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1
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.
2
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)
3
The Differential Equation(s)
KVL around the loop:
vr(t) + vc(t) = vs(t)
R+
-Cvs(t)
+
-
vc(t)
+ -vr(t)
4
Differential Equation(s)
)()(1
)( tvdxxiC
tRi s
t
dt
tdvC
dt
tdiRCti s )()(
)(
dt
tdvRC
dt
tdvRCtv sr
r
)()()(
5
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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The Differential Equation(s)
KCL at the top node:
v(t)is(t) R L
+
-
)()(1)(
tidxxvLR
tvs
t
dt
tdiL
dt
tdv
R
Ltv s )()()(
7
)0();(
0,)(
211
21
vKKvK
teKKtv RC
t
)0();(
0,)(
211
21
iKKiK
teKKti L
Rt
)(ti
)0();(
0,)(
211
21
iKKiK
teKKti RC
t
Why? (Superposition)
8
Solving First Order Circuits1. Draw the circuit for t=0- and find v(0-), or i(0-)
2. Use the continuity of the capacitor voltage, or inductor current, draw the circuit for t=0+ to find v(0+), or i(0+)
3. Find v( ), or i( ) at steady state
4. Find the time constant – For an RC circuit, = RC– For an RL circuit, = L/R
5. The solution is:/)]()0([)()( texxxtx
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The Time Constant
• For an RC circuit, = RC
• For an RL circuit, = L/R• -1/ is the initial slope of an exponential with an
initial value of 1• Also, is the amount of time necessary for an
exponential to decay to 36.7% of its initial value
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Implications of the Time Constant
• Should the time constant be large or small:– Computer RAM
– The low-pass filter for the envelope detector
– The sample-and-hold circuit
– The electrical motor
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2nd Order Circuits
• Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2.
• Any voltage or current in such a circuit is the solution to a 2nd order differential equation.
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A 2nd Order RLC Circuit
• The source and resistor may be equivalent to a circuit with many resistors and sources.
R+
-Cvs(t)
i (t)
L
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Applications Modeled by a 2nd Order RLC Circuit
• Filters– A bandpass filter such as the IF amp for the
AM radio.– A lowpass filter with a sharper cutoff than can
be obtained with an RC circuit.
14
The Differential Equation
KVL around the loop:
vr(t) + vc(t) + vl(t) = vs(t)
R+
-Cvs(t)
+
-
vc(t)
+ -vr(t)
L
+- vl(t)
i (t)
15
Differential Equation (cont’d)
)()(1)(
)( tvdxxiCdt
tdiLtRi s
t
dt
tdv
Lti
LCdt
tdi
L
R
dt
tid s )(1)(
1)()(2
2
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The Differential Equation
Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form:
)()()(
2)( 2
002
2
tftidt
tdi
dt
tid
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Example response: Over Damped
0
0.2
0.4
0.6
0.8
1
-1.00E-06
t
i(t)
-0.2
0
0.2
0.4
0.6
0.8
-1.00E-06
ti(t)
18
Example Response: Under Damped
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1.00E-05 1.00E-05 3.00E-05
t
i(t)
