ECE 2006 LABORATORY 8
RLC TRANSIENT RESPONSE
OBJECTIVES
The learning objectives for this laboratory are to give the student the ability to:
•
•
•
use the function generator to generate a step input with an appropriate repetition rate.
use the oscilloscope to measure RLC overdamped and underdamped transient response
values.
use the PSPICE transient response tool.
REFERENCES
•
Alexander/Sadiku, “Fundamentals of Electric Circuits – 2nd Edition”, 2003, McGrawHill
BACKGROUND
•
•
See above reference, Appendix D.4: pp.A-38 thru A-46, for PSPICE analysis of RLC
transient response.
The analysis of series RLC circuits can be summarized as follows:
Finding the solution to a second order linear differential equation with a unit-step forcing
function involves finding the roots of its characteristic equation. The characteristic equation is
quadratic and thus has two roots. The roots may be 1) real and equal, 2) real but unequal or 3)
complex conjugates.
A series RLC circuit can be modeled as a second order differential equation. When its roots are
real and equal, the circuit response to a step input is called “Critically Damped”. When its roots
are real but unequal the circuit response is “Overdamped”. When roots are a complex conjugate
pair, the circuit repsonse is labeled “Underdamped”.
To find the solutions for voltage and current in an RLC circuit, such as in Figure A, it is
Figure A: Series RLC Circuit with Step Input
necessary to first determine its damping by finding the roots of the characteristic equation:
Series RLC Characteristic Equation:
S2 + S(R/L) + 1/LC = 0
Putting the above equation into standard form: S2 + 2 S +
= R/2L
where:
and
0
0
2
= 0, it follows that:
= 1 / LC
is the Damping Coefficient and
0 is the Natural or Resonant Frequency
The roots of the Characteristic Equation, by using the Quadratic Formula, are:
S1,2 = -
± (
2
-
0
2
)
The damping is determined by the ratio of / 0. If the ratio is greater than one, i.e. > 0, the
circuit is Over Damped. If = 0, the circuit is Critically Damped. Otherwise, the circuit is
UnderDamped. For each type of damping condition, the voltage and current solutions take a
different form:
CAPACITOR VOLTAGE FOR A STEP INPUT TO A SERIES RLC CIRCUIT:
Vc(t) = Vc( ) + A1e S1t + A2e S2t Volts
Overdamped:
>
0,
Critically Damped:
=
0
, Vc(t) = Vc( ) + (A1 + A2t)e
- t
Underdamped:
<
0
, Vc(t) = Vc( ) + (A1cos
+ A2 sin
dt
d=
(
2
Volts
-
0
2
)e -
dt
t
Volts
)
INDUCTOR CURRENT FOR A STEP INPUT TO A SERIES RLC CIRCUIT:
IL(t) = IL ( ) + B1e S1t + B2e S2t Amps
Overdamped:
>
0,
Critically Damped:
=
0
, IL (t) = IL ( ) + (B1 + B2t)e
- t
Underdamped:
<
0
, IL (t) = IL ( ) + (B1cos
+ B2 sin
d=
dt
(
2
Amps
-
0
2
)e -
dt
t
Amps
)
Once the damping condition is known and the form of the solution is determined, it necessary
only to determine the values of the coefficients, A1, A2, B1, and B2 in order to form a complete
solution.
It requires two independent equations to solve for two unknowns. Since it is possible to
determine initial and final values for Capacitor Voltage and Inductor Current, evaluating the
capacitor voltage equation and its first derivative at time, t = 0, will form an independent pair of
equations:
Find Vc(0), Vc( ) by inspecting the circuit.
Find d[Vc(0)]/dt by using the equation, Ic(0+) = C dVc(0+)/dt
Solve for A1, A2:
For example, in an overdamped circuit with Zero initial conditions:
Vc(0) = 0 = Vc( ) + A1e 0 + A2e 0
d[Vc(0)]/dt = 0 = S1A1e 0 + S2A2e 0
Solve the two equations simultaneously to find the A constants.
EQUIPMENT
Oscilloscope
Function generator
Resistor, 100 Ω
Resistor, 1.0 kΩ
Capacitor, 1.0 µF
Inductor, 220 mH
PROCEDURE
1.
Overdamped RLC circuit capacitor voltage transient response to a step input.
1.1
With the RLC circuit disconnected, adjust the function generator to produce a repetitive
pulse that is -5 volts for about 10 msec, then +5 volts for about 10 msec. (i.e. 10 Volts
peak-to-peak, 0 Volts of DC offset, 20 msec Period or 50 Hz)
1.2
For the circuit in Figure 1, calculate the output response, VC(t), t > 0, to an input step,
from -5 to +5 Volts.
FIGURE 1: Series RLC with Step Input, Measuring Vc
First determine and 0. Calculate the roots of the characteristic equation, S1,2.
Determine Vc(0), Vc( ) , and d[Vc(0)]/dt. Calculate A1 and A2. Fill in the calculated
values in the Data Table for Figure 1 below:
Quantity
DATA TABLE 1: OVERDAMPED RLC
Calculated Value
Measured Value
N/A
N/A
0
Over,Under or
Critical Damping?
S1,2
N/A
Vc(0)
Vc( )
d[Vc(0)]/dt
= IL(0+)/C =
0.000 Amps/Coulomb
A1 and A2
N/A
Equation for Vc(t):
N/A
Vc(0.5ms):
Vc(1.0ms):
Vc(2.0ms):
1.3
Connect the circuit in Figure 1. Measure the final value, VC(t=∞), and the initial value,
VC(t=0+), from the oscilloscope and record in the Data section. Also measure the
voltages VC(t=0.5 msec), VC(t=1.0 msec), and VC(t=2.0 msec)from the oscilloscope and
record in the Data section.
2.
Underdamped RLC circuit capacitor voltage transient response to a step input.
2.1
Keep the function generator settings used in Part 1.
2.2
For the circuit in Figure 2, calculate the output response, VC(t), t > 0, to an input step,
from -5 to +5 Volts. Note that the only change to the circuit is replacing the 1 k-Ohm
resistor with a 100 Ohm resistor.
FIGURE 2: Underdamped RLC Circuit with Step Input
First determine and 0. Calculate the roots of the characteristic equation, S1,2.
Calculate d.Determine Vc(0), Vc( ) , and d[Vc(0)]/dt. Calculate A1 and A2. Fill in the
calculated values in the Data Table for Figure 2 below:
Quantity
DATA TABLE 2: UNDERDAMPED RLC
Calculated Value
Measured Value
N/A
N/A
0
Over,Under or
Critical Damping?
Continued on NEXT PAGE
S1,2
and
d
d
= 2 /T =
Vc(0)
Vc( )
d[Vc(0)]/dt
= IL(0+)/C =
A1 and A2
0.000 Amps/Coulomb
N/A
Equation for Vc(t):
Vc(0.5ms):
Vc(1.0ms):
Vc(2.0ms):
2.3
Connect the circuit in Figure 2. Measure the final value, VC(∞), the initial value, VC(0+),
and T/2 (one half the period of oscillation of the output waveform) from the oscilloscope
and record in the Data section. Also measure the voltages V2(t=0.5msec), V2(t=1.0msec),
and V2 (t=2.0msec) from the oscilloscope and record in the Data section.
3.
Underdamped RLC circuit resistor voltage transient response to a step input.
3.1
Keep the function generator settings used in Part 2.
3.2
For the circuit in Figure 3, calculate the output response, VR(t), t > 0, to an input step,
from -5 to +5 Volts. Note that the only change to the circuit is changing the position
of the 100 Ohm resistor and capacitor (in order to measure Vr).
FIGURE 3: Underdamped Series RLC Circuit with Step Input, Measuring
Resistor Voltage
First determine and 0. Calculate the roots of the characteristic equation, S1,2.
Calculate d. Realize that in order to calculate the voltage across the 100 Ohm resistor,
one must determine the current in the series circuit and use Ohm’s Law. The current in
the circuit is the inductor current, IL(t). Determine IL(0), IL( ) , and d[IL(0)]/dt. Multpily
these by 100 Ohms to make Vr(0), etc. Calculate B1 and B2. Fill in the calculated values
in the Data Table for Figure 2 below:
DATA TABLE 3: UNDERDAMPED RLC, Resistor Voltage
Calculated Value
Measured Value
N/A
Quantity
N/A
0
Over,Under or
Critical Damping?
S1,2 and d
d
= 2 /T =
IL(0) and Vr(0)
Vr(0) =
IL( ) and Vr( )
Vr( ) =
d[Vr(0)]/dt
= R*VL(0+)/L =
B1 and B2
N/A
Equation for Vr(t):
Vr(0.5ms):
Vr(1.0ms):
CONTINUED on NEXT PAGE
Vr(2.0ms):
3.3
Connect the circuit in Figure 3. Measure the final value, V2(∞), the initial value, V2(0+),
the initial derivative, dV2(0+)/dt and T/2 from the oscilloscope and record in the Data
section. Also measure the voltages V2(t=0.5msec), V2(t=1.0msec), and V2(t=2.0msec)
from the oscilloscope and record in the Data section.
4. PSPICE ANALYSIS
4.1
Write and run a PSPICE program for the circuit in Figure 1. Use the VPULSE
model to represent the step input voltage. In the VPULSE model, make V1=-5V,
V2=5V, PW=10ms and PER= 15ms. Have the program plot the voltage across
the capacitor from 0 to 5 msec in 0.1 msec increments (i.e. in transient analysis,
make the print step = .1 ms and final step = 5ms). Print the Netlist output file, and
the Output waveform and attach.
4.2
Write and run a PSPICE program for the circuit in Figure 2. Have the program
plot the voltage across the capacitor from 0 to 5 msec in 0.1 msec increments.
Print the Netlist output file, and the Output waveform and attach.
4.3
Use the program from 4.2 above to also analyze the resistor voltage. Have the
program plot the voltage across the resistor from 0 to 5 msec in 0.1 msec
increments. Print the Output waveform and attach.
CONCLUSIONS:
Standard conclusions based on experience gained.
SAMPLE CALCULATIONS
None.