## Description

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