Transcript First-Order Circuits (7.1-7.2)

First-Order Circuits (7.1-7.2)
Dr. Holbert
April 12, 2006
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1st Order Circuits
• Any circuit with a single energy storage
element, an arbitrary 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 to a 1st order differential
equation.
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Important Concepts
•
•
•
•
The differential equation
Forced and natural solutions
The time constant
Transient and steady-state waveforms
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A First-Order RC Circuit
+
vr(t) –
R
vs(t)
+
–
+
vc(t)
C
–
• One capacitor and one resistor
• The source and resistor may be equivalent
to a circuit with many resistors and sources.
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Applications Modeled by
a 1st Order RC Circuit
• Computer 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.
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The Differential Equation(s)
+
vr(t) –
R
vs(t)
+
–
+
vc(t)
C
–
KVL around the loop:
vr(t) + vc(t) = vs(t)
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Differential Equation(s)
t
1
Ri(t )   i( x)dx  vs (t )
C 
dv s (t )
di (t )
RC
 i (t )  C
dt
dt
dv s (t )
dv r (t )
RC
 vr (t )  RC
dt
dt
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What is the differential equation
for vc(t)?
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A First-Order RL Circuit
+
is(t)
R
L
v(t)
–
• One inductor and one resistor
• The source and resistor may be equivalent
to a circuit with many resistors and sources.
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Applications Modeled by
a 1st Order LC Circuit
• The windings in an electric motor or
generator.
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The Differential Equation(s)
+
R
is(t)
L
v(t)
–
KCL at the top node:
t
v(t ) 1
  v( x)dx  is (t )
R L 
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The Differential Equation
dis (t )
L dv (t )
v(t ) 
L
R dt
dt
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1st Order Differential Equation
Voltages and currents in a 1st order circuit
satisfy a differential equation of the form
dv (t )
 a v(t )  f (t )
dt
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Important Concepts
• The differential equation
• Forced (particular) and natural
(complementary) solutions
• The time constant
• Transient and steady-state waveforms
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The Particular Solution
• The particular solution vp(t) is usually a
weighted sum of f(t) and its first derivative.
– That is, the particular solution looks like the
forcing function
• If f(t) is constant, then vp(t) is constant.
• If f(t) is sinusoidal, then vp(t) is sinusoidal.
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The Complementary Solution
The complementary solution has the following
form:
vc (t )  Ke
a t
 Ke
t / 
Initial conditions determine the value of K.
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Important Concepts
• The differential equation
• Forced (particular) and natural
(complementary) solutions
• The time constant
• Transient and steady-state waveforms
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The Time Constant ()
• The complementary solution for any 1st
order circuit is
vc (t )  Ke
t / 
• For an RC circuit,  = RC
• For an RL circuit,  = L/R
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What Does vc(t) Look Like?
 = 10-4
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Interpretation of 
• The time constant, , is the amount of time
necessary for an exponential to decay to
36.7% of its initial value.
• -1/ is the initial slope of an exponential
with an initial value of 1.
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Implications of the Time
Constant
• Should the time constant be large or small:
– Computer RAM
– A sample-and-hold circuit
– An electrical motor
– A camera flash unit
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Important Concepts
• The differential equation
• Forced (particular) and natural
(complementary) solutions
• The time constant
• Transient and steady-state waveforms
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Transient Waveforms
• The transient portion of the waveform is a
decaying exponential:
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Steady-State Response
• The steady-state response depends on the
source(s) in the circuit.
– Constant sources give DC (constant)
steady-state responses.
– Sinusoidal sources give AC (sinusoidal)
steady-state responses.
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LC Characteristics
Element
V/I Relation
DC Steady-State
Resistor
V(t) = R I(t)
V=IR
Capacitor I(t) = C dV(t)/dt
I=0; open
Inductor
V=0; short
V(t) = L dI(t)/dt
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Class Examples
• Learning Extension E7.1
• Learning Extension E7.2
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