Lecture 14. Transient Circuits

• Transient & steady state waveforms1 t d i it• 1st order circuits

• The time constant• Solving 1st order circuitsg• Applications

1

Solving Differential Equationsg q

• Example: For zero initial conditions, solve

)(4)(30)(11)(2

2

tutydt

tyddt

tyd=++

• Laplace transform approach automatically includes initial conditions in the solution

dtdt

in the solution

)0()()( yssdt

tyd−=⎥⎦

⎤⎢⎣⎡ YL

)0(')0()()( 22

2

yysssdt

tyd−−=⎥

⎦

⎤⎢⎣

⎡⎦⎣

YL

2

A RC Circuit

+ vr(t)

R+

+

+ -r( )

-Cvs(t)

-

vc(t)

• One capacitor and one resistorOne capacitor and one resistor• The source and resistor may be equivalent to a circuit

with many resistors and sources.

Lecture 14 3

A RC Circuit: Loop Analysisp y

v (t)

R+

+

+ -vr(t)

+

-Cvs(t)

-

vc(t)

KVL around the loop:p

vr(t) + vc(t) = vs(t)

Lecture 14 4

A RC Circuit

Rtitv )()( = Rtitvr )()( =

tdvCti c )()( =dt

Cti )( =

)(tdv )()()( tvtvdt

tdvRC scc =+

This is a 1st order differential equation.

Lecture 14 5

A RL circuit

( )

+

v(t)is(t) R L

-

KCL at the top node: )()()( tititi sLr =+

)()(1)( tidttvLR

tvs=+ ∫

dttditv

Ldttdv

Rs )()(1)(1

=+

Lecture 14 6

dtLdtR 1st order differential equation.

What is a 1st Order Circuit?

• Voltages and currents in a 1st order circuit satisfy a 1stg yorder differential equation

• Any circuit with a single energy storage element, an bit b f d bit b farbitrary number of sources, and an arbitrary number of

resistors is a circuit of order 1.• Any voltage or current in such a circuit is the solution toAny voltage or current in such a circuit is the solution to

a 1st order differential equation.

Lecture 14 7

Solving 1st order circuitsSolving 1 order circuits

Lecture 14 8

1st Order Differential Equationq

tvttdv sc )()(1)(Th RC i it

RCtv

RCdts

cc )()()(

=+The RC circuit:

The RL circuit: dtdiRtv

LR

dtdv s )()()(

=+The RL circuit: dtLdt)(

1)(d )()(1)( tftydt

tdy=+

τBoth circuits can be expressed by:

Lecture 14 9

The Particular SolutionThe Particular Solution

1)(tdy )()(1)( tftydt

tdy=+

τ

• The particular solution yp(t) is usually a weighted sum of f(t) and its first derivative.( )

• If f(t) is constant, then yp(t) is constant.• If f(t) is sinusoidal, then yp(t) is sinusoidal.

Lecture 14 10

The Complementary Solutionp y

The complementary solution has the following form:p y gτ/)( t

c Kety −=What is K? K=yc(0) – initial condition.

For an RC circuit t = RC

What is τ?

For an RC circuit, t = RCFor an RL circuit, t = L/R

Lecture 14 11

Laplace Methodp

)()(1)( tftytdy=+ )()( tfty

dt=+

τ

1 )()(1)0()( sFsYyssY =+−τ

)()0()( +=sFysY

ττ11)(

++

+ sssY

Lecture 14 12

Laplace Methodp

)()0()( +=sFysY

ττ11)(

++

+ sssY

}1)({}1

)0({)( 11

++

+= −−

s

sFLs

yLty

ττ++ ss

⎪⎫

⎪⎧

⎪⎭

⎪⎬

⎪⎩

⎪⎨

++= −− τ

1)()0()( 1/

s

sFLeyty t

Lecture 14 13

⎪⎭

⎪⎩

+τ

s

The Time Constant

10 4τ = 10-4

y(t)

Lecture 14 14

Interpretation of τp

• τ is the amount of time necessary for an exponential to y pdecay to 36.7% of its initial value.

• -1/τ is the initial slope of an exponential with an initial value of 1

Lecture 14 15

value of 1.

Applications of a 1st Order RC CircuitApplications of a 1st Order RC Circuit

• Computer RAMComputer RAM– A dynamic RAM stores ones as charge on a

capacitor.– The charge leaks out through transistors modeled

by large resistances.The charge must be periodically refreshed

Lecture 14 16

– The charge must be periodically refreshed.

Implications of the Time Constantp

• Should the time constant be large or small:g– Computer RAM– The low-pass filter for the envelope detector– The sample-and-hold circuit– The electrical motor

Lecture 14 17

Transient Waveforms

• The transient portion of the waveform is a decaying p y gexponential:

Lecture 14 18

Steady State Responsey p

• The steady state response depends on the source(s) in y p p ( )the circuit.– Constant sources give DC (constant) steady state

responses.– Sinusoidal sources give AC (sinusoidal) steady state

responses.responses.

Lecture 14 19

More Applicationspp

• The low-pass filter for an envelope detector in a p psuperhetrodyne AM receiver.

• A sample-and-hold circuit for a PCM encoder:– The capacitor is charged to the voltage of a waveform

to be sampled.– The capacitor holds this voltage until an A/D– The capacitor holds this voltage until an A/D

converter can convert it to bits.

Lecture 14 20

Low Pass Filter

• The voltage in the filter may look like this:g y

Lecture 14 21

Sample and Holdp

• The voltage in the sample and hold circuit might look like g p gthis:

Lecture 14 22

Class Examplesp

• Example 6-4(page 210)

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