Physical Systems Analysis Laboratory 1: Impulse Response

Jeff Santner, Nick Villagra, Rio Akasaka Swarthmore College Engineering

Performed January 31st, 2007 Turned in February 14, 2006

Abstract

In the experiment, the impulse response of two circuits was determined; an RC

circuit with output across the resistor and an RLC circuit with output across the capacitor

were used. The circuits behaved as predicted, although the results show that there was

significant resistance in the pulse generator. The RC circuit responded with a first order

exponential output at a negative voltage that approached zero. The RLC circuit gave a

second order exponential output that was positive, increased to a maximum, and then

approached zero.

Introduction

In this experiment, the response of first and second order circuits to impulses was

found. An ideal impulse is a signal that is zero for all time, and infinity when t=0. It also

has the property that the integral of the function is one over any boundary including t=0.

A circuit’s response to an impulse is very useful, because any signal can be represented

as the sum of many impulses that are time shifted and multiplied by constants. However,

ideal impulses do not exist, because a pulse must have some time duration. But, as long

as the duration of the pulse is much smaller than the time constant of the circuit, it can be

approximated by an impulse with the same area.

An impulse can be thought of as the electrical equivalent of a hammer strike.

When a physical object is hit with a hammer, it will respond with some sort of motion

(unless it breaks). This motion could be harmonic, like hitting a bell, causing oscillations

that eventually decay. Or, the motion could be overdamped, like hitting a car’s shock

absorber; it will start to move when hit, and reach maximum displacement shortly after

impact, and then return to its original location. Lastly, the motion could be first order,

like hitting a concrete wall, which displaces a small amount immediately (approximated

as instantaneously) and then returns to its original state. In this experiment, the concrete

wall is a series RC circuit with output across the resistor, and the shock absorber is a

series RLC circuit with output across the capacitor. The cause for these responses lies in

energy storage. The energy from the impulse is stored in the capacitor in the RC circuit

and inductor in the RLC circuit, and is then released after the impulse is over.

Procedure

First, we connected the pulse generator to the oscilloscope to select a pulse

duration that was short, but not noisy. We then selected a resistor and capacitor for an

RC circuit such that the time constant, RC, was much less than the pulse duration. We

chose a pulse of 10.04µs, a 1.1kΩ resistor, and a 0.112µF capacitor. We connected the

circuit in series with the output across the resistor going into the oscilloscope. We then

obtained two printouts of the output; one in close enough detail to show the length of the

pulse and the behavior of the circuit during the pulse, and another with a longer time span

to show the response of the circuit after the pulse.

For the second-order RLC circuit, we wanted a slightly overdamped circuit so we

chose a resistor, capacitor, and inductor such that R2C>4L and the time constant was still

much less than the pulse duration. We used the same pulse duration and resistor as the

RC circuit, but we changed to a 0.375µF capacitor, and added a 112mH inductor. We set

up the RLC circuit in series with the output across the capacitor connected to the

oscilloscope. Again, we obtained two printouts of the output; one in close enough detail

to show the length of the pulse and the behavior of the circuit during the pulse, and

another with a longer time span to show the response of the circuit after the pulse.

Theory

The unit step function, denoted by U(t), represents an input that is zero for and

unity thereafter.

t ≤ 0

⎭⎬⎫

⎩⎨⎧

≥<

=0 t1,0 t,0

)(tU

It serves as a convenient method to treating inputs that are non-zero after a certain time t,

since it would simply require shifting the unit step function and multiplying its amplitude

by some constant A. Furthermore, any function that consists of vertical and horizontal

lines can be modeled by one or more unit step functions.1 Whenever the input to a system

is the unit step function, the output is referred to as the unit step response.

s

Figure 1 The unit step function, U(t)

1 Close, Frederick, Newell. Modeling and Analysis of Dynamic Systems. pp. 208

The unit impulse

The unit impulse, denoted by δ(t), is an idealized pulse of area 1 and infinite height and

infinitesimal width.

⎭⎬⎫

⎩⎨⎧

≠=∞

=0 x0,0 x,

)(xδ

and

∫∞

∞−

= 1)( dxxδ

Since a pulse of infinite height (infinite voltage) and infinitesimal width (infinitesimal

time) cannot be recreated in a laboratory setting, it is approximated by having a pulse

width very small compared to the time constant of the circuit.

Figure 2 The response of an RC circuit due to different pulse widths

Image courtesy of Signals and Systems: models and behaviour by Charles Raymond Dillon, M. L. Meade.

With such an approximation it can be noted that the response of the circuit to that pulse

would depend on the area underneath the pulse, and not on its shape. This is because the

charge stored in a capacitor is dependent on pulse duration and its amplitude, or the area

beneath the pulse. It can be shown that the charge stored by the capacitor due to a very

short pulse is closest to the charge stored when a steady current is flowed through the

circuit for time T.

For example, an RC circuit is constructed with voltage input V0 and a resistor R, so a

current IRV= flows through, charging a capacitor, C. If the input voltage is applied as a

current for a time t, say half the length of the period T, or 0.5T = 0.5RC, the charge stored

would be

00

0 5.0)5.0(5.0 CVRCR

VTIQ ==⋅=

If instead a rectangular pulse were applied (of width 0.5T), we obtain

005.0

0 787.0393.0)1( QCVeCVCVQ c ==−== −

We shorten the pulse width, to 0.1T, and multiply the voltage accordingly by 5 so that the

area is maintained:

001.0

0 95.04758.0)1( QCVeCVCVQ c ==−== −

We can see that the actual charge stored approaches the charge stored by a flowing

current.

Figure 3 The unit impulse, δ(t)

The unit impulse is also the derivative of the unit step.

For the first order circuit, the following circuit was constructed, and the output across

the resistor was measured.

Figure 4 A first-order RC circuit

Using KVL:

010 ( )v Ri idt VC

tδ= = + −∑ ∫

01 ( )Vdq q tdt RC R

δ+ =

0 0 00

0 0 0

1 ( )Vdq q tdt RC R

δ+ + +

− − −

+ =∫ ∫ ∫

⎪⎪

⎩

⎪⎪

⎨

⎧

=

=

=+

∫

∫∫∫ →+

→−

+

−+

−

+

−

RTV

dtR

V

RV

tR

V

qRCdt

dqT

00

00

0

00

0

00

0

0

0

)(1

δ

Since we are approximating using a pulse rather than an impulse, we obtain:

RTV

oqq 00

0)( == ++

−

Using the s-substitution for the homogeneous solution:

1 0sRC

+ =

1sRC

= −

The test solution: 1 t

RCq Ae−

= at 0t = we have RTV

q 0= so RTV

A 0=

RCt

eRTV

q−

= 0

Using cqVC

=

RCt

c eRC

TVV

−= 0

0 ( )C RV V V tδ+ = , so

00 ( )

tRC

RV TV V t eRC

δ−

= − , where 0 ( )V tδ = 0 whenever 0t ≠ and T is the width of the pulse

0

0

( ) t=0

t >0t

R RC

V tV V T e

RC

δ−

⎧⎪= ⎨−⎪⎩

Figure 5 The RC unit impulse response

Figure 6 The unit impulse response approximated. The strikethroughs indicate that the height is considerably taller than indicated.

Figure 7 A detailed image of the pulse with the response of the resistor and capacitor.

For a second order impulse response, the following circuit was made:

Figure 8 The second-order RLC circuit constructed in lab.

Using KVL:

010 ( )div L Ri idt V

dt Ctδ= = + + −∑ ∫

Knowing dq dvi Cdt dt

= = the formula is rewritten as:

20

2

1 ( )Vd q R dq q tdt L dt LC L

δ+ + =

For the homogeneous solution, the source is set to 0 and using s-substitution:

2 1 0Rs s qL LC

+ + =

Applying the test solution:

2 1 0stRs s AeL LC

⎛ ⎞+ + =⎜ ⎟⎝ ⎠

2

1 21 ,

2R RsL L LC

α α⎛ ⎞= − ± − = − −⎜ ⎟⎝ ⎠

In order to obtain a response that would be easy to fit, without oscillations, a slightly

overdamped circuit response was constructed. By choosing a slightly overdamped circuit

(so that the two real solutions would be close to each other) the software we used to curve

fit could also better approximate the curves.

1 21 2

t thq A e A eα α− −= +

For an impulse, direct integration is used for initial conditions:

0 0 0 020

20 0 0 0

1 ( )Vd q R dq q tdt L dt LC L

δ+ + + +

− − − −

+ + =∫ ∫ ∫ ∫

00 00

0

0 (Vdq R qdt L L

++

−−

+ + = 1)

0

0

(0 )t

Vdq R qdt L L+

+

=

+ =

(0 )q + = 0, so , 0 01 2 0A e A e+ = 1 2A A= −

Hence

0

0t

Vdqdt L+=

=

0 0 01 1 2 2

hdq Vi A e A edt L

α α= = − − =

Using the substitution 1 2A A= −

( )1 2i Aα α= − + 1

( )0

11 2

VAL α α

=− +

( )1 20

2 1

( ) ( )t tVq t e e u tL

α α

α α− −⎡ ⎤= −⎣ ⎦−

Using the substitution q CV=

( )1 20

2 1

( )t tc

Vv e eLC

α α

α α− −⎡ ⎤= −⎣ ⎦−

u t

Results

The following table identifies the components used in this lab, measured using an RLC

meter:

Table 1: Values of components used

Capacitor Resistor Inductor Pulse width Input Voltage

First-order RC 0.112µF 1.1kΩ N/A 10.04µs 9.000V

Second-order RLC 0.375µF 1.1kΩ 112mH 10.04µs 9.375V

Figure 9: Experimental RC Voltage Response; 6789*( 0.00011)0.1001 0.3295 tV e− −= −

Figure 10: Multisim RC Response; 8121*( 0.0101)0.4615 tV e− −= −

Figure 11: Experimental RLC Response; 3633*( 0.00016) 6412*0.1345 0.4515 0.6166t tV e− − −= + − e

Figure 12: Multisim RLC Response; 2708*( 0.01) 7938*( 0.01)0.1744 0.3879 0.4932t tV e e− − − −= + −

RC circuit impulse response

Figure 13: Detail View of RC Response V0T : -6 -5 10.04 (10) 9.1875 9.22425 (10)⋅ ⋅ = ⋅

Figure 14: Full View of RC Response

Response form: it can be seen that the impulse response of the circuit is of the form

. The experiment provided the equation /t RCAe−

6789( 0.0001144)0.1 0.33 te− −−

RLC circuit impulse response

Figure 15: Detail View of RLC Response V0T: -6 -5 10.04 (10) 9.563 9.60125 (10)⋅ ⋅ = ⋅

Figure 16: Full View of RLC Response Response form: 3633*( 0.00016) 6412*0.1345 0.4515 0.6166t tV e− − −= + − e

MultiSim8 Simulation

Figure 17. The first order RC circuit modeled using MultiSim.

Figure 18. The second RLC circuit modeled using MultiSim.

Discussion

The response from the RC circuit was very different from the RLC circuit. As the

graph illustrates, the RC output immediately jumps up, and then becomes negative after

the impulse, and slowly returns to zero. This is because the impulse charges the capacitor

a little bit over the duration of the pulse. KVL dictates that the total voltage drop around

a closed loop must be zero, so the voltage drop across the resistor must be large. Then,

after the impulse, the capacitor discharges through the resistor in the opposite direction of

the impulse. This causes a negative voltage across the resistor that decays to zero.

However, the RLC output stays low during the impulse, and then starts to rise after the

impulse, has a maximum value, and then decays to zero. This is because the inductor

limits current during the impulse by absorbing almost all of the voltage. By KVL, this

means that the output rises a very small amount during the pulse. Then, after the pulse,

the inductor and capacitor create a second order response, where the inductor releases its

stored energy as a current, and this energy builds up in the capacitor as the voltage

output. If there were no resistor, the energy in the circuit would just oscillate between

voltage and current, but the resistor quickly dissipates this energy because we chose a

high enough resistance for an overdamped circuit. So, after the voltage reaches a

maximum, the capacitor discharges its energy through the resistor, and the output

approaches zero.

Using the equations derived in the theory section, we can compare predicted,

experimental, and simulated values for the circuits. For V0, we use the area under the

pulse, which was in the RC circuit and for the RLC circuit. -59.22425 (10)⋅ -59.60125 (10)⋅

Table 2: Experimental, MultiSim and Theoretical Values

Circuit Value Predicted from Theory Multisim Curve Fit

Experimental Curve Fit

1/RC 8117 8121 6789 1st Order V0/RC 0.7487 0.46149 0.3295

α1 -4358 -2702 -3633 α2 -5464 -7938 -6412

2.067 A10.8226 (using

experimental α’s)

0.38794 0.45154

-2.067

2nd Order

A2-0.8226 ((using

experimental α’s)

-0.4932 -0.61662

In the first order circuit, the results were below the prediction, and the error in the

2nd order circuit suggests that the term under the square root in the calculation for alpha

was too large. We calculated A1 and A2 in two different ways. The calculation involves

α1 and α2, we first used the predicted α1 and α2. However, the error was very large, so we

also calculated A1 and A2 using the α1 and α2 found experimentally. One should also note

that A1 should be -A2, but neither MultiSim nor the experimental values produced this

result.

The errors could be caused if the resistance in the circuit was more than the

measured resistance of 1.1kΩ. So, there was probably some resistance in the pulse

generator. This explains the error because after the impulse, the pulse generator is still

included in the circuit, and all current has to flow through it because it is in series with

the rest of the circuit.

The error in the experiment can be summarized in the table below:

Table 3: Summary of Error

Circuit Value Multisim Error (%)

Experimental Error (%)

1/RC .05 16.4 1st Order V0/RC 38.6 44

α1 38 16.6 α2 31.1 14.8 A1

(experimental)52.8 45

2nd Order

A2 (experimental)

40 25

A large source of error was manifested in the curve fitting process using

Kaleidagraph. In our experiment, data were collected on a very small time scale, and

therefore, the oscilloscope was more sensitive of picking up small fluctuations in the

voltage value leading to significant noise. The curve fit therefore took into account the

noise, which lead to the deviation from the theoretical values obtained. To limit the

source of this strain of error, we could repeat the experiment using a larger voltage to

minimize the effect of the noise (assuming that the noise does not increase with greater

voltage). We could also remove errant data points that strayed far from the trend by only

accepting data points within the range of no more than two standard deviations.

Error in the MultiSim values can also be partly attributed to the curve fitting

process in Kaleidagraph, because the fit we used did not completely align with the data

points acquired. This may have occurred because our guess for the constant values in the

curve fit equation were not accurate enough in relation to the ideal fit equation. This

error might have been diminished if we had attempted a greater variation of guesses for

the equation and narrowed the incongruity between the data and the curve. Since noise

does not appear in the MultiSim simulation, additional accuracy in the fit might have

been obtained if we collected data at smaller time intervals to make an even smoother

curve of points.

Conclusion

This experiment allowed for the verification of the impulse response in a first-

order RC and a second-order RLC circuit by approximating the impulse by a pulse of a

width that is very small in comparison to the time constant. Thus the charge held by the

capacitor and then released after the pulse was significantly close to the charge it would

hold if a steady current were flowing in a standard circuit. By measuring the response

across the resistor and the capacitor, extracting the data using Agilent, and curve-fitting

the result using Kaleidagraph, we were able to obtain values that agreed with the

theoretical values within error.

Future Work

Future work could involve analysis on output measured across the capacitor in the

first-order circuit, so as to compare against the output of the resistor to ensure that they

add up to the input voltage. The impulse response of other circuits, such as those

including op-amps, could be analyzed and compared with that of the RLC circuit to find

how the op-amp affects the response. We could also collect current data to see how it

would be affected by the impulse or how charge behaves as a result.

Bibliography

Smith, Ralph Judson; Dorf, Richard C. ‘Circuits, Devices and Systems’ 5th Edition.