between the
Vs and VL denoted as D2. Now determine the phase shift, 3 in degrees using Eq.
(5.1). Insert all the measured values in Table 1. From the results of Figure R3 and
Table 1(a) represent the signal VL in both polar and rectangular forms and insert the
answer in Table 2.
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Name: ________________________________ Matrix No: ______________ Date: _________
________
RESULT
Horizontal sensitivity = _______
Vertical sensitivity = _______
Figure R1: Vs versus VR
Horizontal sensitivity = _______
Vertical sensitivity = _______
Figure R2: Vs versus VC
Instructor Approval: _________________________________
7
Date: ______________
Name: ________________________________ Matrix No: ______________ Date: _________
________
Horizontal sensitivity = _______
Vertical sensitivity = _______
Figure R3: Vs versus VL
VR
Measured
Resistance
()
D1
(div)
D2
(div)
VC
1
(degree)
D1
(div)
D2
(div)
VL
2
(degree)
D1
(div)
D2
(div)
3
(degree)
Table 1(a): Experimental results of phase shift measurement between Vs and VR, Vs and
VC and Vs and VL.
Phase Angle
1
(degree)
2
(degree)
3
(degree)
1 + 2
(degree)
3 - 1
(degree)
Experimental Result
Theoretical Results (PRE-LAB)
Percentage Difference (%)
Leading or Lagging Reference
Table 1(b): Phase shift deviation between experimental and theoretical results.
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Instructor Approval: _________________________________
Date: ______________
Name: ________________________________ Matrix No: ______________ Date: _________
________
MATHEMATICAL
REPRESENTATION
POLAR
RECTANGULAR
XC ()
XL ()
Theoretical
Result
(PRE-LAB)
ZT = Rmea ± jX
()
Ip-p = Vs/ZT
(mA)
VR (V)
VC (V)
VL (V)
VR (V)
VC (V)
Experimental
Result
VL (V)
Ip-p = VR/Rmea
(mA)
ZT = Vs/Ip-p ()
Table 2: Phasor representation of circuit variables (all values in peak-peak).
Instructor Approval: _________________________________
9
Date: ______________
Name: ________________________________ Matrix No: ______________ Date: _________
________
PRE-LAB CALCULATION (Show your WIP)
(All calculations should be done in peak-peak values and at the operating frequency 5
kHZ.)
1. Calculate the phasor current, Ip-p and phasor impedance, ZT
2. Obtain the phasor voltages for VR, VC and VL.
3. Calculate the phase angle of overall circuit.
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Instructor Approval: _________________________________
Date: ______________
Name: ________________________________ Matrix No: ______________ Date: _________
________
4. Determine phase angle between the reference signal Vs and the following and specify
whether it is leading or lagging the reference:
(i) VR
(ii) VC
(iii) VL
5. Calculate the phase angle between:
(i) VR and IR
(ii) VC and IC (iii) VL and IL
Instructor Approval: _________________________________
11
Date: ______________
Name: ________________________________ Matrix No: ______________ Date: _________
________
EVALUATION QUESTION
1. What do you think will happen to the phase angle of the current in this experiment if
the value of resistance increases?
Answer:
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
2. Using the measured values of all the voltages and current, sketch the phasor diagram
in Figure E1.
Figure E1
3. Based on the series RLC circuit in Figure 5.6, suggest the possible modification to the
circuit to achieve overall phase angle of :
(i) 0o
(ii) 900
(iii) -90
Answer:
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Instructor
Instructor
Approval:
Approval:_________________________________
_________________________________
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Date:
Date:
______________
______________
Name: ________________________________ Matrix No: ______________ Date: _________
________
4. In this experiment (based on consideration that none of the circuit elements being
omitted in Figure 5.6) can we directly measure the phase angle between V L and IL or
VC and IC? Justify your answer.
Answer:
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
5. Using the measured voltage levels and Rmeasured, determine the actual inductance L of
Figure 5.6 at frequency 5 kHz.
Answer:
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
6. Using the measured voltage levels and Rmeasured, determine the actual inductance C of
Figure 5.6 at frequency 5 kHz.
Answer:
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Instructor Approval: _________________________________
13
Date: ______________