ECED 2000 – Electric Circuits
Chapter 7 – Response of First-Order RL & RC Circuits
- Natural Response
- Step Response
- General Solution
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ECED 2000 – Electric Circuits
7.1, 7.2 – Natural Response of RL and RC Circuits
Natural Response / Source-Free (undriven) or Zero-Input Response:
Connection of a resistor with an inductor or a capacitor without a source.
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ECED 2000 – Electric Circuits
L
7.1, 7.2 – Natural Response of RL and RC Circuits
di
+ Ri = 0
dt
i (t )
C
t
v(t ) = v(0) ⋅ e
di
R
=
−
∫i (0 ) i L ∫0 dt
ln
i(t )
R
=− t
i(0)
L
dv v
+ =0
dt R
−t RC
, t≥0
v(t ) = v(0 ) ⋅ e −t τ , t ≥ 0
where τ = RC is the time constant
i (t ) = i (0) ⋅ e − ( R L )⋅t , t ≥ 0
i (t ) = i(0 ) ⋅ e −t τ , t ≥ 0
L
where τ = is the time constant
R
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ECED 2000 – Electric Circuits
Time constant
1
XK
0.9
0.8
x(t ) = Ke
0.7
−t /τ
0.6
0.5
0.4
0.368K
0.3
0.135K
0.2
0.0498K
0.1
t
0
0
1
2
3
4
5
Xτ
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ECED 2000 – Electric Circuits
Example 7.1
a) Find iL(t) for t ≥ 0
b) Find io(t) for t ≥ 0+
c) Find vo(t) for t ≥ 0+
d) Find % of initial inductor energy
dissipated in 10Ω resistor
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ECED 2000 – Electric Circuits
Example 7.2
a) Find i1(t), i2(t) & i3(t) for t ≥ 0
b) Find initial energy in Inductors, E0
c) Find energy in Inductors as t → ∞, E∞
d) Show that the total Energy delivered to
Resistors is E0 ̶ E∞
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ECED 2000 – Electric Circuits
Example 7.3
a) Find vC(t) for t ≥ 0
b) Find vo(t) for t ≥ 0+
c) Find io(t) for t ≥ 0+
d) Find the total Energy dissipated
in the 60kΩ resistor
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ECED 2000 – Electric Circuits
L
i (t )
di
+ Ri = Vs
dt
t
di
R
∫i (0 ) i − (Vs R ) = − L ∫0 dt ,
ln
7.3 – The Step Response
i (t ) − i(∞ )
R
=− t
i (0) − i (∞ )
L
C
Vs
= i (∞ )
R
dvC vC
+
= Is ,
dt
R
I s R = vC (∞ )
vC (t ) = vC (∞ ) + (vC (0) − vC (∞ )) ⋅ e −t RC , t ≥ 0
vC (t ) = vC (∞ ) + (vC (0 ) − vC (∞ )) ⋅ e −t τ , t ≥ 0
i (t ) = i (∞ ) + (i(0) − i (∞ )) ⋅ e − ( R L )⋅t , t ≥ 0
i (t ) = i (∞ ) + (i (0 ) − i(∞ )) ⋅ e −t τ , t ≥ 0
L
where τ = is the time constant
R
τ = RC is the time constant
For vC (0 ) = 0 :
(
)
vC (t ) = vC (∞ ) 1 − e −t τ , t ≥ 0
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ECED 2000 – Electric Circuits
7.3 – The Step Response
RL circuit curves
For i (0) = 0 :
(
)
i (t ) = i (∞ ) 1 − e −t τ , t ≥ 0
v(t ) = L
di(t )
dt
v(t ) = Vs ⋅ e −t τ , t ≥ 0 +
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ECED 2000 – Electric Circuits
Example 7.5
a) Find i(t) for t ≥ 0
b) Find v(0+)
c) Verify v(0+) another way
d) Find t for v(t) = 24V
e) Plot i(t) & v(t)
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ECED 2000 – Electric Circuits
Example 7.6
a) Find vo(t) for t ≥ 0
b) Find io(t) for t ≥ 0+
Norton Equivalent Circuit, t > 0
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ECED 2000 – Electric Circuits
Graphical interpretation of the step response
x(∞)
For the case x(∞)>x(to)
0.95 [x(∞)-x(to)]
0.632 [x(∞)-x(to)]
Elapsed time (t-to)
x(to)
0
1τ
2τ
3τ
4τ
5τ
General solution for natural and step response of RL and RC circuits:
−
x(t ) = [ Final _ value] + ([ Initial _ value] − [ Final _ value])e
Elapsed _ time
Time _ Cons tan t
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1. Identify variable of interest;
2. Determine initial value at t=to+
3. Calculate the final value
4. Calculate the time constant
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ECED 2000 – Electric Circuits
Example
The current and voltage at the terminals of the inductor in the circuit are:
i (t ) = (4 + 4e −40t ) A, t ≥ 0;
v(t ) = −80e − 40tV , t ≥ 0 +.
a) Specify the numerical values of Vs, R, Io, and L.
b) How many milliseconds after the switch has been closed does the energy stored
in the inductor reach 9 J?
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