Experiment 5
Resonant Circuits and Inductors
____________________
Name: Kenny Lu
UID: 903801866
Partner: Michael Chow
1. Introduction:
The purpose of this lab is to develop an understanding of how to effectively use an
oscilloscope, resonance, bandwidth, RLC circuits, frequency response, and how to calculate
quality factor and mutual inductance.
2. Objectives:
The main objectives of this lab are to measure the frequency response RLC circuits,
resonance and bandwidth, how to calculate the quality factor, how to find mutual inductance
between two inductors, and to become more familiar with the digital oscilloscope, its operation,
and applications.
3. Theory:
1. Resonance
Resonance describes the strong response of a circuit when it is excited by a sinusoidal
source at a frequency close or equal to the natural frequency of the circuit. A resonant
frequency is the frequency at which the imaginary part of the impedance of the circuit goes to
zero.
2. Series Resonant Circuits
At low frequencies, the capacitive reactance is much larger than the resistance of the circuit,
so very little current flows through the circuit. At high frequencies, the inductive reactance is
much larger than the resistance of the circuit, so very little current flows through the circuit.
Thus, only at mid-range frequencies, will the series reactance cancel and be low enough that
a fraction of current will pass through the resistor, causing a voltage V0.
Resonant frequency
w0 = 1/((LC))^(1/2)
Cutoff frequencies
wc1 = -R/2L +(((R/2L)2 + 1/(LC)))^(1/2)
wc2 = R/2L +(((R/2L)2 + 1/(LC)))^(1/2)
Bandwidth
β = wc2 – wc1 = R/L
Quality factor
Q = (L/CR2)^(1/2)
3. Parallel RLC Circuits
Similar to the series RLC circuits, only at mid-range frequencies, will the parallel admittance
cancel and be low enough that a fraction of the current will pass through the resistor, causing
a voltage V0.
Resonant frequency
w0 = 1/((LC))^(1/2)
Cutoff frequencies
wc1 = -1/2RC +(((1/2RC)2 + 1/(LC)))^(1/2)
wc2 = 1/2RC +(((1/2RC)2 + 1/(LC)))^(1/2)
Bandwidth
β = wc2 – wc1 = 1/RC
Quality factor
Q = (R2C/L)^(1/2)
4. Bandwidth
Bandwidth is a range of frequencies which a system will pass or attenuate. It is the difference
between the upper and lower frequencies at which the system passes half of the input power
to the output. These upper and lower frequencies are those frequencies when the magnitude
of the voltage or current gain is 0.707 of the maximum voltage or current gain.
Bandwidth β = whi - wlo
5. Quality factor Q
Quality factor is the ratio between the resonant frequency and the bandwidth. A network with
a large Q is very frequency selective and has a sharp peak in its gain vs. frequency curve. A
network with a low Q has a wide bandwidth and has a low peak. In most real-life situations,
high values of Q are desirable.
Quality factor Q = wresonnant/β
6. Mutual Inductance
Faraday’s law states that the induced voltage in a coil equals the rate of change of the flux
times the number of turn in that coil.
v(t) = N*dΦ(t)/dt
Mutual induction is a means of energy transmission without the needing for contact between
terminals by conductors. Examining the voltage induced across one coil by the flux generated
by another coil,
V1(t) = Ldi1/dt
V2(t) = N1N2α21di1/dt = M21di1/dt
V2(t)/V1(t) = M21/L1
where M is the mutual inductance of the system.
4. Procedures, Results, and Discussions:
4.1 Part 1: Series LCR Circuit
4.1.1 Procedure:
Create the circuit shown below and measure the input/output characteristics across
the resistor. Measure the frequency, input and output voltages, time delay between
the two waveforms, the period of the two waveforms, and calculate the phase shift
and Vo/Vi. Find the resonant frequency of the circuit.
4.1.2 Results
Measured Values
R
101.8 Ohms
C
10.5 nF
L
3.4 mH
Calculated resonant frequency using measured L and C:
1
𝑓=
= 26.637 𝑘𝐻𝑧
2𝜋√𝐿𝐶
f (kHz)
Vi (V)
Vo (V)
14.21
21.2
3.2
16.75
21.0
3.65
18.00
20.5
4.32
20.27
20.4
5.76
24.75
16.6
10.6
26.28
15.2
12.7
27.22
15.4
12.6
28.00
16
11.6
29.44
17.2
10.2
31.45
18.4
8.80
34.82
19.6
6.8
38.16
20.2
5.4
41.80
20.2
4.6
Vo/Vi
(mdB)
td (x10^-6 s) T (s)
Phase (Degrees)
0.151
16.8 0.0003932
85.94
0.175
14 0.0001982
84.42
0.234
12.2 0.0001173
79.05
0.31
8.8 0.0000996
64.22
0.646
4 0.0000765
35.64
0.83
0.8 0.0000826
7.56
0.818
1.2 0.0000955
-11.76
0.725
2.6 0.0000907
-26.21
0.593
4
0.000089
-42.39
0.478
5 0.0000868
-56.61
0.347
5.4
0.000084
-67.69
0.267
5.4 0.0000835
-74.18
0.228
5.5 0.0000815
-82.76
45.8
20.6
4.0
0.194
5.5
0.0000788
-90.68
Sample Vo/Vi calculation for f = 26.28 kHz:
𝑉𝑜 12.7
=
= 0.8355
𝑉𝑖 15.2
Sample Phase Calculation for f = 26.28 kHz:
𝑡𝑑
0.8𝑥10^ − 6
𝜃 = 𝑥360𝑜 =
𝑥360𝑜 = 7.56𝑜
𝑇
1/26280
Vo/Vi
Vo/Vi vs Frequency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
40
50
Frequency (kHz)
Phase
Phase vs Frequency
100
80
60
40
20
0
-20 0
-40
-60
-80
-100
4.1.3 Discussion
10
20
30
Frequency (kHz)
For this circuit, we calculated the resonant frequency to be 26.637 kHz. Out closest
frequency 26.637kHz in the data set is 26.28kHz, where the measured phase was
7.56 degrees. At this frequency, the phase is minimum and the gain is maximum.
After substituting different resistors, we found that the resonant frequency did not
change, as it should not because the formula for calculating resonant frequency does
not contain a term for resistance.
4.2 Part 2: Series LRC Circuit
4.2.1 Procedure:
Using the same resistor, capacitor, and inductor as in part 4.1.1, create the circuit
shown below to measure the input/output characteristics across the capacitor.
Measure the frequency, input and output voltages, time delay between the two
waveforms, the period of the two waveforms, and calculate the phase shift and
Vo/Vi. Find the resonant frequency of the circuit.
4.2.2 Results
f (kHz)
15.78
17.9
19.23
21.05
22.85
24.81
25.84
26.18
27.51
28.58
30.04
30.70
34.71
38.50
40.61
Vi (V)
19.2
20.0
20.6
20.0
19.0
16.8
16.0
15.2
16.0
16.8
18.0
19.6
20.0
21.2
21.0
Vo (V)
30.4
36.0
40.8
48.0
57.6
70.4
73.6
72
69.6
62
52
40.8
29.6
20.8
18.2
Vo/Vi
1.58
1.8
1.98
2.4
3.03
4.19
4.6
4.74
4.35
3.69
2.89
2.08
1.48
0.98
0.87
td (x10^-6 s)
-1.7
-2.1
-2.2
-2.4
-3.6
-6
-7.8
-8.6
-10.6
-12.0
-12.4
-12.6
-12.6
-12.2
-12.2
T (s)
6.33714E-05
5.58659E-05
5.20021E-05
4.75059E-05
4.37637E-05
4.03063E-05
3.86997E-05
3.81971E-05
3.63504E-05
3.49895E-05
3.32889E-05
3.25733E-05
2.88101E-05
2.5974E-05
2.46245E-05
Phase (Degrees)
-9.65736
-13.5324
-15.23016
-18.1872
-29.6136
-53.5896
-72.55872
-81.05328
-104.9782
-123.4656
-134.0986
-139.2552
-157.4446
-169.092
-178.3591
Gain (Vo/Vi)
Gain vs Frequency
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
10
20
30
40
50
40
50
Frequency (kHz)
Phase (deg)
Phase vs Frequency
0
-20 0
-40
-60
-80
-100
-120
-140
-160
-180
-200
10
20
30
Frequency (kHz)
4.2.3 Discussion
We found that this circuit behaved as it was predicted to. The phase started at 0o and
became larger (in the negative direction) as the frequency was increased. We also
found that the calculated resonant frequency was close to the measured resonant
frequency as the closest measured data point was 26.18 kHz and the phase was 81.05o which is close to -90o because the phase was changing quickly relative to the
frequency.
4.3 Part 3: Series RCL Circuit
4.3.1 Procedure:
Create the circuit shown below using the same elements to measure the input-output
characteristics across the inductor. Measure the frequency, input and output
voltages, time delay between the two waveforms, the period of the two waveforms,
and calculate the phase shift and Vo/Vi. Find the resonant frequency of the circuit.
f (kHz)
Vi (V)
Vo (V)
Vo/Vi
17.07
19.21
20.25
22.22
23.32
24.51
25.96
26.99
28.14
29.84
31.22
33.31
35.85
37.56
39.42
42.6
20.8
20.8
20.2
19.6
18.4
17.4
16
15.4
16.4
18
18.8
19.8
20.4
20.6
20.8
20.8
14
20.4
25
36.6
45.6
56.4
68.4
72.2
71.2
64
57.2
49.4
43.2
40
37.2
34
0.673077
0.980769
1.237624
1.867347
2.478261
3.241379
4.275
4.688312
4.341463
3.555556
3.042553
2.494949
2.117647
1.941748
1.788462
1.634615
td (x10^-6
s)
27.6
23.6
21.6
17.6
17.2
14
10.8
8.4
5.6
3.6
2.8
2
1.4
1.2
1.1
0.9
T (s)
5.85823E-05
5.20562E-05
4.93827E-05
4.50045E-05
4.28816E-05
4.07997E-05
3.85208E-05
3.70508E-05
3.55366E-05
3.35121E-05
3.20307E-05
3.0021E-05
2.7894E-05
2.66241E-05
2.53678E-05
2.34742E-05
Phase
(Degrees)
169.6075
163.2082
157.464
140.7859
144.3974
123.5304
100.9325
81.61776
56.73024
38.67264
31.46976
23.9832
18.0684
16.22592
15.61032
13.8024
Gain (Vo/Vi)
Gain vs Freqeuncy
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
10
20
30
40
50
40
50
Frequency (kHz)
Phase (deg)
Phase vs Freqeuncy
180
160
140
120
100
80
60
40
20
0
0
10
20
30
Frequency (kHz)
4.3.3 Discussion
We found that this circuit behaved as it was predicted to. The phase started close to
180o and decreased to 0o as the frequency was increased. We also found that the
calculated resonant frequency was close to the measured resonant frequency as the
closest measured data point was 26.99 kHz and the phase was 81.62o which is close
to 90o because the phase was changing quickly relative to the frequency.
4.4 Part 4: Parallel RLC Circuit
4.4.1 Procedure:
Create the circuit shown below. Measure the input voltage provided by the source,
and the voltages across RL and RC. Using the voltages measured and the values of
the two resistors, calculate the current through the inductor and capacitor.
4.4.2 Results
Measured Values
R
988 Ohms
C
10.5 nF
L
3.4 mH
RC
2.6 ohms
RL
2.5 ohms
Calculated resonant frequency using measured L and C:
1
𝑓=
= 26.637 𝑘𝐻𝑧
2𝜋√𝐿𝐶
The current in the capacitor is found by dividing the voltage Vrc by RC:
Ic = Vrc/RC
Similarly, the current in the inductor is found by:
IL = Vrl/RL
frequency
(kHz)
Vi (v)
Vrc (mv)
Vrl (mv)
13.22
20.2
44
Ic (A)
IL (A)
84
0.0176
0.0336
16.64
18.55
19.16
21.15
23.1
24.63
25.5
26.6
27.37
34.76
37.57
41.69
44.21
20.2
20.4
20.4
20.4
20.8
20.8
20.6
20.8
20.8
20.4
20.4
20.2
20
52
60
60
72
84
92
104
104
104
100
64
52
48
88
82
96
100
104
108
104
104
100
96
94
90
84
0.0208
0.024
0.024
0.0288
0.0336
0.0368
0.0416
0.0416
0.0416
0.04
0.0256
0.0208
0.0192
0.0352
0.0328
0.0384
0.04
0.0416
0.0432
0.0416
0.0416
0.04
0.0384
0.0376
0.036
0.0336
Capacitor and Inductor Current vs
Frequency
Current (amperes)
0.05
0.04
0.03
capacitor current Ic
0.02
inductor current IL
0.01
0
0
10
20
30
40
50
Frequency (kHz)
4.4.3 Discussion
For this circuit, the calculated resonant frequency was 26.637 kHz. Looking at the
graph, we see that the inductor current and capacitor currents reach a peak value
around 27kHz as expected. According to the data, at 26.6kHz, the curves meet and
both currents have a value of 0.0416 amperes.
4.5 Part 5: Mutual Inductance
4.5.1 Procedure:
Create the circuit shown below. Measure the distance between the two inductors,
the input and output voltages and the mutual inductance.
4.7.2 Results
Measured Values
L1
3.34 mH
L2
3.34 mH
𝑀=
𝑀12
√𝐿1 𝐿2
𝑉𝑜
∗𝐿
𝑉𝑖
= 0.0065
1
𝑀 = 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 ( )𝑛
𝐷
D (cm)
Vi (V)
Vo (mV) M (H)
1.4
8.72
56.8
0.0000221
1.5
8.48
44
0.0000176
1.8
8.4
38.4
0.0000155
2.1
8.4
33.6
0.0000136
2.7
8.4
32
0.00001295
3.9
8.4
28.8 0.000011657
5.6
8.4
26.4 0.000010686
6.1
8.4
22.4
9.0667E-06
Through the calculations of solving for n, the value of n was found to be approximately 3. It is
the value that best fits the data.
Inductance vs. Distance
0.000025
Inductance (H)
0.00002
0.000015
0.00001
0.000005
0
0
1
2
3
4
5
6
7
Distance (cm)
4.7.3 Discussion
The mutual inductance was found to scale cubically with the reciprocal of the
distance between the two coils. In addition, the constant α was found to be 0.0065.
Mutual inductance is a safer alternative than plugging a power cord into the car
because the same effect is achieved without direct connection which may cause fire.
5. Conclusion:
The objectives set for this experiment were accomplished by verifying the theoretical
values of the resonant frequencies of various series and parallel RLC circuits and calculate
bandwidth and quality factor. In addition, the relationship between mutual inductance and
distance was verified.