lecture 17 1
First-Order Circuits (6.1-6.2)
Prof. Phillips
March 24, 2003
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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
• One capacitor and one resistor• The source and resistor may be equivalent
to a circuit with many resistors and sources.
R
Cvs(t)
+
–
vc(t)
+ –vr(t)
+–
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Applications Modeled bya 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)
KVL around the loop:
vr(t) + vc(t) = vs(t)
R
Cvs(t)
+
–
vc(t)
+ –vr(t)
+–
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Differential Equation(s)
)()(1
)( tvdxxiC
tRi s
t
dt
tdvCti
dt
tdiRC s )(
)()(
dt
tdvRCtv
dt
tdvRC s
rr )(
)()(
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What is the differential equation for vc(t)?
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A First-Order RL Circuit
• One inductor and one resistor• The source and resistor may be equivalent
to a circuit with many resistors and sources.
v(t)is(t) R L
+
–
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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)
KCL at the top node:
)()(1)(
tidxxvLR
tvs
t
v(t)is(t) R L
+
–
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The Differential Equation
dt
tdiL
dt
tdv
R
Ltv s )()()(
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1st Order Differential Equation
Voltages and currents in a 1st order circuit satisfy a differential equation of the form
)()()(
tftvadt
tdv
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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:
Initial conditions determine the value of K.
/)( ttac KeKetv
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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
• For an RC circuit, = RC
• For an RL circuit, = L/R
/)( tc Ketv
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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 = I R
Capacitor I(t) = C dV(t)/dt I=0; open
Inductor V(t) = L dI(t)/dt V=0; short
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Class Examples
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