Download Response of First-Order Circuits

Response of First-Order Circuits
RL Circuits
RC Circuits
ECE 201 Circuit Theory I
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The Natural Response of a Circuit
• The currents and voltages that arise when
energy stored in an inductor or capacitor is
suddenly released into a resistive circuit.
• These “signals” are determined by the
circuit itself, not by external sources!
ECE 201 Circuit Theory I
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Step Response
• The sudden application of a DC voltage or
current source is referred to as a “step”.
• The step response consists of the voltages
and currents that arise when energy is
being absorbed by an inductor or
capacitor.
ECE 201 Circuit Theory I
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Circuits for Natural Response
• Energy is “stored” in an inductor (a) as an initial
current.
• Energy is “stored” in a capacitor (b) as an initial
voltage.
ECE 201 Circuit Theory I
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General Configurations for RL
• If the independent
sources are equal to
zero, the circuits
simplify to
ECE 201 Circuit Theory I
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Natural Response of an RL Circuit
• Consider the circuit shown.
• Assume that the switch has been closed
“for a long time”, and is “opened” at t=0.
ECE 201 Circuit Theory I
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What does “for a long time” Mean?
• All of the currents and voltages have
reached a constant (dc) value.
• What is the voltage across the inductor
just before the switch is opened?
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Just before t = 0
• The voltage across the inductor is equal to
zero.
• There is no current in either resistor.
• The current in the inductor is equal to IS.
ECE 201 Circuit Theory I
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Just after t = 0
• The current source and its parallel resistor
R0 are disconnected from the rest of the
circuit, and the inductor begins to release
energy.
ECE 201 Circuit Theory I
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The expression for the current
di
L  Ri  0
dt
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di
L  Ri  0
dt
A first-order ordinary differential equation with
constant coefficients.
How do we solve it?
di
R
dt   idt
dt
L
ECE 201 Circuit Theory I
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di
R
dt   idt
dt
L
di
R
  dt
i
L
dx
R
   dy

x
L
i (t )
R
ln
  (t  t )
i (t )
L
i (t )
R
ln
 t
i (0)
L
i (t )
t
i ( t0 )
t0
0
0
i (t )  i (0)e
(
R
)t
L
ECE 201 Circuit Theory I
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The current in an inductor cannot change
instantaneously
• Let the time just before switching be called
t(0-).
• The time just after switching will be called
t(0+).
• For the inductor,
i(0)  i(0)  I
ECE 201 Circuit Theory I
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The Complete Solution
i (t )  I e
0
R
 t
L 
,t  0
ECE 201 Circuit Theory I
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The voltage drop across the resistor
v  iR
v  I Re
R
 t
L 
0
,t  0  .
v(0)  0
v(0)  I R
0
ECE 201 Circuit Theory I
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The Power Dissipated in the Resistor
v
p  vi  i R 
R
2
2
p  I Re
2
0
R
2 t
L 
,t  0 
ECE 201 Circuit Theory I
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The Energy Delivered to the Resistor
w   pdx   I R e
t
t
2
0
0
R
2 x
L 
0
1
w
I R (1  e
R
2
L
1
t  , w  LI
2

2
0
dx
R
2 t
L 
), t  0.
2
0
ECE 201 Circuit Theory I
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Time Constant
• The rate at which the current or voltage
approaches zero.
L
 
R
ECE 201 Circuit Theory I
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Rewriting in terms of Time Constant
i (t )  I e

t

0
v(t )  I R e

t

0
p  I Re
2
2
t

0
1
w  LI (1  e )
2
2
2
t

0
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Table 7.1 page 217 of the text
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Graphical Interpretation of Time Constant
• Determine the time constant from the plot
of the circuit’s natural response.
i (t )  I e

t

0
di
1
 I e
dt

di
I
(0)  
dt

I
i (t )  I  t

t

0
0
0
0
Straight Line Approximation
ECE 201 Circuit Theory I
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Graphical Interpretation
Tangent at t = 0 intersects the time axis at
the time constant
ECE 201 Circuit Theory I
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Procedure to Determine the Natural
Response of an RL Circuit
• Find the initial current through the inductor.
• Find the time constant,τ, of the circuit
(L/R).
• Generate i(t) from I0 and τ using
i (t )  I e

t

0
ECE 201 Circuit Theory I
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