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Lab 3 Transient Lab

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Lab 3 Transient Lab
VE215 Intro to Circuits

1 Lab goals
1. Apply the theory you learned on the step responses in first- and secondorder circuits to series RC and RLC circuits, which you will build in the lab.
2. Build a series RC circuit, observe its responses to input square wave
signal of varied frequency, and explain them based on the theory you learned:
• Relate the observed capacitor voltage and resistor voltage as functions of
time to your pre-lab calculations

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Lab 3 Transient Lab
VE215 Intro to Circuits

1 Lab goals
1. Apply the theory you learned on the step responses in first- and secondorder circuits to series RC and RLC circuits, which you will build in the lab.
2. Build a series RC circuit, observe its responses to input square wave
signal of varied frequency, and explain them based on the theory you learned:
• Relate the observed capacitor voltage and resistor voltage as functions of
time to your pre-lab calculations
• Explain the changes of both output waveforms in response to the increase
of the frequency of the input square wave signal
• Explain the amplitudes of the capacitor voltage and the resistor voltage
related to the amplitude of the input square wave
3. Build a series RLC circuit, observe the three types of its responses to input
square wave signal, and relate them to the theory you have learned. For the
under-damped/ over-damped/ critical damped response, compare the resistance
in the circuit measured in the lab with the critical resistance you calculated in
the pre-lab.
4. Build the simplest second-order circuit, an LC tank, and observe oscillations.
2 Theoretical background
2.1 First-order circuits
Theoretically, the transient responses in electric circuits are described by
differential equations. The circuits, whose responses obey the first-order differential equation
dx(t)
dt +
1
τ
· x(t) = f(t)
are called first-order circuits. Their responses are always monotonic and
appear in the form of exponential function
x(t) = K1 · e
−(
t
τ
) + K2
1
A first-order circuit includes the effective resistance R and one energy-storage
element, an inductor L or a capacitor C.
In an RC circuit, the time constant is
τ = RC.
In an LC circuit, the time constant is
τ =
L
R
.
The fall time of a signal is defined as the interval between the moment when
the signal reaches its 90% and the moment when the signal reaches its 10% level.
Note that the 10% level is reached between 2τ and 3τ . Approximately, you can
assume f alltime ≈ 2.2τ . After t = 5τ , the exponent practically equals zero.
2.2 Second-order circuits
Many circuits involve two energy-storing elements, both an inductor L and a
capacitor C. Such circuits require a second-order differential equation description
d
2x(t)
dt2
+ 2 · α ·
dx(t)
dt + ω
2
0
· x(t) = f(t)
thus they are called second-order circuits.
We will consider only second-order circuits with one inductor and one capacitor. The differential equation includes two parameters: the damping factor
α and the undamped frequency ω0 which are determined by the circuit and its
components.
For example, in the series RLC circuit, which you will build and study in
this lab,
α =
R
2·L
, and ω0 = √
1
L·C
,
while in the parallel RLC circuit,
α =
1
2·R·C
, and ω0 = √
1
L·C
.
Depending on the two parameters α and ω0, second-order circuits can exhibit
three types of responses.
2.2.1 The underdamped response
If α < ω0,
x(t) = e
−αt(K1cos(ωt) + K2sin(ωt))
where ω =
p
ω
2
0 − α2.
The underdamped circuit response involves decaying oscillations, which may
last for many periods or for less than one period, depending on the damping
ratio ξ =
α
ω0
, which for the series RLC circuit ξ =
R
2L

LC =
R
2
·
q
C
L
. Varying
the values of R, L, C, affects the damping ratio ξ.
2
2.2.2 The critically damped response
If α = ω0,
x(t) = e
−αt(K1 + K2t)
and the circuit has the critically damped response.
The critically damped response does not involve oscillations.
For the series RLC circuits, α = ω0 corresponds to R
2L = √
1
LC or R =
Rcritical = 2q
L
C
.
If L = 1mH and C = 10nF, then Rcritical ≈ 632Ω.
2.2.3 The overdamped response
If α > ω0,
x(t) = K1 · e
s1t + K2 · e
s2t
where s1 = −α +
p
α2 − ω
2
0
and s2 = −α −
p
α2 − ω
2
0
.
In the series RLC circuits, the overdamped solution is obtained if the resistance is larger that the critical resistance, such that R > Rcritical = 2 ·
q
L
C
.
Notice that the larger resistance corresponds to the longer delay, and even
the faster decay has a much longer fall time than the critically damped response.
One of the most interesting features of series RLC circuits is that increasing
the resistance above the critical value results in much longer fall time, or longer
delays of responses in digital circuits. Among all monotonic responses, the
critically damped is the fastest.
3 Pre-lab assignments
1. Calculate the time constant τ and fall time values for the maximal and
minimal resistances (the two limit positions of the potentiometer) in the circuit
on this diagram using the following nominal values:
R1 = 1kΩ, RP = 10kΩ, C = 100nF = 0.1µF.
For simplicity, assume F alltime = 2.2τ
3
2. In the circuit shown on this diagram, assume:
L = 1mH, R1 = 100Ω, RP = 10kΩ, C = 820pF.
Depending on the position of the potentiometer’s tap, RX varies from zero to
RP .
Calculate:
A. The range of RX that ensures under-damped response
B. The range of RX that ensures critically damped response
C. The range of RX that ensures over-damped response
3. In this lab, you will use a function generator to produce a square wave. The
function generator voltage abruptly changes from +V0 to −V0 and back again,
which is quite similar to the switch flipped back and forth. Therefore, the
impulse response of the circuit can be studied. Assume the circuit in question 1
is being studied, and the your function generator sets a square wave at 1V ppk
(peak-to-peak). For circuits with τmin and τmax, calculate the fastest frequency
that allows the output signal to reach saturation.
4. Prove that for a first-order circuit, the rise time/ fall time is approximately
2.2 times of the time constant τ .
5. Simulate the circuit in question 1 with P Spice, assuming the smallest time
constant τmin. Analyze the transient response of the circuit, with proper frequency as you calculated in question 3. Generate plots of both input and output
signal waveforms. Label the intervals in which rise time and fall time are defined.(Hint: Refer to section 7.8 and 8.9 for instructions of transients analysis
with P Spice.)
6. Generate plots for the three types of transient responses, assuming the
circuit in question 2 is being studied. You can do it with MATLAB or by
simulating the circuit in P Spice. Both input and output and output signal
waveforms should be monitored. Suppose that the function generator produces
a square wave at 1V ppk at 10kHz.
4