# First-Order Circuits

```First-Order Circuits
First-Order Circuits Overview
• RC &amp; RL natural response
• RC &amp; RL step response
ECE 221
• First-order circuits the easy way
• Examples
Portland State University
t=0
+
R v
-
C
iC
Ver. 1.65
RC Natural Response: Initial Conditions t = 0−
V
0
iR
For t &lt; 0, the switch has been closed for a long time.
Introduction: RL &amp; RC Circuits
• First-Order Circuits: Circuits that contain a single capacitor or
inductor and a network of DC sources, resistors, and switches
• Can also analyze circuits containing multiple capacitors and
inductors if we can combine them into a single equivalent
capacitor/inductor
• Will analyze using KCL &amp; KVL
• Generates a ﬁrst-order diﬀerential equation
• Simple enough we can use ordinary calculus to solve
• Eventually we will discuss the easy way
• Circuits which contain no sources are called source-free
ECE 221
First-Order Circuits
Ver. 1.65
• If there is no source in the circuit, is called the natural response
1 Portland State University
t=0
+
R v
-
C
iC
Ver. 1.65
RC Natural Response: Initial Conditions t = 0+
V
0
iR
For t &lt; 0, the switch has been closed for a long time.
First-Order Circuits
• What is the voltage across the capacitor at t = 0+ ?
ECE 221
• What is the voltage across the capacitor at t = 0− ?
3 Portland State University
• How much energy is stored in the capacitor at t = 0+ ?
Ver. 1.65
• How much energy is stored in the capacitor at t = 0− ?
First-Order Circuits
• What is the current in the capacitor for t = 0+ ?
ECE 221
• What is the current in the capacitor for t = 0− ?
Portland State University
2
4
t=0
=
=
=
=
=
+
R v
-
C
iC
First-Order Circuits
Ver. 1.65
Ver. 1.65
RC Natural Response: Final Conditions t → ∞
V0
iR
As t → ∞, the switch has been opened for a long time.
• What is the current in the capacitor as t → ∞?
• What is the voltage across the capacitor as t → ∞?
First-Order Circuits
• How much energy is stored in the capacitor as t → ∞?
ECE 221
• Where did the energy go?
Portland State University
v(t)
V0
v(t) =
ECE 221
v(t)
−
RC
1
−
dt
RC
1
−
dv
RC
t
1
−
dv
RC 0
1
(t − 0)
−
RC
−t
RC
t
V0 e− RC t ≥ 0
t≤0
V0
RC Natural Response Continued (1)
dv(t)
dt
1 dv(t)
dt
v(t) dt
du
u
v(t)
1
du
u
V0
ln
ln (v(t)) − ln (V0 ) =
Portland State University
V0
t=0
iR
+
R v
-
= 0
= 0
= 0
First-Order Circuits
C
iC
RC Natural Response General Equation
Solve for v(t).
V
0
t=0
+
R v
-
C
iC
V0
= − e−t/RC
R
= 21 CV02 e−2t/RC
First-Order Circuits
iR
RC Natural Response Summary
ECE 221
iC (t) + iR (t)
dv(t) v(t)
C
+
dt
R
dv(t) v(t)
+
dt
RC
We know for t ≤ 0, v(t) = V0 . For t ≥ 0,
5 Portland State University
For t &gt; 0,
v(t) = V0 e−t/RC
ECE 221
v(t)
i (t) = −
C
R
2
w(t) = 21 C (v(t))
7 Portland State University
Ver. 1.65
Ver. 1.65
8
6
0
t≤0
t≥0
First-Order Circuits
V0
t
V0 e− RC
2τ
3τ
Time (s)
RC Circuit Natural Response
1τ
v(t) =
ECE 221
4τ
RC Natural Response Continued (3)
V0
0
V0
0
Portland State University
2
Time (s)
3
4
τ = 1.0 s
τ = 2.0 s
τ = 0.5 s
RC Circuit Natural Response
1
ECE 221
First-Order Circuits
5τ
5
Ver. 1.65
Ver. 1.65
The time constant τ determines how quickly the voltage settles to its
ﬁnal value.
0
First Order Response &amp; Time Constants
Portland State University
Voltage (V)
Voltage (V)
V0
t≤0
t
V0 e− RC t ≥ 0
v(t) =
• The voltage response of the RC circuit is an exponential decay
• Called the natural response of the circuit
– Natural response: Behavior of the circuit with no external
sources of excitation
First-Order Circuits
+
R v
-
C
iC
Ver. 1.65
• The rate at which the voltage decreases is measured by the time
constant, τ
• τ = RC
• v(t) = V0 e−t/τ
• In 5 time constants v(t) is within 1% of its ﬁnal value
ECE 221
• We will treat this as the steady-state value
9 Portland State University
t=0
iR
RC Natural Response Energy
V
0
wR (t) =
First-Order Circuits
Ver. 1.65
Solve for the energy stored in the capacitor, wC (t) for t ≥ 0 and the
energy dissipated by the resistor, wR (t).
wC (t) =
1
2C
2
1
2 Cv(t)
2
V0 e−t/τ
=
2 −2t/τ
1
2 CV0 e
ECE 221
=
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10
12
wR (t) =
2
1
2 CV0
First-Order Circuits
First-Order Circuits
Ver. 1.65
Ver. 1.65
RC Natural Response Energy Key Ideas
1 − e−2t/τ
wC (t) =
2 −2t/τ
1
2 CV0 e
• How much energy is dissipated in the resistor?
• How much energy is initially stored in the capacitor?
ECE 221
Example 1: Workspace (1)
ECE 221
• How much energy is stored in the capacitor as t → ∞?
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18 mA
t=0
0.2 &micro;F
20 kΩ
0.125m v1
First-Order Circuits
First-Order Circuits
-
v1
+
Example 1: RC Natural Response
5 kΩ
ECE 221
Example 1: Workspace (2)
ECE 221
Find the time constant for t &gt; 0. Find v1 (t).
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Ver. 1.65
16
14
10 kΩ
Ver. 1.65
t=0
+
R v
-
L
i
RL Natural Response: Initial Conditions t = 0−
t=0
I
0
For t &lt; 0, the switches have not changed for a long time.
t=0
L
I
0
t=0
t=0
di(t)
dt
v(t) + Ri(t)
di(t)
+ Ri(t)
dt
L
ECE 221
+
R v
-
+
-
R v
= 0
= 0
L
i
L
i
= −Ri(t)
First-Order Circuits
Ver. 1.65
Ver. 1.65
20
18
RL Natural Response: Initial Conditions t = 0+
t=0
I
0
For t &lt; 0, the switches have not changed for a long time.
First-Order Circuits
• What is the current in the inductor at t = 0+ ?
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We know for t &lt; 0, i(t) = I0 . For t &gt; 0,
Solve for i(t).
RL Natural Response General Equation
ECE 221
• What is the current in the inductor at t = 0− ?
+
L
i
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• What is the voltage across the inductor at t = 0+ ?
Ver. 1.65
• What is the voltage across the inductor at t = 0− ?
First-Order Circuits
• How much energy is stored in the inductor at t = 0+ ?
ECE 221
• How much energy is stored in the inductor at t = 0− ?
Portland State University
t=0
R v
-
Ver. 1.65
RL Natural Response: Final Conditions t → ∞
t=0
I
0
As t → ∞, the switches have not changed for a long time.
• What is the current in the inductor as t → ∞?
• What is the voltage across the inductor as t → ∞?
First-Order Circuits
• How much energy is stored in the inductor as t → ∞?
ECE 221
• Where does this energy go?
Portland State University
=
=
=
=
=
R
− i(t)
L
R
− dt
L
R
− dv
L
R t
−
dv
L 0
R
− (t − 0)
L
R
−t
L
R
I0 e−t L
I0
t≥0
t≤0
RL Natural Response Continued (1)
di(t)
dt
di(t)
i(t)
du
u
i(t)
1
du
u
I0
ln
=
First-Order Circuits
t≤0
t≥0
ECE 221
i(t)
i(t)
I0
ln (i(t)) − ln (I0 ) =
Portland State University
For t &gt; 0,
I0
i(t) =
I0 e−t/τ
Ver. 1.65
Ver. 1.65
• The current response of the RL circuit is an exponential decay
First-Order Circuits
• Called the natural response of the circuit
L
R
ECE 221
• Time constant: τ =
Portland State University
For t &gt; 0,
t=0
+
R v
-
L
i
RL Natural Response Summary
t=0
I0
R
i(t) = I0 e−t L
R
= −RI0 e−t L
+
R
v(t) = −i(t)R
First-Order Circuits
= 21 LI02 e−2t L
ECE 221
w(t) = 21 L (i(t))2
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t=0
R v
-
L
i
RL Natural Response Energy
t=0
I
0
wL (t) =
wR (t) =
ECE 221
2 −2t/τ
1
2 LI0 e
First-Order Circuits
Ver. 1.65
Ver. 1.65
Solve for the energy dissipated by the resistor, wR (t), for t ≥ 0.
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22
24
First-Order Circuits
First-Order Circuits
Ver. 1.65
Ver. 1.65
RL Natural Response Energy Key Idea
wR (t) = 21 LI02 1 − e−2t/τ
wL (t) = 21 LI02 e−2t/τ
• How much energy is dissipated in the resistor?
• How much energy is initially stored in the inductor?
ECE 221
Example 2: Workspace (1)
ECE 221
• How much energy is stored in the inductor as t → ∞?
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54 V
3Ω
4.5 Ω
t=0
+
v
-
200 mH
iL
50 i1
Example 2: RL Natural Response
9Ω
ECE 221
First-Order Circuits
First-Order Circuits
Example 2: Workspace (2)
ECE 221
Find iL (t) for t ≥ 0, v(t) for t &gt; 0, and i1 (t) for t &gt; 0.
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200 Ω
Ver. 1.65
28
100 Ω
i1
Ver. 1.65
t=0
R
+
v(t)
-
i(t) t = 0
C
V
0
RC Step Response: Initial Conditions t = 0−
V
s
For t &lt; 0, the switches have not changed for a long time.
t=0
R
R
+
v(t)
-
+
v(t)
-
i(t) t = 0
C
i(t) t = 0
C
V0
Ver. 1.65
RC Step Response: Initial Conditions t = 0+
V
s
For t &lt; 0, the switches have not changed for a long time.
First-Order Circuits
• What is the current in the capacitor for t = 0+ ?
t=0
RC Step Response Solution
ECE 221
• What is the current in the capacitor for t = 0− ?
V
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• What is the voltage across the capacitor at t = 0+ ?
Ver. 1.65
• What is the voltage across the capacitor at t = 0− ?
First-Order Circuits
i(t) t = 0
V
s
Solve for v(t).
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Vs − v(t)
− v(t)
RC
Vs − v(t)
Vs
ECE 221
= Ri(t)
dv(t)
= RC
dt
dv(t)
dt
=
First-Order Circuits
Ver. 1.65
We know v(t) = V0 for t ≤ 0. We need to solve for v(t) for t ≥ 0.
V0
• How much energy is stored in the capacitor at t = 0+ ?
ECE 221
R
+
C
First-Order Circuits
Ver. 1.65
• How much energy is stored in the capacitor at t = 0− ?
Portland State University
t=0
v(t)
-
0
RC Step Response: Final Conditions t → ∞
V
s
As t → ∞, the switches have not changed for a long time.
• What is the current in the capacitor as t → ∞?
• What is the voltage across the capacitor as t → ∞?
ECE 221
• How much energy is stored in the capacitor as t → ∞?
Portland State University
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32
dv
V0
= ln
v(t) − Vs
V0 − Vs
= ln (v(t) − Vs ) − ln (V0 − Vs )
=
dv(t)
1
dt =
RC
V − v(t)
s
dv(t)
−1
dt =
RC
v(t) − Vs
du
−1
dv =
RC
u − Vs
t
v(t)
0
−t
RC
−t
RC
First-Order Circuits
= Vs + (V0 − Vs ) e−t/RC
ECE 221
= (V0 − Vs ) e−t/τ
= Vs
= Vs + (V0 − Vs ) e−t/τ
= vf + vn (t)
Natural &amp; Forced Response
v(t)
du
u − Vs
RC Step Response Solution Continued
−1
RC
Portland State University
For t &gt; 0
v(t)
vf
vn (t)
Ver. 1.65
• Forced response is what the response eventually is forced to
• Also called the steady-state response
• Natural response is the part of the response due to the change
ECE 221
First-Order Circuits
Ver. 1.65
• Also called the transient response because it is temporary
Portland State University
V
s
1
ECE 221
R
+
v(t)
-
=
i(t) t = 0
C
First-Order Circuits
3
RC Step Response
2
Time (s)
Forced
Natural
Total
4
First-Order Circuits
5
Ver. 1.65
Ver. 1.65
36
34
1
= C − (V − Vs ) e−t/τ
0
τ
V0
Vs
−
e−t/τ
R
R
V0
RC Step Response General Equations
t=0
v(t) = Vs + (V0 − Vs ) e−t/τ
ECE 221
dv
i(t) = C
dt
1
(V0 − Vs ) e−t/τ
R
=−
0
RC Step Response Graphed
= i(0+ )e−t/τ
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Vs
V0
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Voltage (V)
Vs
R
+
L
i(t)
t=0
I0
t=0
RL Step Response: Initial Conditions t = 0−
t=0
v
-
For t &lt; 0, the switches have not changed for a long time.
Vs
R
+
L
i(t)
t=0
I0
t=0
RL Step Response: Initial Conditions t = 0+
t=0
v
-
For t &lt; 0, the switches have not changed for a long time.
R
+
t=0
I0
First-Order Circuits
• What is the current in the inductor at t = 0+ ?
t=0
v
-
L
i(t)
RL Step Response Solution
ECE 221
• What is the current in the inductor at t = 0− ?
t=0
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• What is the voltage across the inductor at t = 0+ ?
Ver. 1.65
• What is the voltage across the inductor at t = 0− ?
First-Order Circuits
t=0
V
s
Solve for i(t).
39 Portland State University
Vs − Ri(t)
Vs − Ri(t)
Vs
− i(t)
R
ECE 221
= v(t)
di(t)
= L
dt
L di(t)
R dt
=
First-Order Circuits
Ver. 1.65
Ver. 1.65
We know i(t) = I0 for t ≤ 0. We need to solve for i(t) for t ≥ 0.
t=0
• How much energy is stored in the inductor at t = 0+ ?
ECE 221
R
+
I
0
First-Order Circuits
Ver. 1.65
• How much energy is stored in the inductor at t = 0− ?
t=0
v
-
L
i(t)
RL Step Response: Final Conditions t → ∞
Portland State University
V
s
As t → ∞, the switches have not changed for a long time.
• What is the current in the inductor as t → ∞?
• What is the voltage across the inductor as t → ∞?
ECE 221
• How much energy is stored in the inductor as t → ∞?
Portland State University
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40
=
I0
du
u − Vs /R
RL Step Response Solution Continued
dv
i(t) − Vs /R
I0 − Vs /R
0
di(t)
R
dt =
L
Vs /R − i(t)
R
di
− dt =
L
i − Vs /R
R
du
− dv =
L
u − Vs /R
t
i(t)
−R
L
R
−t
L
R
−t
L
= Vs /R + (I0 − Vs /R) e−t L
= ln
ECE 221
R
First-Order Circuits
First-Order Circuits
Ver. 1.65
Ver. 1.65
= ln (i(t) − Vs /R) − ln (I0 − Vs /R)
i(t)
Portland State University
= Vf + (V0+ − Vf )e−t/τ
RC &amp; RL Circuits the Easy Way
v(t)
i(t) = If + (I0+ − If )e−t/τ
• Every circuit we have seen follows this pattern
• It is true, in general
ECE 221
• This leads to a general approach to analysis
Portland State University
=
I0
R
Vs /R + (I0 − Vs /R) e−t L
t≤0
t≥0
RL Step Response General Equations
i(t)
v(t)
t≤0
t≥0
First-Order Circuits
di(t)
= L
dt
0
R
(Vs − RI0 ) e−t L
=
ECE 221
There is a pattern here . . .
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t→∞
xf ≡ lim x(t)
Ver. 1.65
→0
x0+ ≡ lim x(||)
First-Order Circuits the Easy Way
x(t) = xf + (x0+ − xf )e−t/τ
1. Combine any networks of inductors (capacitors) into their single
inductor (capacitor) equivalents.
2. Use DC analysis to solve for the current (voltage) ﬂowing through
the inductor (across the capacitor) at t = 0−
3. Use DC analysis to ﬁnd the initial value of the variable of interest
at t = 0+
4. Use DC analysis to ﬁnd the steady state value of the variable of
interest as t → ∞
First-Order Circuits
Ver. 1.65
5. Find the equivalent resistance seen by the inductor (capacitor) for
t&gt;0
6. Solve for the time constant τ
ECE 221
7. Plug into the general equation
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44
100 &micro;A
20 kΩ
0.4 &micro;F
10 kΩ
i1(t)
5 kΩ
Example 3: RC Circuit
5 kΩ
i2(t)
t=0
5. Find i1 (t).
10 V
1. Find i1 (0− ) and i2 (0− ).
6. Find i2 (t).
First-Order Circuits
2. Find i1 (0+ ) and i2 (0+ ).
3. Explain why i1 (0− ) = i1 (0+ ).
ECE 221
First-Order Circuits
Example 3: Workspace (2)
ECE 221
4. Explain why i2 (0− ) = i2 (0+ ).
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Portland State University
Ver. 1.65
Ver. 1.65
45 Portland State University
20 mA
48 Ω
First-Order Circuits
Example 3: Workspace (1)
ECE 221
t=0
12 mH
i1
First-Order Circuits
8 mH
i2
Example 4: RL Circuit
15 Ω
ECE 221
Find vo (t), i1 (t), and i2 (t).
47 Portland State University
-
vo
+
Ver. 1.65
10 mA
Ver. 1.65
46
48
Portland State University
i2
0.4 H
+
v
-
100 Ω
First-Order Circuits
i1
Example 5: RL Circuit
ECE 221
Example 4: Workspace (1)
0.6 H
At t = 0− , i1 = 1 mA and i2 = −1 mA.
1. Find v(t) for t ≥ 0+ .
ECE 221
First-Order Circuits
Ver. 1.65
Ver. 1.65
2. What percentage of the energy initially stored in the inductors is
dissipated in the 100 Ω resistor?
Portland State University
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First-Order Circuits
Example 4: Workspace (2)
ECE 221
First-Order Circuits
Example 5: Workspace
ECE 221
Ver. 1.65
Ver. 1.65
52
50
0.6 H
0.4 H
i1
+
v
-
Example 6: RL Circuit
i2
At t = 0− , i1 = 2 mA and i2 = −1 mA.
1. Find v(t) for t ≥ 0+ .
12 kΩ
ECE 221
20 V
4 kΩ
100 Ω
First-Order Circuits
t=0
200 Ω
240 kΩ
Example 7: Hybrid Circuit
12 kΩ
800 Ω
First-Order Circuits
iL
500 mH
Ver. 1.65
vL
Ver. 1.65
2. What percentage of the energy initially stored in the inductors is
dissipated in the 100 Ω resistor?
C
vC
Portland State University
i
33.33 &micro;F
ECE 221
Solve for the following for t &gt; 0: vC (t), iC (t), vL (t), iL (t).
Portland State University
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First-Order Circuits
Example 6: Workspace
ECE 221
First-Order Circuits
Example 7: Workspace (1)
ECE 221
Ver. 1.65
Ver. 1.65
56
54
Portland State University
Portland State University
First-Order Circuits
Example 7: Workspace (2)
ECE 221
First-Order Circuits
Example 8: Workspace (1)
ECE 221
Ver. 1.65
Ver. 1.65
vL
iL
2H
+
-
5 kΩ
10 V
t=0
4 kΩ
Example 8: Hybrid Circuit
5 kΩ
t=0
20 mA
ECE 221
6 kΩ
First-Order Circuits
First-Order Circuits
Example 8: Workspace (2)
ECE 221
+
5 mF
58
vc
Ver. 1.65
60
ic
Ver. 1.65
-
Solve for the following for t &gt; 0: vC (t), iC (t), vL (t), iL (t).
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t=0
5 kΩ
125 kΩ
8 nF
20 kΩ
10 V
-10 V
12 kΩ
vo
Example 9: First-Order Op Amp Circuit
3V
ECE 221
First-Order Circuits
Ver. 1.65
There is no energy stored in the capacitor at t = 0. How long does it
take for the op amp to saturate? Repeat this question assuming there
is an initial voltage of 2 V on the capacitor.
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First-Order Circuits
Example 9: Workspace
ECE 221
Ver. 1.65
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