Transcript First-Order Circuits cont'd
First-Order Circuits Cont’d
Dr. Holbert April 17, 2006 ECE201 Lect-20 1
Introduction
• In a circuit with energy storage elements, voltages and currents are the solutions to linear, constant coefficient differential equations.
• Real engineers almost never solve the differential equations directly.
• It is important to have a qualitative understanding of the solutions.
ECE201 Lect-20 2
Important Concepts
• • The differential equation for the circuit
Forced
(particular) and
natural
(complementary) solutions •
Transient
and
steady-state
• 1st order circuits: the responses
time constant
( ) ECE201 Lect-20 3
The Differential Equation
• Every voltage and current is the solution to a differential equation.
• In a circuit of order
n
, these differential equations have order
n
.
• The number and configuration of the energy storage elements determines the order of the circuit.
n
# of energy storage elements ECE201 Lect-20 4
The Differential Equation
• Equations are linear, constant coefficient:
a n d n x
(
t
)
dt n
a n
1
d n
1
x
(
t
)
dt n
1 ...
a
0
x
(
t
)
f
(
t
) • The variable
x
(
t
) could be voltage or current.
• The coefficients
a n
through
a
0 depend on the component values of circuit elements.
• The function
f(t)
depends on the circuit elements and on the sources in the circuit.
ECE201 Lect-20 5
Building Intuition
• Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition to be developed: – Particular and complementary solutions – Effects of initial conditions ECE201 Lect-20 6
Differential Equation Solution
• • • The total solution to any differential equation consists of two parts:
x
(
t
)
= x p
(
t
)
+ x c
(
t
)
Particular
(
forced
) solution is
x p
(
t
) – Response particular to a given source
Complementary
(
natural
) solution is
x
– Response common to all sources, that is, due to the “passive” circuit elements
c
(
t
) ECE201 Lect-20 7
The Forced Solution
• The forced (particular) solution is the solution to the non-homogeneous equation:
a n d n x
(
t
)
dt n
a n
1
d n
1
x
(
t
)
dt n
1 ...
a
0
x
(
t
)
f
(
t
) • The particular solution is usually has the form of a sum of
f(t)
and its derivatives.
– If
f(t)
is constant, then
v p (t)
is constant ECE201 Lect-20 8
The Natural Solution
• The natural (or complementary) solution is the solution to the homogeneous equation:
a n d n x
(
t
)
dt n
a n
1
d n
1
x
(
t
)
dt n
1 ...
a
0
x
(
t
) 0 • Different “look” for 1 st and 2 nd order ODEs ECE201 Lect-20 9
First-Order Natural Solution
• The first-order ODE has a form of
dx c
(
t
)
dt
1
x c
(
t
) 0 • The natural solution is
x c
• Tau ( ) is the
time constant
(
t
• For an RC circuit, =
RC
• For an RL circuit, =
L/R
)
Ke
t
/ ECE201 Lect-20 10
Initial Conditions
• The particular and complementary solutions have constants that cannot be determined without knowledge of the initial conditions.
• The initial conditions are the initial value of the solution and the initial value of one or more of its derivatives.
• Initial conditions are determined by initial capacitor voltages, initial inductor currents, and initial source values.
ECE201 Lect-20 11
Transients and Steady State
• The steady-state response of a circuit is the waveform after a long time has passed, and depends on the source(s) in the circuit.
– Constant sources give DC steady-state responses • DC SS if response approaches a constant – Sinusoidal sources give AC steady-state responses • AC SS if response approaches a sinusoid • The transient response is the circuit response minus the steady-state response.
ECE201 Lect-20 12
Step-by-Step Approach
1.
2.
3.
4.
Assume solution (only dc sources allowed):
x
(
t
) = K 1 + K 2 e
-t/
At
t
=0 – , draw circuit with
C L
as open circuit and as short circuit; find I L (0 – ) or V C (0 – ) At
t
=0 + , redraw circuit and replace
C
or
L
with appropriate source of value obtained in step #2, and find
x
(0)=K 1 +K 2 At
t
= , repeat step #2 to find
x
( )=K 1 ECE201 Lect-20 13
Step-by-Step Approach
5.
Find time constant ( ) Looking across the terminals of the
C
or
L
element, form Thevenin equivalent circuit; =R Th C or =L/R Th 6.
Finish up Simply put the answer together.
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Class Examples
• Learning Extension E7.3
• Learning Extension E7.4
• Learning Extension E7.5
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