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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)
23
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