Laboratory of the circuits and signals
Laboratory work No. 5
TIME RESPONSES OF THE RC AND RL CIRCUITS
1. Objectives:
„ to get acquainted with time and transient responses measurement technique using oscilloscope,
„ to investigate integrating and differentiating RC and RL circuits,
„ to investigate relations between circuit frequency response and signals integrating and differentiating
processes.
2. Basic definitions.
Integrating circuit - A circuit in which output voltage is directly proportional to the integral of the input
is known as an integrating circuit. An integrating circuit is a simple RC series circuit with output taken
across the capacitor C.
For the circuit to render good integration, the following conditions should be fullfilled.
1. The time constant RC of the circuit should be very large compared to the time period of the input
wave.
2. The value of R should be 10 or more times larger than Xc.
Let ei be the input alternating voltage and let i be the resulting alternating current. Since R is very large
compared to capacitive reactance Xc of the capacitor, it is reasonable to assume that the voltage across
R(i.e. eR) is equal to the input voltage, i.e. ei = e R . Now
i=
e R ei
=
.
R R
∫
The charge on the capacitor at any instant time is q = idt and output voltage
1
ei dt = a ∫ ei dt .
RC ∫
Here a =1/RC is constant. So Output _ voltage = a ∫ input _ voltage or the output is proportional to
=
the time integral of the input.
Example – time responses of the RC integrating circuit with R=1kOhms, C=100nF.
The input and output waveforms are shown in the figure 1.
Fig. 1. Input and output waveforms of the RC integrating circuit
Differentiating circuit - A circuit in which output voltage is directly proportional to the derivative of
input is known as a differentiating circuit. A differentiating circuit is a simple RC series circuit with
output taken across the resistor R.
In order to achieve good differentiation, the following conditions should be satisfied:
1. The time constant RC of the circuit should be smaller than the time period of the input wave.
2. The value of Xc should be 10 or more times larger than R at operating frequency.
Let ei be the input alternating voltage and i be the resulting alternating current. The charge q on the
capacitor at any instant is q = CeC . So
i=
dq d
d
= CeC = C eC .
dt dt
dt
Since capacitive reactance is very much larger than R, the input voltage can be considered to be equal to
the capacitor voltage with negligible error. i.e. ec = ei
and i = C
Output Voltage, e0 = iR = RC
∴ Output _ voltage =
d
ei
dt
d
d
ei = a e i
dt
dt
d
input _ voltage
dt
Hence the output is proportional to the time derivative of the input.
Example – time responses of the differentiating RC circuit with R=10kOhms, C=25nF
The input and output waveforms are shown in the figure 2.
Fig.2. Input and output waveforms of the RC differentiating circuit
Basic definition of the Time constant - The interval required for a system or circuit to change a
specified fraction from one state or condition to another.
Note 1: The time constant is used in the expression
where A (t) is the value of the state at time t, A (0) is the value of the state at time t = 0, a is the time
constant, and t is the time that has elapsed from the start of the exponential decay.
Note 2: When t = a, A (t)/A (0) = 1/e, or approximately 0.37, and the system has changed about 63%
toward its new value in one time constant. A system is considered to have changed its state after the
elapse of three time constants, which corresponds to a 95% change in state. For example, if an electrical
capacitor, having a capacitance of C farads, is discharged through a resistor, having a resistance of R
ohms, the capacitor will be approximately 95% discharged after the elapse of 3RC seconds.
Note 3: Time constants are expressed in seconds, such as 3.5 × 10-6 seconds, i.e., 3.5
s.
RC Time constant - The time required to charge a capacitor to 63 percent (actually 63.2 percent) of full
charge or to discharge it to 37 percent (actually 36.8 percent) of its initial voltage is known as the time
constant (TC) of the circuit. The charge and discharge curves of a capacitor are shown in figure 3.
Fig. 3. RC time constant
Fig. 4. L/R time constant
The value of the time constant in seconds is equal to the product of the circuit resistance in ohms and the
circuit capacitance in farads. The value of one time constant is expressed mathematically as τ = RC .
L/R Time Constant - is a valuable tool for use in determining the time required for current in an
inductor to reach a specific value. As shown in figure 4, one L/R time constant is the time required for the
current in an inductor to increase to 63 percent (actually 63.2 percent) of the maximum current. Each time
constant is equal to the time required for the current to increase by 63.2 percent of the difference in value
between the current flowing in the inductor and the maximum current. Maximum current flows in the
inductor after five L/R time constants are completed. The following example should clear up any
confusion about time constants. Assume that maximum current in an LR circuit is 10 amperes. As you
know, when the circuit is energized, it takes time for the current to go from zero to 10 amperes. When the
first time constant is completed, the current in the circuit is equal to 63.2% of 10 amperes. Thus the
amplitude of current at the end of 1 time constant is 6.32 amperes. During the second time constant,
current again increases by 63.2% (.632) of the difference in value between the current flowing in the
inductor and the maximum current. This difference is 10 amperes minus 6.32 amperes and equals 3.68
amperes; 63.2% of 3.68 amperes is 2.32 amperes. This increase in current during the second time constant
is added to that of the first time constant. Thus, upon completion of the second time constant, the amount
of current in the LR circuit is 6.32 amperes + 2.32 amperes = 8.64 amperes.
Eyeball Method. One can eyeball the value of τ by estimating the time at which 1/e (~0.368)~1/3 of the
total charge remains, or equivalently the time when (1-e-1)~0.632~2/3 of the total charge has occured.
3. Work procedures
3.1. Connect x input of the oscilloscope (figure 6) to the function generator (figure 7). Adjust your
generator output to produce sine type signal with frequency fs = 2 kHz and peak-to-peak amplitude equal
to 10 V. (DC offset voltage must be equal to zero). Set up control of oscilloscope to see signal in the
scope display and measure its parameters: amplitude and period.
Fig. 6. Laboratory oscilloscope
Fig. 7. Laboratory generator
3.2. Connect function generator (its output with 50 Ohms resistance) to the input of the RC-RL
circuit model (Fig. 8 – clamps 2 and 3). Connect y input of the oscilloscope to the model‘s 4 and 7 clamps
(clamps 3 and 7 must be connected as ground). Change frequency till model output peak-to-peak voltage
becomes equal to ~7 V. Using this frequency and data shown in the model calculate inductance L1.
4
R1
100
C1
5
1
R2
1k
L1
6
2
R3
10k
3
7
Fig. 8. RLC circuit model and its equivalent circuit
3.3. Adjust your function generator output to produce square wave (pulse type) signal (example is shown
in the Fig. 9 and signal parameters are given in the table 1). Note that this require adjusting both the
signal amplitude and DC offset of the generator output. Plot the time dependent voltage waveform you
see on the scope’s screen to the graph (preferable to use graph paper – example is shown in the figure 10).
Measured output signal must be plotted together with input signal (see examples of such graphs in the
figures 1-4).
T
v(t)
A
0
ti
t
Fig.9. Square wave signal (usually called
pulse type or meander type) parameters
Version
ti, ms
Ap-p, V
AD, V
1
1
10
5
2
1.5
12
6
3
0.75
14
7
Fig. 10. Plot of the signal’s waveform seen on the scope screen
4
1.25
16
8
5
2
18
9
6
0.5
20
10
7
1
22
11
8
1.5
24
12
9
2
10
5
Table 1
10
1
12
6
(Here Ap-p – generator signal’s peak-to-peak voltage amplitude, AD - DC offset voltage)
3.4. Investigate pulse propagation in the integrating circuit RC at various time constant τ values: when
means that τ could be in the same values range as ti). For this connect
osilloscope input y to the clamps 4-1. Connect generator to the model clamps 7-1, later to the clamps 6-1
and final to the clamps 5-1. Adjust the oscilloscope to see signal waveforms as shown in the example
(Fig.11). Plot the time dependent voltage waveforms you see to the graph. Measured output signal must
τ << ti , τ ≈ ti , τ >> ti. ( τ ≈ ti
be plotted together with input signal (see examples of such graphs in the figures 1-4). Define from these
graphs circuit time constants and put results into table 2.
3.5. Investigate pulse propagation in the differentiating LR circuit at various time constant τ values:
when τ << ti , τ ≈ ti , τ >> ti. For this connect oscillator to the model clamps 2-3. Connect generator to the
model clamps 7-1, later to the clamps 6-1 and final to the clamps 5-1. Adjust the oscilloscope to see
signal waveforms as shown in the example (Fig.12). Plot the time dependent voltage waveforms you see
to the graph. Measured output signal must be plotted together with input signal (see examples of such
graphs in the figures 1-4). Define from these graphs circuit time constants and put results into table 2.
3.6. Investigate pulse propagation in the integrating LR circuit at various time constant τ values:
when τ << ti , τ ≈ ti , τ >> ti. For this connect generator to the model clamps 2-3. Connect oscilloscope to
the model clamps 4-7, later to the clamps 4-6 and final to the clamps 4-5. Plot the time dependent voltage
waveforms you see to the graph. Define from these graphs circuit time constants and put results into table
2.
3.7. Investigate pulse propagation in the differentiating RC circuit at various time constant τ values:
when τ << ti , τ ≈ ti , τ >> ti. For this connect generator to the model clamps 2-3. Connect oscilloscope to
the model clamps 4-7, later to the clamps 4-6 and final to the clamps 4-5. Plot the time dependent voltage
waveforms you see to the graph. Define from these graphs circuit time constants and put results into table
2.
Fig.11. Signals in the integrating circuit
Fig.12. Signals in the differentiating circuit
3.8. Calculate time constant for all investigated circuits. Results put into table 2 also.
Table 2
C=
nF
L=
mH
RΣ1,
kΩ
Integrating circuit
RC
RL
RΣ2, RΣ3 RΣ1, RΣ2,
kΩ kΩ kΩ kΩ
RΣ3
kΩ
Time constant Measured
Calculated
τ , ms
3.9. Repeat of all measurements using EWB package is welcome.
RΣ1,
kΩ
Differentiating circuit
RC
RL
RΣ2, RΣ3 RΣ1, RΣ2,
kΩ kΩ kΩ kΩ
RΣ3
kΩ
4. Work report:
4.1. Work objectives.
4.2. List of used equipment.
4.3. Equivalent circuits of the investigated model for each measurement. Graphs of the measured
oscillograms. Measured oscillograms of the output signals must be presented together with the input
signal oscillogram (desirable in different colour).
4.4. Table with the results of time constant measurement and calculation.
4.4. Conclusions.
5. Control questions
5.1. Define time constants of the RC and RL circuits.
5.2. Explain physical processes in the RC and RL circuits when voltage step is acting.
5.3. Definition of the RC and RL circuits parameters to obtain:
- input pulse differentiation,
- input pulse integration,
- output pulse without distortions.
5.4. Sketch example of the integrating circuit magnitude frequency response.
5.5. Sketch example of the differentiating circuit magnitude frequency response.
5.5. Sketch circuit diagrams of the RC or LR low-pass and high-pass filters.
5.6. Explain RC or LR circuits time constant definition using pulse transient responses.
5.7. Explain RC or LR circuits time constant definition using frequency responses.
References:
Raymond A.DeCarlo/Pen-Min Lin. Linear Circuit Analysis (Ch.8 pp.217-240: –First Order RL and RC
Circuits).