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LaPlace Transforms in Design and Analysis of Circuits Part 1: Basic Transforms Course No: E04-015

Credit: 4 PDH

Thomas G. Bertenshaw, Ed.D., P.E.

Continuing Education and Development, Inc. 9 Greyridge Farm Court Stony Point, NY 10980 P: (877) 322-5800 F: (877) 322-4774 info@cedengineering.com

LaPlace Transforms in Design and Analysis of Circuits

Part 1 - Basic Transforms

by Tom Bertenshaw

Why Use the LaPlace Transform?? In a short synopsis; using the LaPlace transform method of solving circuit differential equations allows the building of simple algebraic transfer functions that mathematically model the actual circuit; provides a quick method for calculating transfer function amplitude and phase as a function of frequency; and creates a foundation for the rapid calculating and graphing of circuit loop behavior with respect to stability. Summing the above, the use of transforms provides a simple procedure for performing an essential engineering function; i.e., predict circuit output as a function of input. We will get to all of these as a matter of course, but first comes the fundamentals. We are all familiar with the concept of calculating the output by using a voltage division network, i.e.,

because the circuit is series, t oI I= , then:

1 2 2

2

1 2

or

in o

o i

V Vz z z

zV Vz z

=+

n= +

which is a very handy and simple way of predicting the output for any given input. It is, in fact, a very simple transfer function. Be aware that we are slanting the whole further development of analysis and design around the transfer function technique as illustrated in the simple example above.

Rev. A.2 1

Notice that the equation has the general form of

( ) inXferout *=

where the term means "transfer function". Re-arranging this equation isolates and defines the transfer function as

Xfer

Xferin

out=

Transfer functions have the innate ability to allow prediction of output as a function of input, and as such are extremely valuable engineering tools. Getting there when the circuit equations are integral/differential can be cumbersome, and error fraught. Nevertheless circuit equations are generally integral/differential that can be masked and "worked around" by LaPlace's technique.

A Series Circuit Consider a low-pass filter;

The characteristic equation describing the voltage/current relationship for a resistor is:

V I R= or riv = (lower case denotes time changing variables)

whereas for a capacitor the relationship is:

dvi Cdt

=

Because a capacitor is an open to DC, the vast majority of circuit problems in design or analysis that include capacitors occur in alternating or changing current situations. i.e.,

there is a dtdv .

Forming the ratio ino VV for an RC network is not as straightforward as developing the impedance ratio as in the first case mentioned for the general, or frequency independent, case. However a general method of solving the problem follows, and that is how we will proceed.

Rev. A.2 2

Observing that the circuit is a series circuit, and realizing that the current through the resistor MUST equal the current through the capacitor, we can write:

Cr ii = or

dtdv

CR

vv ccin =

where and represent the same voltage. In other words, cv oV oV vc= ; and is whatever the case may be: DC, changing DC or AC. The notation for voltage is changed to denote that it can be time variable.

inv

After suitable rearranging we get:

c inc

v Vdv dt dtRC RC

+ =

This relationship has an integrating factor of:

1

or tdt

RC RCe e . After both sides are multiplied by the integrating factor, the equation becomes:

tRC

cd v e

+ dtRCeV

dtRCv

eRCt

incRCt

=

Integrating both sides gives us:

where is the constant of integration.t t

inRC RCc nst nst

vv e e c cRC

= +

Isolating the dependent variable ( )cv the relationship becomes:

RCt

nstin

c ecRCv

v

+=

at ;0=tRCv

c innst = ; therefore the complete solution is:

=

RCt

inc eRC

vv 1

Rev. A.2 3

After learning LaPlace Transform pairs and their applications, and having appealed to the use of the LaPlace Transform instead of using Ordinary Differential Equations, the process devolves to simple Algebra.

dtdv

CR

vv ccin =

becomes by direct application:

( ) ( ) ( )ssCv

Rsvsv

ccin =

(Step 1)

rearranging yields:

( ) ( )

+

=

RCsRC

svsv inc 1

(Step 2)

which directly inverts to:

=

RCt

inc eRC

vv 1 (Step 3)

a far simpler process. Because converting differential circuit equations into their LaPlace Transform pairs is so labor saving (and by extension, error saving) it is well worth while to become familiar with the process.

The Definition Learning to convert expressions to their LaPlace equivalent is straightforward. In every case we apply the definition of the LaPlace Transform:

( ) ( ) dtetfsF st= 0

This expression says that the LaPlace Transform, ( )sF , equals the integral of the time function, , times the transform function . ( )tf ste Ultimately the utility of the LaPlace Transform is to predict circuit behavior as a function of time, and by extension, using Bode's technique, to predict output amplitude and phase as a function of frequency. Further, the transform of the transfer function provides for plotting the poles and zeros of the transfer function, which in turn, lays the foundation for the Root Locus method of analyzing circuit stability as a function of amplitude and frequency. These topics will be covered in some detail as progression through the modules develops an ever increasing sophistication in the uses of the LaPlace Transform.

Rev. A.2 4

The Basic Transform Pairs Suppose we have a constant DC voltage of amplitude K . The task is to apply the definition and develop the LaPlace Transform of the constant K . Directly applying the definition:

( ) = Ktf ( )sK

seKdteKdteKsF

ststst ====

000|

Recall that stst ee

1= and therefore, as t , 01 ste and also that .10 =e

sKK (Transform Pair #1)

For example a constant voltage of 10vdc transforms to s10 ; 413.32 transforms to

s32.413 ; -22.87 s87.22 ; etc.

Quite often in the physical world we are confronted with signals that exponentially decay over time can and be expressed as:

( ) tetf =

where represents some physical parameter and having the units of sec1 or (radians). In electronics

rads is usually a time-constant generated by either an RC

(resistance/capacitance) or an RL (resistance/inductance) network. Recall that a single time-constant is defined when 1=t ; the value of the time-constant being 1 . When

1=t , meaning that the signal decays to 37% of its peak value in one time-constant.

37.= te

Graphically, it looks like this:

Rev. A.2 5

Exponential Decay

1.200

1.000

0.800

0.600

0.400

0.200

0.000 Time

This particular form occurs so often in nature, especially when applied to signals that decay over time, that it is also assigned a transform. Suppose there is an exponentially decaying voltage of value . atKe

To find the transform we again directly apply the definition:

( ) ( ) ( )( )

( ) ( )asK

asKedteKdteeKsFKetf

tastasstatat

+=

+====

+ + 000|

asKKe at+

(Transform pair #2)

For example:

31010 3+

s

e t

78.636.236.2 78.6

+

s

e t

and so forth.

Rev. A.2 6

At this point we need to take a side excursi into Euler's Identities, as the use of these

ll of the following four identities can be developed and established by using the

onidentities does two things: greatly simplifies the calculus of trigonometric functions by avoiding integration by parts and familiarizes us with the notational shorthand found in the literature. Aprocesses of Maclaurin's infinite series, which is found in any beginners Calculus text. Then by definition:

( ) ( )tjte tj sincos += ( ) ( )tjte tj sincos =

Recall that the units on is in rads (radians), and that f 2= , also that tf 1= (units on f is in Hertz and t is in seconds), and as usual 1=j ther of the above identities:

. Also note that from ei

ubtracting the lower from the upper, and rearranging, we get:

1)0sin()0cos(0 == je j

S

( )jeet

tjtj

2sin

=

then adding the two, and rearranging, we get:

2)cos(

tjtj eet

+

=

Using the Identities

Suppose we have a simple sinusoid, such as:

)sin()( tKtf =

then,

Ktf =)(jee tjtj

2

finding the LaPlace Transform by direct application then,

( ) ( )

=

+

dtedtej

KdtejeeK tjstjsst

tjtj

000 22

erforming the integration, and evaluation at the limits: p

Rev. A.2 7

( )

( )( )

( ) ( ) ( ) 220011

2||

2

+=

+

=

+

+

+

sK

jsjsjK

jse

jse

jK tjstjs

so

22)sin( +

s

KtK (Transform pair #3)

For example:

( )( )( )2