ECE 3550 - Practicum
Fall 2006
Lab 1 - Single Time Constant RC Circuits
Objective
The objective is to measure the time constant of typical RC circuits, and to examine the
correlation between time constant and frequency response using the oscilloscope as the
measuring device.
Parts
Resistors: 10 kΩ 5 % tolerance
Capacitors: 0.01 μF (notation: 103Z, typically measure around 9 nF to 10.5 nF)
Preparation:
Refer to the basic equations and definitions shown in Appendix 3.
1. Determine the time constant, τ, of the circuit in Fig. 1. Give result in μs (microseconds).
2. Suppose a capacitor of equal value is placed in series with the given capacitor.
Determine the new time constant.
3. Suppose a capacitor of equal value is placed in parallel with the given capacitor.
Determine the new time constant.
4. Compute the corner or break frequency, fb, of the circuits in parts 1, 2 and 3 above. Give
the unit for each result.
5. The output voltage divided by the input voltage (or gain) of the circuit in Fig. 1 is 0.447
when the input sinusoid has a frequency twice the break frequency. Determine the gain,
in decibels (dB).
Procedure
1) We wish to measure time constant for the circuit in Fig. 1, using as an input, a 0 to 3
volt square wave obtained from the function generator (FG) at a frequency of 1 kHz. Collect
components and measure all component values. Refer to Appendix 2 to properly set the FG
display.
Set the FG to externally trigger the oscilloscope (scope) by connecting a coax cable from
the SYNC terminal of the FG to the Ext Sync terminal on the rear panel of the scope. Then in
the Trigger section of the scope press Edge and select Ext.
R1
10 k
C1
Input
Output
0.01uF
Figure 1
Lab 1 Single Time Constant RC Circuits july 28 2006
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Place both scope probes on the input of the circuit. Select Auto Scale. Set the vertical
position controls so that ground for both channels is in the middle of the screen. Signal ground is
indicated by the following symbol shown at the left of the screen – it is very small.
You should see two cycles of a square wave; each wave starting at ground and going
positive, with an amplitude of 3 divisions.
Now move the CH 2 probe to the output of the circuit. You should see a signal
resembling the input but rising to the 3-volt input-voltage level, and then falling to the 0-volt
input-voltage level. The time it takes to rise (or fall) to within 1/e of the final voltage level is the
time constant of the circuit. Use Cursors for the following measurements:
a) Set the X cursors to measure the half-period of the input signal. Record X1 and X2.
b) Determine the time constant, τ, as follows. Set Y1 at 0.00 V and Y2 at 1.90 V1.
Record the value of X1 and X2. The time constant can be determined from X2 minus
X1.
2) Increase the input frequency to 10 kHz. Sketch the output signal and its relationship to
the input signal. Record the peak-to-peak (p-p) value and the slope of the signal. What is the dc
level of the input? Of the output? There are two ways to determine dc level: (1) rotate the
vertical position knob until the signal straddles the ground line – read the value in the little
window, or (2) use the Average function within the Quick Meas menu.
3) Change the input to a triangle wave. Sketch the waveform noting the p-p value and the
dc level of the output. Is the output a sinusoidal wave?
4) Reset the input to a 1 kHz, 0 to 3 volt square wave. Measure the rise time of the
output two ways: one, using Cursors, and the other by selecting Rise Time from the Quick Meas
menu. Determine the time constant from the rise time measurement.
5) Change the input to a 3 volt p-p sine wave. Looking at both the input and output
signals with the scope probes, vary the input frequency from 100 Hz to 10 kHz. You should
observe that the output signal decreases in amplitude as the frequency increases. This is the
reason this circuit is called a low pass filter. You should also observe that the output lags behind
the input – this is called phase lag.
Set the input frequency at 1.6 kHz. From the Quick Meas menu, select Peak-Peak to
determine the p-p amplitude, and select Phase (1→2) to determine the phase. Note: The
displayed phase will be positive; however, you should record it as a negative value, meaning that
the output of this circuit lags the input.
Next measure the amplitude (p-p value) and phase of the output as the frequency of the
sinusoidal input is varied from 50 Hz to 100 kHz. Note: Keep the input amplitude constant as
1
The theoretical value is 3(1 – e-1) = 1.896 ≈ 1.9
2
you vary the frequency of the FG. Make measurements at these frequencies: 50 Hz, 1kHz, 2
kHz, 5 kHz, 10 kHz, 20 kHz, 40 kHz, 50 kHz, and 100 kHz. Note: As you increase the input
frequency it becomes more difficult to determine the phase using Quick Meas. Instead, I
recommend using the Cursors method (see Appendix 1).
Voltage gain equals output voltage divided by input voltage. Use your data to determine
gain in volts/volt as well as in decibel (dB) for each frequency measurement. Put the data in
tabular form. Make a sketch, in your notebook, of the dB gain versus frequency plot as well as
the phase versus frequency plot. The plots should have a log axis for frequency. From your dB
gain plot, determine the slope (in dB per decade or dB per octave) of the high-frequency gain
roll-off. How does it compare with the theoretical value?
6) Interchange the resistor with the capacitor in your circuit. The output now is across
the resistor. Drive the circuit with a 0 to 3 volt square wave at a frequency of 1 kHz. The output
waveform now shows a positive spike and then a negative spike. Sketch the input and output
waveforms in your notebook, showing key amplitudes and times. Can you explain the shape of
the output waveform? Use Cursors to measure the 37 % time, t37%, which is the time required for
the output to fall from its peak value to 37 % of its peak value. What is the significance of this
time?
Change the input to a triangle wave. Make a sketch of the input and output waveforms.
You will have to increase the vertical sensitivity of your scope to observe the output.
7) Change the input to a 3 volt p-p sine wave (zero offset). Looking at both the input and
output with the scope probes, vary the input frequency from 5 kHz down to 100 Hz. You should
observe that the output signal decreases in amplitude as the frequency decreases. This is the
reason this circuit is called a high pass filter. You should also observe that the output leads the
input – this is called phase lead.
Keep the input amplitude at 3 volts p-p. Measure the amplitude (p-p) and phase of the
output at 1 kHz, 500 Hz, 200 Hz, 100 Hz and 50 Hz.
Experimentally determine the frequency where the phase is 45 °. This frequency is the
break frequency, fb. Measure the amplitude (p-p) and frequency.
Use your data to determine the gain in volts/volt as well as in decibel (dB) for each
frequency measurement. Put your data in tabular form. Make a sketch, in your notebook, of the
dB gain versus frequency plot as well as the phase versus frequency plot. The plots should have
a log axis for frequency. From your dB gain plot, determine the slope (in dB per decade or dB
per octave) of the low-frequency gain roll-off. How does it compare with the theoretical value?
Report
The report consists of four sections, corresponding to items A), B), C) and D) below.
Number these sections that way.
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A) In part 1, the input voltage jumps from 0 to 3 volts. Write the differential equation for
the circuit and express the output voltage as a function of time, i.e., vout(t) = ?. Evaluate this
expression at t = τ and t = 4τ.
Compare the predicted and experimental values of time constant and comment on
discrepancies.
B) Derive an expression that shows that the output, vout(t), is proportional to the integral
of the input for the circuit in Fig. 1, if the duration of each half cycle of the square wave input is
much smaller than the time constant.
From the previous result, compare the p-p amplitude with the measured p-p amplitude of
the output from part 2.
C) Give the transfer function TLP(jω) = Vout / Vin = A(ω) ejφ(ω) for the circuit in Fig. 1.
Derive the formula fb = 0.35/tr where fb is the break frequency and tr is the rise time.
How do the measured values of break frequency and rise time compare?
Compute theoretical values for A(ω) (in dB) and φ(ω) for these frequencies: 50 Hz, fb, 20
kHz, 40 kHz and 80 kHz. Make up a table comparing these predicted values with the measured
values from part 5.
Produce, on semi-log graph paper, the dB magnitude response plot and the phase
response plot for the data in part 5. Comment on the slope of the magnitude response curve for
frequencies greater than the break frequency.
D) Give the transfer function, THP(jω) = A(ω) ejφ(ω), for the high pass filter. Compute
theoretical values for A(ω) (in dB) and φ(ω) for frequencies 80 kHz, fb, 200 Hz, 100 kHz and 50
Hz. Make up a table comparing these predicted values with the measured values from part 7.
Produce, on semi-log graph paper, the dB magnitude response plot and the phase
response plot for your data in part 7. Comment on the slope of the magnitude response curve for
frequencies less than the cutoff frequency.
Appendix 1 Using Cursors to measure phase
Information on using cursors is found, starting with page 5-31 of the manual. To
measure phase, follow these instructions.
- Display the two sinusoidal signals centered on ground. Use no more than two cycles and
expand the vertical scale as large as possible.
- Select Cursors. Set X1 cursor at a zero crossing (positive slop) for signal 1. Set X2
cursor at the next zero crossing (positive slope) of signal 2. Record values as t1 and t2.
- Set X2 for the next zero crossing (positive slope) for signal 1. Record the value as tp.
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Appendix 2: How to Change the FG Display for the Usual Peak-to-Peak Amplitude.
The default display for the Agilent 33120A function generator shows the peak-to-peak
amplitude of the output, provided that the FG output terminals are shunted (or terminated) by 50
ohms. Typically, in our lab, we do not have that situation.
So, in order to have the FG display read the actual peak-to-peak amplitude, enter the
following commands on the front panel of the FG. [Ref.: page 40 of manual.]
Keys to Press
Shift Menu
On/Off
→ → →
↓
↓
→
Enter
What is in the Display
A: MOD MENU
D: SYS MENU
1: OUT TERM
50 OHM
HIGH Z
ENTERED
Appendix 3 Definitions and equations used in your work.
•
Time constant, τ, equals RC or L/R.
•
Corner or break frequency, fb, equals 1/τ. The unit for fb is hertz (Hz).
•
Voltage gain in dB equals 20 log (Vout/Vin).
•
The quantity t63% is the time for the output to go from zero value to 63 % of the maximum
value of the output.
•
Rise time is given by the equation tr = t90% - t10%. The quantity t90% is the time for the output
to equal 90 % of the maximum value of the output, and t10% is the time for the output to equal
10 % of the maximum value of the output.
•
Rise time is related to time constant by the formula tr = 2.2 τ.
•
Fall time is related to time constant by the formula tf = 2.2 τ.
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