Experiment 20: RL Transient
Response
(Student Name)
EET-112
Procedure Performed: 2-26-2011
Report Turned in: 2-28-2011
Introduction:
An inductor is a circuit element that stores energy in its magnetic field. Current
cannot change through an inductor instantaneously, and the change in current must
follow an exponential function.
i(t) = If + (Ii - If) e-t/τ, where τ = L/R
Equation 1
Ii is the initial current in the inductor and If is the final current in the inductor after
steady state is reached. This lab will demonstrate how current begins to flow through an
inductor (forced response), that the inductor stores energy, and how this energy is given
from the inductor back to the circuit (natural response). A series RL circuit will be built
and the circuit’s current will be monitored by measuring the voltage across a known
resistance. This measurement of voltage will be taken at multiples of the time constant,
τ, and compared to values predicted by the voltage’s exponential time function.
Experimental Procedure:
A series RL circuit was built (figure 1) and measurements of the resistor, the
inductor’s inductance, and the inductor’s resistance were taken. [Note: Schematic
drawings from LTSpice can be captured by going to ToolsCopy bitmap to clipboard,
and then paste the resulting image into MS Paint. Save the file and drag it into Word]
Figure 1: Series RL Circuit
A 0-to-8 volt, 10kHz square wave was applied to the circuit to simulate the
switching in and out of a 8 volt DC source. The frequency of 10kHz has a period of
100µs, which allows the RL circuit enough time to reach steady state during its forced
response (source is at 8V) and enough time to reach steady state in its natural response
(source returns to 0V).
Since we are interested in the current of the circuit, the voltage across the known
resistor, R, was monitored by the oscilloscope. This is the actual measured quantity that
was expected to follow the exponential expression.
V(t) = Vf + (Vi - Vf) e-t/τ, where τ = L/R
Equation 2
A measurement of Vf , the final voltage across the resistor, was made. Once the
final voltage is known, the values of the voltage across the resistor can be theoretically
predicted at any moment of time by plugging in values of t into the above exponential
expression for V(t).
At integer multiples of τ, this can be done without any knowledge of the value of
τ. For the forced response, theoretical values for V(t) can be obtained with knowledge of
only Vf (using a Vf of 7.12 V):
At t = 1τ, V(τ) = Vf (1 - e- τ /τ ) = Vf (1 - e- 1 ) = 4.501 V
At t = 2τ, (2τ) = Vf (1 - e- 2τ /τ ) = Vf (1 - e- 2 )= 6.156 V
Likewise, theoretical values for V(t) can be obtained for the natural response(using a Vfi
of 7.12 V):
At t = 1 τ, V(τ) = Vi e- τ /τ = Vi e- 1 = 2.619 V
At t = 2τ, V(τ2) = Vi e- 2τ /τ = Vi e- 2 = 0.964 V
After these voltages are calculated at t = τ, 2 τ, 3 τ, 4 τ, and 5 τ, the time can be
measured at which the actual voltage across the resistor reaches these values. For
example, in the forced response, the calculations above reveal that V(t) should be at
4.501 V when t =1τ . With the natural response on channel 2 of the oscilloscope, the
level of 4.501 V is located with the oscilloscope’s voltage cursor. The time at which that
voltage occurs is measured and recorded with the time cursor. The voltage source on
channel 1 gives a reference for time, or when t = 0 seconds, when the source goes from 0
to 8 volts. The same process is repeated for the natural response, but here the time at
which the source drops from 8 to 0 volts is used as the reference time.
These measured times will be compared to multiples of τ as a measure of how
well the circuit’s behavior matches the exponential in equation 2 for both the forced and
natural responses.
Results:
Below are the measurements for the measured component values, and for the times at
which predetermined values of resistor voltage were observed to occur.
Rth (Ω)
(given)
RL (Ω)
R1 (Ω)
L (H)
Final Voltage
50
77.38
986.9
0.008767
Vf
7.12
Table 1: Final Voltage Across Resistor
Table 2: Measured Component Values
Multiples of
τ
1
2
3
4
Expected
Voltage
Vf (1-e-t/τ)
4.501
6.156
6.766
6.990
Measured Time
(us)
7.70
15.50
23.00
34.20
Calculated
Time (us)
7.868
15.736
23.604
31.471
% Error
-2.134
-1.498
-2.557
8.670
Circuit
Current
(mA)
4.56
6.24
6.86
7.08
% Error
-0.227
-3.404
0.027
9.941
Circuit
Current
(mA)
2.65
0.98
0.36
0.13
Table 3: Forced response
Multiples of
τ
1
2
3
4
Expected
Voltage
Vf e-t/τ
2.619
0.964
0.354
0.130
Measured Time
(us)
7.85
15.20
23.61
34.60
Calculated
Time (us)
7.868
15.736
23.604
31.471
Table 4: Natural Response
For both the forced and natural responses, the measurements of time were within +/- 4%
of the theoretical values for the first 3 measurements of time. These match well with the
behavior predicted by equation 2. The last measurements of time for both responses have
large errors. This was due to the difficulty in determining the exact location in time at
which the waveform reaches a value after the waveform has flattened out. The thickness
of the line impedes the process of obtaining an accurate measurement. Normally,
zooming in with the oscilloscope would help minimize this error, however, this would
cause the reference time (the source’s transitions) to disappear from the screen. Without
the reference time on the screen, the time cannot be measured, so this error is an
unfortunately necessity.
Conclusion:
The circuit’s voltage waveform matched well within acceptable error bounds for
what was predicted by equation 2 for both the forced and natural responses. In tables 3
and 4, the circuit current was calculated using the voltage across the circuit’s resistor and
Ohm’s law. For the forced response, the voltage source goes from 0V to 8 V. The
current starts at lower values and approaches its maximum value as the circuit reaches
steady state. At this point, the inductor has stored its maximum amount of energy. When
the voltage source goes from 8 V to 0 V, the circuit’s natural response begins. Current is
observed to start at a maximum and approach zero amperes. During this time, the energy
stored in the inductor’s magnetic field is bled out and dissipated by the circuit’s
resistances. Overall, the circuit behaved as predicted by the exponential time expression
in equation 2.