Munashe Benhura
BNHMUN001
UNIVERSITY OF CAPE TOWN
Frequency Response of R-L
Circuits
EEE2041F/EEE2049W Online Teaching (OT)
Aim: To verify the predicted behaviour of first order R-L
circuits.
Deliverables: A report, consisting of the procedure for the
determination of the frequency response of RL circuit and
the experimental results, along with any tables and graphs
obtained from the laboratory session.
Brief Theory:
After initial conditions have died out, the theoretical response of a first order R-L circuit shown
⃗ 𝑖𝑛 = 𝑉𝑚 ∠0° is,
below to a sinusoidal input, 𝑣𝑖𝑛 = 𝑉𝑚 sin(𝜔𝑡) , 𝑉
√2
𝑉 𝐴
⃗ 𝐿 = 𝑚 ∠𝜙
𝑣𝐿 (𝑡) = 𝑉𝑚 𝐴 sin(𝜔𝑡 + 𝜙) , or 𝑉
√2
𝑗𝜔𝐿
𝐴=|
|
𝑗𝜔𝐿 + 𝑅
𝜔𝐿
𝜙 = 90° − 𝑡𝑎𝑛−1 ( )
𝑅
To express magnitude, we sometimes use a logarithmic measure called the deciBell – named
after the famous Bell laboratories 𝑨𝒅𝑩 = 𝟐𝟎 𝐥𝐨𝐠 𝟏𝟎 𝑨 (if it helps you to remember it, the original
definition of a Bell was log10 A2 ). Because of the wide frequency range in many applications,
we often use semilog graph paper to plot things. (Note that young humans can hear from 20 Hz
to 20 kHz (3 decades) and that hearing is also logarithmic:
0 dB
30 dB
60 dB
90 dB
120 dB
softest sound you can hear
whisper, quiet library
Normal conversation
Machinery, loud traffic noise (8 hours exposure per day without protection)
Very loud music in a club (15 minutes exposure per day without protection)
The angle ωt is in radians but we often express the phase shift ϕ in degrees, just make sure that
you sort things out before doing calculations!
1
The time constant of the circuit 𝐿/𝑅 = 𝜏 and the corner frequency is 𝛼 = 𝜏 = 𝑅/𝐿. Consider
the following frequencies
𝜔
0
𝛼 = 𝑅/𝐿
∞
A
0
1/√2
1
A in dB
-∞
-3 dB
0 dB
ϕ
90
45
0
In the experiment, you will connect up the circuit in Multisim, supply the input voltage using
variable frequency sinusoidal AC voltage source and measure the input and output
behaviour. Use a 2 V peak to peak (i.e. 0.707 Vrms ) sinusoid with no offset to drive the circuit.
Pre-Practical
Using the equations given above, populate the calculated values (Pre-Practical results) of A,
A (db) and ϕ (calc) in Table 2 using Vm = 1 V (peak), R = 1 kΩ and L = 10 mH.
Experimental Set-up:
Set-up the Multisim environment as directed from the practical video
(https://www.multisim.com)
Connect the circuit up on Multisim using the circuit diagram (Figures 1 and 2). Supply the
input of the circuit with a variable frequency sinusoidal wave from 2 kHz to 100 kHz, with a
peak amplitude Vm = 1 V (peak).
i)
ii)
iii)
iv)
v)
Components:
Resistor = (1 kΩ)
Inductor = (10 mH)
Probe
Ground
Variable frequency sinusoidal AC voltage
source = (1 V peak)
Circuit Diagrams:
Figure 1: The circuit diagram
Figure 2: The circuit connection in Multisim
Experiment:
1) Connect the circuit diagram in Multisim as in Figure 2
2) At each frequency from 2kHz to 100kHz, record the values ∆T, Vm(peak) and VL(peak)
using Figure 3 and Figure 4 as a guide and complete Table 1. Take note of the units and
use them appropriately (refer to the lab video for clarification)
𝐕(𝐩𝐞𝐚𝐤) =
𝑽(𝒑𝒆𝒂𝒌−𝒑𝒆𝒂𝒌)
𝟐
Figure 3: Recording values from measurement (∆T)
Figure 4: Recording values from measurement (V(peak)
Table 1
Frequency
Input
(f) kHz
T = 1/f [s] Vm(peak) [v]
2
4
8
10
20
40
80
100
0.0005
0.00025
0.000125
0.0001
0.00005
0.000025
0.0000125
0.00001
0.99555
0.99505
0.99625
0.9951
0.9962
0.99665
0.99545
0.9895
Output
VL(peak) [v]
0.125715
0.245445
0.44745
0.50955
1.4740
0.92565
0.9772
0.98285
∆T(Us)
0.00011454
0.000053375
0.000021978
0.00001684
0.0000054370
0.0000015545
0.00000042394
0.00000039530
From Table 1,
3) Calculate the phase difference, i.e. ϕ (measured) at each frequency using the formula;
Δ𝑇
𝜙(𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑) = 𝑇 ∗ 360°
4) Calculate A (measured) using the formula; 𝐴(𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑) =
𝑉𝐿(𝑝𝑒𝑎𝑘)
𝑉𝑚 (𝑝𝑒𝑎𝑘)
5) Complete the experimental results part of Table 2 from (3) and (4)
6) Plot a logarithm plot of the experimental results using excel or any appropriate means.
Make plots of (ϕmeas vs ω {k rad/s}) and (Ameas {dB} vs ω {k rad/s}).
7) Make appropriate comments on your observations and how it compares with the theoretical
analysis.
Table 2
Frequency
Pre-Practical Results
Frequency,ω
[k rad/s]
A (calc)
A (calc)
dB
2
12.57
0.1257
98.01
89.23
0.125715
4
25.13
0.2513
91.996
88.56
8
50.27
0.5027
85.97
10
62.83
0.6283
20
125.66
40
VL
(rms)
[V]
ϕ (measured)
[degrees]
A (meas)
A (meas)
dB
0.0888
82.47
0.1263
17.97
0.245445
0.1735
76.86
0.2467
12.15
87.122
0.44745
0.316
63.30
0.449
6.95
84.03
86.40
0.50955
0.360
60.62
0.512
5.81
0.012566
78.01
82.83
1.4740
1.04
39.15
1.47
3.34
251.32
0.25132
72.00
75.89
0.92565
0.65
22.38
0.928
0.64
80
502.65
0.50265
65.97
63.31
0.9772
0.693
12.21
0.98
0.18
100
625.32
0.62532
64.08
57.85
0.98285
0.694
10.23
0.9932
0.06
Frequency, f
[kHz]
ϕ (calc)
[degrees]
Experimental Results
VL(peak)
[V]
Δ𝑇
𝑇
∗ 360° (with the correct
ΔT
Measuring phase: The 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 [𝐻𝑧] = 1/𝑇. The phase is 𝜙 =
sign, the output below LEADS the input and ϕ is positive 30)
T
Comments
The measured results were coherent with the calculated but with small differences .The
theoretical predictions about this experiments were obeyed in the practical .As the frequency of
the AC voltage source is increased it was observed that the Voltage graphs had a very small
phase differences which was predicted from the calculations.