Electric Circuits Laboratory 5:

AC Signals and RC Circuits

ELEE 2790U - Fall 2022

Teacher Assistant: Leon Wu

Nov. 20th, 2022

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Objectives:
● Measure capacitive reactance using Ohm’s law.
● Show how to calculate impedance and phase angle in series RC circuits.
● Calculate impedance, current, and voltages in series RC circuits.
● Discuss the behaviour of sinusoidal waves in series RC circuits.
● Study the characteristics of passive low-pass filters by obtaining the frequency response.
● Understand the time domain behaviour of RC circuits.

Components:
● Oscilloscope
● Function Generator
● DT Digital Multimeter
● Advanced breadboard
● Resistors: 220 Ω
● Capacitors: 0.01 nF, 1 uF
● Connecting wires

Instructions and Steps:

Section 5.4.1
1. Set up the circuit shown in Figure 5-2 with component values R = 200 Ω, C = 1 μF.
2. Connect the Function Generator with probe selection at X1. Apply 1 V peak-peak
sinusoidal signal of 10 kHz as the input voltage to the circuit. (Note: look at the front
panel of the Function Generator and identify the buttons you need to press in order to
select the signal type and its characteristics such as amplitude and frequency). Make sure
the Output button of the Function Generator is on. Otherwise, no signal will be sent to the
Oscilloscope.
3. Connect the Oscilloscope channel 1 to the output terminals across the capacitor, measure
the output signal Vpp voltage, and record the measured RMS (convert to RMS value
from peak to peak value) in Table 1 (columns 1 and 2). When using the Oscilloscope,
make sure the CH 1 button is on. Then, press the Autoscale button. Now, you should
have the sinusoidal signal appear on the screen. Next, press the Single button to freeze
the signal and press the Quick meas button to measure Vc, pp (peak to peak voltage
across the capacitor) and Vc, RMS (RMS value across the capacitor). Now, you should
have the measurements appear just above the Toolbar on the screen. If not, press the
Select: soft button (second button from the left on the Toolbar) to choose the desired
measurement (i.e. Amplitude). Then, press the Measure Ampl soft button (third button
from the left on the Toolbar) to bring the measurement to the screen. Note: Since you are
using X1 as the Function Generator probe setting, (ONLY) the amplitude measurement
that appears on the Oscilloscope screen is twice as much as the actual amplitude.
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Therefore, you have to divide every measured voltage value (both Vpp and Vrms) by a
factor of 2. Repeat the same procedure for other desired measurements.
4. Repeat the above two steps for another signal of 8 V peak-peak signal with a frequency
of 20KHz.
5. Repeat the above steps with the components reversed to measure the peak-peak and RMS
voltages across the resistor and record these values in Table 5-2 (columns 3 and 4).

Prelab and Data:

MultiSim

2.1 Low Pass Filter:

Plot 2.1

It has a magnitude of -3.00 dB and a phase angle of -45 degrees around 796 Hz, which is about
the cutoff frequency.
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2.3 High Pass Filter:

The magnitude is around -3.00 dB and the phase angle is around 45 degrees at the cutoff
frequency of roughly 796 Hz.

Comparisons between plots 2.1 and 2.3

In both cases, frequency vs. magnitude plots are similar, with the high-pass filter plot flipping the
low-pass filter plot vertically. In contrast, there is no difference between the frequency vs. phase
angle plots in either case except that the low pass filter phase angle values are negative, while the
high pass filter phase angle values are positive.
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3.3 Calculate rise and fall time

Section 5.4.2

1. Set up the circuit shown in Figure 5.2 with the component values R = 200 Ω, C = 1μF.
Compute the 70 % of Vpp and obtain the frequency at which this occurs on the
Oscilloscope. This gives the cut-off (roll-off) frequency for the constructed Low Pass RC
filter. What is the cut-off frequency? Apply a 1 V peak-peak sinusoidal signal to the
circuit as input voltage using the Function Generator. Make sure the Output button of the
Function Generator is on. Otherwise, no signal will be sent to the Oscilloscope.
2. Connect the Oscilloscope to the Source (input) on Channel 1 and the output signal on
Channel Press the Autoscale button to measure the peak-to-peak values of the two
signals. Note: Since you are using X1 as the Function Generator probe setting, (ONLY)
the amplitude measurement that appears on the Oscilloscope screen is twice as much as
the actual amplitude. Therefore, you have to divide every measured voltage value (both
Vpp and Vrms) by a factor of 2.
3. Vary the frequency of the function generator panel according to Table 2. Observe the
response on the Oscilloscope in each case and note down the output voltage (Vpp) for the
corresponding input frequency in a tabular column as shown in Table 5-3.

Section 5.4.3
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1. Set up the circuit shown in Figure 5.2 and select the square wave signal of 1V peak to
peak on the function Generator. Make sure the Output button of the Function Generator is
on. Otherwise, no signal will be sent to the Oscilloscope.
2. Observe the output signal waveform by superimposing it on the input signal. Locate the
charging and discharging parts of the output signal for each frequency 200 Hz, 700 Hz,
800 Hz, and 1 kHz. Note: Since you are using X1 as the Function Generator probe
setting, (ONLY) the amplitude measurement that appears on the Oscilloscope screen is
twice as much as the actual amplitude. Therefore, you have to divide every measured
voltage value (both Vpp and Vrms) by a factor of 2.
3. Measure and record the rise and fall times which are defined as the time between the 10
and 90% points of the graph.

Table 5-1: XC and phase angle for various frequencies

Frequency 0.2 0.5 0.6 0.7 0.8 1.0 2.0 3.0 4.0 5.0 10.0
(kHz)
Xc 795.7 318. 265. 212. 188.9 159.16 79.577 53.05 39.789 31.83 15.95
7 31 26 21 4 2 1 1

Phase - - - - - - - - - - -4.549
75.89 57.8 52.9 46.6 44.84 38.51 21.69 14.85 11.251 9.042
angle (θ) 2 58 84 96 8 1 6 5

The plot is an instance of exponential decay

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The plot is an instance of negative exponential decay

5.4 Lab Tasks

Frequency Vpp Vrms Vpp Vrms Load

across the across the across the across the current Capacitive
capacitor capacitor resistor resistor iL=(Vr,rms reactance
(Vc,pp) – (Vc,rms) – (Vr,pp) – (Vr,rms) – / (Xc) –
V V V V R) – mA
(1) (2) (3) (4) (5) Ohms
(6)

10 KHz 300 mV 120 mV 344 mV 117 mV 0.62 mA 15.92 Ω

20 KHz 0.313 V 1.04 V 3.00 V 1.02 V 5.11 mA 7.96 Ω

Table 5-3: Voltage across the resistor using KVL

Frequency Vpp Vpp IL (mA)

source Vpp resistor (V) Vrms resistor
(V) capacitor (V) (V)

10 KHz 1 0.079 0.90 0.312 1.59

20 KHz 8 0.310 7.64 2.72 13.2

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Since capacitors and inductors have opposite phasor angles, a Source Resistor - Inductor is a
Low Pass Filter and a Source - Inductor - Resistor is a High Pass Filter. Calculations and
experiments also indicate that capacitive reactance decreases with frequency. The relationship is
inversely proportional, meaning that faster source cycles create less resistance than capacitors do.
Inductors, on the other hand, resist changes in current and therefore grow in resistance with
frequency. When a conductor undergoes more cycles, there is more change occurring within the
current at any given moment, resulting in more resistance.

Frequency 0.2 0.5 0.6 0.75 0.8 1.0 2.0 3.0 4.0 5.0 10.0
(kHz)
Output 980 815 765 670 645 582 350 255 197 183 82.4
Voltage mV mV mV mV mV mV mV mV mV mV mV
(V)- Vpp

Attenuation 0 - - -3.30 -3.61 -4.52 - - - - -

(dB) 20*log 1.6 2.14 8.90 11.6 14.0 15.7 21.35
0 7 2 6
(Vout/Voutmax)

Discussions:

Section 5.4.1:
1. Calculate load current and enter the value in column 5 in table 5-2. Does the load current
increase or decrease with frequency? Why?

As represented by the calculations above, the load current values increase as the values of
the frequency increase. This indicates that the relationship is proportional since as one
value increases, the other value increase throughout the experiment. This may be because
as frequency increases, more power and energy are required, and thus the value of the
load current (iL) increases.

2. Capacitive reactance Xc (column 6) is calculated by dividing the Vc, RMS value of the
capacitor by the load current (Xc = Vc, RMS/ iL). Calculate the capacitance value in each
case using the formula given in equation 5.2 and then compare it with the nominal value
and also the actual measured value (measure it with the RLC meter available with your
TA). Compare these results. Are they equal?

The values above are very identical to what was observed in the lab experiment session.
The capacitance resistance (Xc) values are roughly half when observing from the 10 kHz
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value to the 20 kHz value. Therefore, the relationship between frequency and capacitive
resistance is inversely proportional.

Section 5.4.2:
1. Determine the slope of the roll-off for 1 decade from the graph. (Slope is calculated by
using the two values of gain at two frequencies which are separated by 10 times. Is it
within the theoretical limit of 20 dB/ decade?

The Slope (m):

= (Gain at 10.0 kHz - Gain at 1 kHz) / 1 decade
= (-2.4756 - (-1.2697)) / 1 decade
= -1.2059/decade
According to the computations done, yes the slope falls within the threshold of 20
dB/decade.

2. Plot the frequency response of the filter (attenuation) and find the cut-off frequency of the
circuit from the plot. Is it comparable with the calculated value?
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Analysis
1. What do you understand by reactance – capacity and inductive? Explain how reactance
changes with respect to frequency.

Reactance is the opposition of an element to current or voltage change. Inductive

reactance is when the inductor is reacting to the current changing and it is measured in
ohms and capacitive reactance is the opposition that changes the voltage across an
element. When looking at the inductance and frequency it is inversely proportional.

2. What is the time constant of a circuit? Calculate the time constant of the RC circuit and
how it is related to rise and fall times.

The time constant of a circuit is the product of the resistance and the capacitance is (C)
this represents the time needed to change a capacitor through a resistor. The time constant
is τ = (200 Ω) (1 μF) = 0.0002 seconds. The time constant is related to rise time and fall
time because rise time happens when the capacitor is starting to charge up and fall time is
when the capacitor discharges.

3. Obtain the frequency response of the RC filter using its transfer function (Use Matlab).

4. Compute the Cut-off frequency for the RC filter using the formulae in equations (6.4).
Compare these theoretical values to the ones obtained from the experiment and provide a
suitable explanation for any difference.
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The cut-off frequency values for the RC filter using the formula are calculated by fc =
1/(2*Pi*Ceq*Req). Obviously, it is apparent that the cut-off frequency is inversely
proportional to the values of the capacitor and the resistor. By utilizing the equation, it is
noticeable that the theoretical values and the obtained values from the experiment are
very equivalent.

5. Graph the Frequency Response of the filter built in the lab using Matlab scripts. (Use the
values recorded in Table 1).

6. Explain the step response of the RC circuit measured in the lab under step 3 of the lab
section.

The step was to observe the effects of the waves of the signals with the 1 V peak. The
Function Generator was used to conduct the experiment as well as changing the variables
and values of the frequencies changing from 200 Hz to 700 Hz, 800 Hz, and even 1 kHz.
While just X1 was utilized as the Function Generator, the amplitude portrayed on the
Oscilloscope screen is seen as twice the actual amplitude. All in all, this step was
conducted to visualize the effects of these variables altering and what occurs as one
variable is changed with the values.

7. Explain any difficulties you had with this lab experiment. (Please include suggestions to
improve the experiment, if you have them).

Some of the difficulties we had with this lab was trying to figure out how to use the
oscilloscope. It took us a good amount of time to get the correct waves. To help with
difficulties there could be a small brief explanation at the beginning of the class on how
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to set up the oscilloscope so it can speed up the correct setup process. Another issue we
had was determining our slope on the X1 machine.

Conclusion:

During this lab, we primarily focused on the first-order RC circuit, the Low Pass Filter. The
function generator and oscilloscope were required to create the AC signals and measure their
output. A capacitor in an RC circuit has a complex reactance value that shifts the circuit's phasor.
Capacitors always produce a fully imaginary reactance causing the Current to leads voltage by
90 degrees but their value is based on the frequency of the source. As the frequency increases in
a Low Pass Filter like the one analyzed, the capacitor offers less reactance and phase angle.
When charging a capacitor, the specific amount of time it takes to reach approximately 70% can
be expressed as a time constant that is unique to each circuit. Capacitors reach positive and
negative charges simultaneously because rise and fall times are equal.