Transient 1st
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Transcript of Transient 1st
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RC and RL Transient Analysis Basics
In these notes, we briefly review the basics of RC and RL circuits and their response
(transient) to a step change in an input voltage or current. The RC and RL circuits
discussed lead to 1st order differential equations in time, and have basic solutions typicalof 1st order differential equations.
A. GENERAL SOLUTION
The general solution for the four basic circuits is
X(t) = 1 + 2/
= (= ) + (= 0+
) (= )[ ] /
Here, X(t) is the transient output voltage or transient output current, depending on the
specific circuit being considered. The time constant is either RC or L/R, depending on
whether one has an RC or an RL circuit.
To obtain the values of X(t = 0+) and X(t = infinity), we use DC analysis. Before t = 0,
the input is constant (current source or voltage source depending on circuit). Under DC
conditions, the CAPACITOR becomes an OPEN CIRCUIT while an INDUCTOR
becomes a CLOSED CIRCUIT. Use these replacements to find the DC currents and
voltages at t < 0, before the input switches.
CAPACITORS: Use the fact that "voltages stick on capacitors" to translate a
capacitor voltage just before t = 0 into a capacitor voltage just after t = 0.
INDUCTORS: Use the fact that "currents stick through inductors" to translate an
inductor current just before t = 0 into an inductor current just after t = 0.
B. RC NETWORKS (time constant is RC)
Figure 1a below illustrates the case of series combination of a voltage source, resistorand capacitor. The voltage source switches between voltage levels at t=0 as shown in
Figure 1b. In this case, the relevant variable (voltage or current) is the common
current I(t) flowing around the loop, through the voltage source, resistor, and
capacitor.
The variable X(t) above is therefore the common current I(t).
Knowing the common current, the voltages across the resistor and
capacitor can be readily calculated.
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Figure 2a below illustrates a parallel combination of a current source, resistor and
capacitor. The current source switches between current levels at t=0 as shown in
Figure 2b. In this case, the relevant variable would be the common voltage V(t)
across the current source, resistor and capacitor.
The variable X(t) above is therefore the common voltage V(t). Knowing the common voltage V(t), the current through the resistor or
capacitor can be readily calculated.
Vin(t)
Vout(t)R
C
(a)
+
Vin(t)
time tt=0
V0
(b)
V1Current
I
Figure 1
Iin(t)
Vout(t)R
C
(a)
Iin(t)
time tt=0
I0
(b)
I1
VoltageV
Figure 2
C. RL NETWORKS (time constant is L/R)
Figure 3a below illustrates the case of series combination of a voltage source, resistor
and inductor. The voltage source switches between voltage levels at t=0 as shown in
Figure 3b. In this case, the relevant variable (voltage or current) is the common
current I(t) flowing around the loop, through the voltage source, resistor, and
capacitor.
The variable X(t) above is therefore the common current I(t).
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Knowing the common current, the voltages across the resistor and
capacitor can be readily calculated.
Figure 4a below illustrates a parallel combination of a current source, resistor and
inductor. The current source switches between current levels at t=0 as shown in
Figure 4b. In this case, the relevant variable would be the common voltage V(t)across the current source, resistor and capacitor.
The variable X(t) above is therefore the common voltage V(t).
Knowing the common voltage V(t), the current through the resistor or
capacitor can be readily calculated.
For RL circuits, the value of X(t = 0+) is based on the value of the current through the
inductor just before the input switches. Since the current through an inductor can not
change instantaneously, the current just after switching is the same as just before
switching. X(t = 0+) = X(t = 0-).
Vin(t)
Vout(t)R
L
(a)
+
Vin(t)
time tt=0
V0
(b)
V1
CurrentI
Figure 3
Iin(t)
Vout(t)R
L
(a)
Iin(t)
time tt=0
I0
(b)
I1
VoltageV
Figure 4