Boise State University
Department of Electrical and Computer Engineering
ECE 212L – Circuit Analysis and Design Lab
Experiment #2: Sinusoidal Steady State and Resonant Circuits
1
Objectives
The objectives of this laboratory experiment are:
• To investigate the sinusoidal steady-state response of a resonant circuit in the phasor domain.
• To compare the timebase and the Lissajous methods for measuring the phase shift between
two sinusoidal waveforms.
2
Theory
Electric circuits containing components like capacitors and inductors can introduce a phase shift
between an exciting (input) sine waveform and a measured (output) sine waveform. This phase
shift may be an important parameter to be measured in certain applications. This experiment
investigates the timebase and Lissajous methods for measuring such a phase shift between two sine
waveforms.
2.1
Timebase Method
v1
v2
D1
D2
Figure 1: Timebase Method for Measuring the Phase Difference Between Two Sine Waveforms
Figure 1 shows two sinusoidal waveforms,
√
v1 (t) = Vm1 cos(ωt + θ1 ) =
2V cos(ωt + θ1 )
√ 1
v2 (t) = Vm2 cos(ωt + θ2 ) =
2V2 cos(ωt + θ2 )
1
(1)
(2)
where v1 (t) is the reference waveform with peak magnitude Vm1 (and rms magnitude V1 ), and v2 (t)
is a secondary waveform with peak magnitude Vm2 (or rms magnitude V2 ) and shifted by an angle
∆θ = θ2 −θ1 with respect to the first waveform. The secondary waveform v2 (t) is said to be lagging
the reference waveform v1 (t) if it peaks later in time as shown in the above figure. In this case, the
phase shift ∆θ (−180o < ∆θ = θ2 − θ1 < 0o ). The waveform v2 (t) is said to be leading the reference waveform v1 (t) if it peaks earlier in time with a positive phase ϕ (0o < ∆θ = θ2 −θ2 < 180o ).
The timebase method of phase measurements consists of displaying both waveforms simultaneously
on the screen and measuring the distance (in scale divisions) between two identical points on the
two traces. In Figure 2(a), this phase shift in degrees is determined from the relation
∆θ = 360o ×
D2
D1
= 360o ×
∆t
T
(3)
where D1 = T is the common period for both waveforms and D2 = ∆t is the time delay between
two zero crossings with rising (or falling) edges on both waveforms.
2.2
Lissajous Method
A
B
A
(a)
(b)
Figure 2: Phase Shift Computation Using Lissajous Patterns
The Lissajous-pattern method of phase measurement is also called the X-Y phase measurement.
To use this method, both signals are applied to two channels and the scope is then switched to the
X-Y mode whereby the reference signal is applied to the horizontal input and the secondary signal
is applied to the vertical input. A pattern known as a Lissajous pattern will appear on the screen.
This pattern can be used to compute the phase shift between the two waveforms.
The patterns shown above indicate phase relationships between the two waveforms. In order to
calculate the phase shift ϕ, it is necessary to center the pattern on the X-Y axis as shown in
Figure 2. The phase angle is obtained as follows for each pattern.
Pattern (a) : 0o ≤ ∆θ ≤ 90o =⇒ ∆θ =
sin−1
A
B
Pattern (b) : 90o ≤ ∆θ ≤ 180o =⇒ ∆θ = 180o − sin−1
2
(4)
A
B
(5)
R
~
I
i(t)
+
+
+
v i(t)
L
C
~
Vi
v o(t)
−
R
−
+
1
jω L
jω C
−
(a)
~
Vo
−
(b)
Figure 3: Resonant RLC Circuit in (a) Time Domain and (b) Phasor Domain
2.3
Resonant Circuit
Consider the above RLC circuit which is excited by a sinusoidal input
√
vi (t) =
2Vi cos(ωt + θ1 )
The output measurement is taken as the voltage across the LC parallel combination
√
vo (t) =
2Vo cos(ωt + θ2 )
The voltage gain or transfer function is obtained as the ratio of the output voltage to the input
voltage in the phasor domain
V˜o
Ṽi
=
|Ṽo | ̸ θ2
|Ṽi | ̸ θ1
=
√
=
jωL ∥ 1/jωC
R + jωL ∥ 1/jωC
ωL
2
(ωL) + R2 (1 − ω 2 LC)2
̸
=
tan−1
ωL
ωL − jR(1 − ω 2 LC)
R(1 − ω 2 LC)
ωL
(6)
(7)
Identifying Equations (6) and (7), the voltage gain and phase shift between the output and input
waveforms are
|Ṽo |
|Ṽi |
ωL
+ R2 (1 − ω 2 LC)2
R(1 − ω 2 LC)
∆θ = θ2 − θ1 = tan−1
ωL
The output voltage reaches a maximum with zero phase shift
=
√
(ωL)2
|Ṽo | = |Ṽi |
∆θ = 0
(8)
(9)
(10)
o
(11)
at the resonant frequency
ωo =
1
√
LC
=⇒ fo =
1
√
2π LC
(12)
This resonance property of circuits with inductors and capacitors can be used for component
measurement. At resonance, the impedance of the parallel LC combination reaches its highest
impedance (ideally an open circuit). The loading effect of the LC circuit is minimized at this
frequency and the output voltage is maximum.
3
3
Equipment
• Agilent DSO5014A Digital Storage Oscilloscope
• Agilent 33220A Function/Arbitrary Waveform Generator
• 330-mH Inductor, 0.068-µF Capacitor, 5.1-kΩ Resistor, Protoboard 942
4
Procedure
Part A: Timebase Method
Build the resonant RLC circuit of Figure 3(a) using R = 5.1 kΩ, L = 330 mH, and C = 0.068 µF.
Hook up vi (t) to channel 1 and vo (t) to channel 2 of the oscilloscope. Energize your circuit with a 2Vpp sine wave with variable frequency and zero offset. First, find the resonant frequency where the
output voltage is maximum and in phase with the input voltage. Record the maximum amplitude
of the output voltage at the resonant frequency and then record the following measurements below
and above the resonant frequency:
f (Hz)
Vi,pp (V)
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
Vo,pp (V)
0.75
1.00
1.25
1.50
1.75
1.75
1.50
1.25
1.00
0.75
4
T (ms)
∆t (µs)
Part B: Lissajous Patterns
Using the same setup as in Part A and for the same frequencies recorded, set up the X-Y (or versus)
mode on the infinium scope and observe a Lissajous pattern. Turn on the two pairs of scope cursors
and measure the quantities A and B for each of the frequencies recorded in Part A.
f (Hz)
Vi,pp (V)
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
Vo,pp (V)
0.75
1.00
1.25
1.50
1.75
A
B
1.75
1.50
1.25
1.00
0.75
Part C: Parameter Measurements
Measure the four parameters above using a shared RLC meter at 1 kHz.
Nominal
Measured
5
f (kHz)
1
1
R (kΩ)
5.1
L (mH)
330
C (µF)
0.068
Data Analysis and Interpretation
1. Using the measured value of C at 1 kHz and the resonant frequency fo , use Equation (12) to
find a value of L in mH. Compare this inductance value with the measured one made with
the shared RLC meter.
2. Compute the voltage gain G(f ) = |Ṽo |/|Ṽi | = Vo,pp /Vi,pp as a function of frequency f and
plot G(f ) as a function of frequency for Part A.
3. Compute the phase shifts ϕ(f ) = 360o × ∆t/T using the timebase method for Part A.
4. Compute the phase shifts ϕ(f ) = sin−1 A/B using the Lissajous method for Part B. (Add
a positive or negative sign to these phase shifts according to your observations in Part A.)
5. Plot both phase shifts ϕ(f ) = 360o × ∆t/T and ϕ(f ) = sin−1 A/B on the same graph.
6. Use the gain and phase plots to find the
√ two phase shifts between the input and output
waveforms where the gain is equal to 1/ 2 = 0.707.
7. Discuss the accuracy of the timebase and Lissajous methods in the discussion section of the
lab report.
5
Boise State University
Department of Electrical and Computer Engineering
ECE 212L – Circuit Analysis and Design Lab
Experiment #2: Sinusoidal Steady State and Resonant Circuits
Date:
Data Sheet Recorded by:
Equipment List
Equipment Description
Agilent DSO5014A Digital Storage Oscilloscope
Agilent 33220A Function/Arbitrary Waveform Generator
BSU Tag Number or Serial Number
Part A: Timebase Method
f (Hz)
Vi,pp (V)
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
Vo,pp (V)
0.75
1.00
1.25
1.50
1.75
T (ms)
∆t (µs)
A
B
1.75
1.50
1.25
1.00
0.75
Part B: Lissajous Patterns
f (Hz)
Vi,pp (V)
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
Vo,pp (V)
0.75
1.00
1.25
1.50
1.75
1.75
1.50
1.25
1.00
0.75
Part C: Parameter Measurements with an RLC Meter
Nominal
Measured
f (kHz)
1
R (kΩ)
5.1
L (mH)
330
C (µF)
0.068