The Impulse Response, the Step Response and Convolution

Background: The unit impulse response of linear, time-invariant, continuous-time (LTIC) system and, in

particular passive RLC circuits, is of significant importance. Knowing the impulse response of an LTIC

system it is possible to determine the output of that system to any applied signal. Given the differential

equation model of the system, the characteristics roots are readily determined and the unit impulse

response can be subsequently calculated using the impulse response matching method.

Consider, as our example, the series RLC circuit shown in Figure 1 below.

Figure 1: Series RLC Circuit

Defining the input voltage as x t( ) and the output or capacitor voltage as y t( ), the differential equation

model of the circuit is readily developedi

. Denoting the mesh current as i t( ) , we can write the following

mesh equation.

( ) ( )

( ) ( ) 1

0

t

di t

Ri t L i d x t

dt C

τ τ

−∞

+ + − = ∫

(1)

Differentiating and writing in operator form yields,

( ) ( )

( ) ( )

( ) ( )

2

2

2

di t di t dx t 1 1 1 R

R L i t D D i t Dx t

dt dt C dt L RC L

+ + = ⇒⇒ + + = (2)

Writing the output voltage as,

( ) ( ) ( ) ( ) 1 1 ( )

t

c

i t

y t v t i d y t

C C D

τ τ

−∞

= = ⇒ = ∫

(3)

We can now write Equation (2) as,

( ) ( ) 2 R 1 1 D D y t x t

L RC LC

⇒ + + = (4)

R

200Ω

L

50mH

C

V1 x(t) 0.1µF y(t)

+

–

+

–

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Equation (4) constitutes the input-output, differential equation model of the circuit and the response of

the circuit is determined by the characteristic roots of the model. Writing the characteristic equation as,

( ) 2 R 1

Q

L RC

λ λ λ

= + + (5)

Then the characteristic roots are,

2

1 2

1

4

,

2

R R

L L RC λ λ

− ± −

=

(6)

And both the unit impulse response and the unit step response of the system are determined by these

characteristic roots.

Analysis of Equation (6) reveals three typical cases:

Case 1:

Over-damped

System

2

1 2

4

: , .

R

are real and distinct

L LC

λ λ

>

(7)

Case 2:

Critically-damped

system.

2

1 2

4

: .

2

R R are real and equal

L LC L

λ λ

= = = −

(8)

Case 3:

Under-damped

System

2

1 2

4

: .

R

j are a complex conjugate pair

L LC

λ λ σ ω

< = = − ±

(9)

Analytical and experimental exploration reveals the following: in the over-damped case, the circuit

response is dominated by two exponentially decaying components; in the under-damped case, the

response is dominated by an exponentially decaying, sinusoidal component; and, as you may expect, the

critically-damped case occurs as we transition from an over-damped to an under-damped response.

It is obvious, that the nature of the response can be readily modified to satisfy the conditions provided

in Equations (7)-(9) by simply changing the circuit resistance. In this context, it is common to re-write the

characteristic equation (5) to define the circuit damping factor ζ , and the undamped natural frequency

ωn

, as follows:

( ) ( ) 2 2

2

1 2

1

2

2

, 1 1

n n n

n n

R C Q where and

LC L

and j when

λ λ ζω λ ω ω ζ

λ λ ζω ω ζ ζ

= + + = =

= − ± − <

(10)

And the conditions in Equation (7)-(9) become the following:

1

1

2

1

Over damped

R C Critically damped

L

Under damped

ζ

> −

= = = −

< −

(11)

This laboratory investigates the impulse response of a series RLC circuit and its relationship to other

forced circuit responses. The circuit pulse response is explored and a scaled version of the unit impulse

response is measured. We then investigate the effect of damping on the circuit pulse response. The

process of convolution is emulated by providing a pulse train as an input to the circuit and a scaled

version of the unit step response is obtained. The unit step response is then measured for comparison

purposes.

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Experimental Procedure:

Impulse Response Characteristics:

1. Construct the RLC circuit shown in Figure 1. As indicated, the output of the system, Vout, is the

voltage across the capacitor. On the Oscilloscope, have Channel 1 be the output from the

waveform generator and Channel 2 be Vout.

2. Use the function generator to create the input voltage V1 as a 10 Vpp, 1 kHz pulse wave with a

DC offset of 5 V. Set the pulse width to 100 µs. Make sure the output characteristic of the source

you are using on the function generator is set to High Z.

3. Vary the source frequency until the pulse response waveform decays to 0 within one-half of the

input signal period. Note this frequency setting. The corresponding half-period is the time it

takes for the circuit to reach a steady-state or equilibrium condition. This is typically referred to

as the settling time ( s

t ) of the circuit.

4. Measure the peak-to-peak voltage and frequency for the input and output signals. Capture a

representative image from the Oscilloscope.

5. Now, set the pulse width to 1.5 ms and lower it down. What do you notice about the amplitude

of the pulse response (amplitude of the output signal)? Provide an explanation of why this

phenomena occurs.

Measure and record the magnitude of the first peak of the pulse response for a range of pulse

widths to identify the relationship between pulse width and the magnitude of the pulse

response. Ensure that sufficient experimental points are recorded in the vicinity of the critical

pulse width (the one below which the magnitude of the first peak starts to diminish). Plot the

magnitude of the first peak versus pulse width for your report.

6. Record a representative pulse response for a pulse width below the critical value. This is an

experimental measurement of the circuit impulse response. Estimate the pulse strength (pulse

area). What experimental challenges prevent you from obtaining the unit impulse response?

7. For the series RLC circuit, the nature of the circuit response is closely related to the circuit

damping factor ζ , as defined in Equations (10) and (11). Select alternate values of R to produce

a circuit that is critically damped and one that is over-damped. Use damping factors of 1 and

2.67.

Record the response of the circuit with the same input pulse width for both alternate values of R

and show the difference between the two circuits by measuring the settling time of the pulse

response. Note that the settling time is the time required for the output signal to reach a steady

state or equilibrium condition, or being in the neighborhood of a final value.

8. The measured impulse response of a circuit depends significantly on the defined output

variable. Reconfigure the circuit in Figure 1 so that (a) the resistor voltage is the output variable,

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and (b) the inductor voltage is the output variable. For each case, repeat tasks 1 through 4.

Comment on the results obtained.

9. Build a model of the circuit in Multisim. Simulate the response of the circuit to produce

analogous results to those obtained experimentally in 6 and 7 above, for the under-damped,

critically-damped and over-damped circuits. For these results, obtain plots of the capacitor

voltage, the inductor voltage, the resistor voltage and the circuit current. The dynamics

observed in these responses should be similar to those observed in the pulse response of the

capacitor voltage, although the steady-state behavior of each response is different. Provide a

qualitative explanation of the steady-state behavior observed in each response.

Step Response Characteristics:

10. Re-construct the RLC circuit as shown in Figure 1. As indicated, the output of the system, Vout, is

the voltage across the capacitor. On the Oscilloscope, have Channel 1 be the output from the

waveform generator and Channel 2 be Vout.

11. To visualize the process or principle of convolution, set the Agilent Function generator to a pulse

waveform with an output frequency at 5.5kHz. Hit the Burst button and turn the burst function

on. Change the number of cycles to 1. Slowly increase the number of cycles until the output

waveform “levels off” and there is a visible maximum voltage over several peaks. Take a screen

capture. Explain your results. Mainly, why do you think the output waveform levels off?

12. The unit step response of the circuit can be measured by using a square wave whose voltage is 1

Vpp, has a duty cycle of 50% and a frequency of 2 kHz. Make sure you turn off the Burst mode

on the function generator. Vary the source frequency so the output waveform has a distinct

transient response during the high and low regions of the square wave. Make sure each

transient response decays to 1 or 0 in half the time that the region is active (1 being when the

waveform is high and 0 when it is low). This is the circuit unit step response. Take a screen

capture and note the source frequency.

13. Using resistance (R) values for damping constants of .385, 1, and 2.67, assuming the inductance

L and capacitance C are 50mH and 0.1 µF respectively. Estimate the Settling Time (i.e. the time it

takes for the response or output signal to get to the same value as the input), and Peak

Overshoot for each of the step responses. Take a screen capture each time.

14. Using your Multisim circuit model, simulate the response of the circuit to produce analogous

results to those obtained experimentally in (13) above, for the under-damped, critically-damped

and over-damped circuits. For these results, obtain plots of the capacitor voltage, the inductor

voltage, the resistor voltage and the circuit current. The dynamics observed in these responses

should be similar to those observed in the step response of the capacitor voltage, although the

steady-state behavior of each response is different. Provide a qualitative explanation of the

steady-state behavior observed in each response.

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Report:

Computational Results: Using the differential equation model of the RLC provided in Equations (1)-(3),

we can express the circuit model using the differential operator D, to obtain the model provided in

Equation (4), and we define the following system polynomials,

( ) ( ) ( ) ( )

( )

( )

2 1

1

Q D y t P D x t where

R

Q D D D and

L RC

P D

LC

=

= + +

=

(12)

Rewriting in terms of the variable s, we define the system transfer function as (Lathi, Pg. 196),

( ) ( )

( ) 2

1

1

P s LC H s

Q s R

s s

L RC

= =

+ +

(13)

The polynomials P s( ) and Q s( ) provide sufficient information to determine the unit impulse and the

unit step response of the system, in this case, of the series RLC circuit. This information can be

represented in Matlab using a special variable type known as a transfer function, where the coefficients

of P s( ) and Q s( ) are stored as the numerator and denominator polynomials of the circuit transfer

function. [We’ll explore transfer functions in significant detail later in the course when we use Laplace

analysis methods for circuit modeling.]

Create transfer function models for the RLC circuit using the Matlab function tf(.) for the under-damped,

critically damped, and over-damped circuits measured in tasks (7) and (13) above. Using the impulse(.)

and step(.) functions in Matlab, calculate and plot the unit impulse and the unit step response of each

circuit. Plot all three impulse responses on the same figure and plot all three unit step responses on the

same figure using the subplot feature. How do these compare with your experimental plots in terms of

the peak overshoot, the settling time and the steady-state behavior?

Write a complete lab report for this experiment using the guidelines provided. As always, work as a

group to prepare your experimental and computational results in final form. Discuss the results together

to clarify your own understanding and to critique the results in light of what you know about linear,

time-invariant systems.