Download Transient Response of a RC Circuit

ECE 213 Electrical Circuits Lab II
LAB #8
Spring 2015
Name ___________________
Section ______________
The Transient RC
OBJECTIVE:
The object of this experiment is to investigate the response of a series resistive-capacative circuit
when excited with a step function and when excited with an initial condition.
COMPONENTS AND EQUIPMENT:
1. Oscilloscope
2. Function Generator
3. Digital Multimeter
BACKGROUND:
Resistor-capacitor (RC) circuits are first-order systems. Consider the circuit diagram shown in Figure 1.
When the capacitor is charging, the voltage drop across the resistor (using KVL) is nonzero and is given
by
Vin  Vout  iR
(1)
The variable i is the current through the resistor. Since the output terminal has no load connected to it,
the variable i is also the current through the capacitor (by KCL). The current through the capacitor is
given by
i C
dVou t
.
dt
(2)
Substituting Eq. (2) into Eq. (1), yields

dVout
 Vout  Vin .
dt
(3)
For this lab, the voltage Vin will be a square wave. The square wave can be represented as a time-domain
step function that either rises from 0V to voltage V1, or falls from voltage V1 to 0V. The time constant of
the RC circuit is τ = RC. For a rising step function input voltage, the solution to equation (3) takes the
form
t
Vout (t )  V1  V1 e

.
(4)
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ECE 213 Electrical Circuits Lab II
LAB #8
Spring 2015
R 3kΩ
= 1.5k
Input
Output
Vout
Vin
i
+
–
Vin
C224nF
= 0.1uF
Figure 1. A low-pass filter is a common and useful RC circuit.
Figure 2 shows the response (output voltage) of the RC circuit when the input is a rising step function
(from 0V to V1). The output voltage reaches 63% of the final value (V1) in one time constant. After five
time constants (5), we say that the response has converged to its final or
steady-state value (V1).
V1
0.368 V1
Vout(t)
0.632 V1
0
0
1
2
3
4
5
6
7
t/
t=
Figure 2. The first-order circuit response to a step input (0 to V1 step) is an exponential that
approaches the input step voltage V1. The current through the resistor charges the capacitor.
Following an input voltage (Vin) step from V1 to 0V, the capacitor discharges through the resistor. By KVL,
when the capacitor is discharging, the voltage drop across the resistor is
2
ECE 213 Electrical Circuits Lab II
LAB #8
Spring 2015
0  Vout  iR .
(5)
The current through the capacitor is
i C
dVou t
.
dt
(6)
Substituting Eq. (2) into Eq. (1), yields

dVout
 Vout  0
dt
.
(7)
The time constant is τ = RC and the initial voltage is Vin. The solution to equation (7) has the form
t
Vout (t )  V1 e

.
(8)
Figure 3 shows the response (output voltage) of the RC circuit when the input is a step function that falls
from V1 to 0V. The output voltage reaches 63% ( = 1 – 0.368) of the final value (0V) in one time constant.
After five time constants (5), we say that the response has converged to its final or steady-state value
(0V).
V1
0.632 V1
Vout(t)
0.368 V1
0
0
1
2
3
4
5
6
7
t/
t=
Figure 3. The first-order circuit response to a step input (V1 to 0V step) is an exponential that falls
from V1 and approaches zero. The capacitor discharges through the resistor.
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ECE 213 Electrical Circuits Lab II
LAB #8
Spring 2015
4
ECE 213 Electrical Circuits Lab II
LAB #8
Spring 2015
PROCEDURE:
1.
Use a square wave to create a sequence of two step functions (0 to +4V and +4V to 0V).
Configure your function generator to be the input voltage to the RC circuit. Create a 0 to 4 volt,
500 Hz square wave (4V peak-to-peak amplitude, 2V DC offset).
2.
Build the circuit in Figure 1. Use one scope probe to monitor the input voltage and the other to
monitor to the output voltage. Capture two periods of the wave on the scope screen. Sketch or
take snapshot of the input and output voltages (label your plots with volts/div and time/div).
3.
Change the frequency of the function generator to 100 Hz and zoom in on the rising portion of
the output voltage. Use the oscilloscope voltage cursors to measure 63.2% of the final value.
What voltages (in units of V) are your cursors set on?
4.
Place oscilloscope time cursors on the intersections of the voltage cursors and the waveform.
The time difference Δt between the cursors is the time constant of the RC circuit (τ). What time
constant did you measure?
5.
Calculate the time constant from the component values in you circuit. What is your calculated τ?
What is the percent error between the measured and calculated values? What are possible
sources of the error?
6.
Zoom in on the falling portion of the output voltage. Use the voltage cursors to measure 36.8%
of the initial value (see Figure 3). What voltages (in units of V) are your cursors set on?
7.
Place time cursors on the intersections of the voltage cursors and the waveform. The time
difference Δt between the cursors is the time constant of the RC circuit (τ). What time constant
did you measure?
8.
What is the percent error between this measured value and the calculated value? What are
possible sources of the error?
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