RLC transients
When there is a step change (or switching) in a circuit with capacitors
and inductors together, a transient also occurs. With some differences:
EE 201
•
Energy stored in capacitors (electric fields) and inductors (magnetic
fields) can trade back and forth during the transient, leading to
possible “ringing” effects.
•
The transient waveform can be quite different, depending on the
exact relationship of the values of C, L, and R.
•
The math is more involved.
RLC transient – 1
series RLC
vR
+
Vf
Vi t = 0
VS
–
R
+
–
i
C
L
–
+
vC
–
=
+
+
vL +
Voltage source makes an abrupt change from Vi to Vf at t = 0.
t < 0:
i = 0.
t >> τ:
i = 0.
vR = 0.
vR = 0.
vL = 0.
vL = 0.
vC = Vi.
vC = Vf.
What is τ? What happens in between?
EE 201
RLC transient – 2
vR
+
Vf
Vi t = 0
VS
R
+
–
i
C
L
=
+
vC
–
vL +
–
t > 0:
–
+
+
=
=
+
+
+
+
=
Second-order differential equation.
EE 201
RLC transient – 3
Diff. Eq. – a bit of a review (or preview)
( )
( )
+
( )=
( )+
+
( )= ( )
g(x) is a “forcing function”
( )
ftr(x) → transient solution (homogenous)
fss(x) → steady-state solution (particular)
fss(x) is any function that you can find that is a solution to the full
differential equation. Usually find it with trial-and-error. The form of
g(x) suggests what to try.
ftr(x) is the solution to the homogenous differential equation.
( )
+
( )
+
( )=
The solution will have two unknown constants that will be specified
using initial conditions, f(0) and df(0)/dx.
EE 201
RLC transient – 4
+
()=
+
=
( )+
()
Since the forcing function is a constant, try using vss(t) = constant = Vss.
+
+
=
Vss = Vf . (Easy!)
(We could have determined this from the physics of the circuit, also.)
()=
( )+
Now we must find the transient part.
EE 201
RLC transient – 5
homogenous solution
+
Guess:
+
=
=
+
+
+
=
Will need to determine s.
exp ( )
+
=
=
so
+
+
=
=
=
()=
EE 201
±
+
=
+
RLC transient – 6
Initial conditions
()=
+
+
Need to determine A and B. Use initial conditions, vc(0) and dvc(0)/dt.
The details will depend on s1 and s2, and, as we will see, there are
three possible cases.
In any case:
vC(0) = Vi
()
=
()
But don’t necessarily know iC(0) since capacitor current can change
abruptly. However, in the series circuit, iC(t) = iL(t). Inductor current
cannot change abruptly. So
()
=
()
EE 201
()
=
=
( )
=
For this particular case. May
not be true in every case.
RLC transient – 7
roots of characteristic equation
Transient behavior depends on the values of s1 and s2.
=
+
=
Rename things slightly: R/2L = α and 1/LC = ωo.
=
+
=
α is the damping factor or decay constant [s–1]$
ωo is the resonant frequency or undamped natural frequency [radian/s].
EE 201
RLC transient – 8
Overdamped response
If
(α > ωo) then s1 and s2 are both real and negative.
>
The capacitor voltage consists of two decaying exponentials.
()=
+
+
A fast, one-way trip from Vi to Vf.
Finally, use initial conditions to find A and B
( )=
=
+
+
solve to give:
=
=
+
=
=
()=
EE 201
=
+
+
RLC transient – 9
Over-damped response: Vi = 5 V, Vf = 20 V, R = 1 k!, L = 15 mH, and
C = 0.5 µF.
25.0!
vc (V)!
20.0!
15.0!
10.0!
5.0!
0.0!
-1.0!
0.0!
1.0!
2.0!
3.0!
4.0!
5.0!
t (ms)!
EE 201
RLC transient – 10
Underdamped response
If
(α < ωo) then s1 and s2 are complex!
<
=
=
+
Write as complex numbers.
s1 =
s2 =
+
=
+
=
=
+
=
(ωd is the damped frequency)
=
()=
EE 201
(For EE/CprEs only.)
=
+
+
RLC transient – 11
()=
+
+
(Euler’s identity.)
= cos + sin
()=
=
()=
[ (cos
+ sin
[( + ) cos
+ (
cos
sin
+
)+
(cos
) sin
sin
)] +
]+
+
There is an oscillation in the response. The voltage changes from Vi to
Vf, but wiggles back and forth a few times in the process. The
oscillation dies out according to the damping factor over about 5 time
constants, where the time constant τ = 1/α.
EE 201
RLC transient – 12
Use initial conditions to find A’ and B’.
( )=
=
=
+
solve to give:
=
=
=
+
=
()=
EE 201
cos
+
sin
+
RLC transient – 13
Under-damped response: Vi = 5 V, Vf = 20 V, R = 300 !, L = 25 mH,
and C = 60 nF.
30.0!
25.0!
vc (V)!
20.0!
15.0!
10.0!
5.0!
0.0!
-0.3!
0.0!
0.3!
0.5!
0.8!
1.0!
t (ms)!
Changing R, L, and C changes the oscillation frequency and the
damping rate.
EE 201
RLC transient – 14
Critically damped response
If
(α = ωo) then s1 = s2 (double root).
=
Note: This will almost never happen. It will be the wildest fluke if the
components have exactly the correct ratios to meet the above requirement.
For the most part, critical damping is only of academic interest.
()=
=
+
exp
+
=
+
+
/
This causes a bit of a problem, because we are left with only one term
in the general solution, and hence only one coefficient – not enough to
satisfy the initial conditions.
This suggests that there must be another solution lurking around in the
math. In the special circumstances for the critically damped case, the
homogeneous equation can be written:
+
EE 201
+
=
RLC transient – 15
+
+
+
+
+
=
where
=
=
+
=
=
+
=
=
+
+
=
+
=
()=(
EE 201
+ )
Now there are two constants.
RLC transient – 16
()=(
+ )
+
Also a fast trip from Vi to Vf.
Use initial conditions to find A and B.
( )=
=
+
=
solve to give:
=
=
=
=
()=
EE 201
( +
)
+
RLC transient – 17
Critically-damped response: Vi = 5 V, Vf = 20 V, R = 1 k!, L = 15 mH,
and C = 60 nF. (Same as overdamped plot of slide 10, except C is
smaller.)
25.0!
vc (V)!
20.0!
15.0!
10.0!
5.0!
0.0!
-50.0!
0.0!
50.0!
100.0!
150.0!
200.0!
250.0!
300.0!
t (µs)!
Note the significantly faster rise time compared to the overdamped
response.
EE 201
RLC transient – 18
parallel RLC
If
IS
iR
Ii t = 0
R
ic
C
iL
+
L v
–
t < 0:
t >> τ:
v = 0.
v = 0.
iR = 0.
iR = 0.
iC = 0.
iC = 0.
iL = Ii.
iL = If.
Current source makes an abrupt change from Ii to If at t = 0.
=
+
=
+
+
+
=
t > 0:
initial conditions (t = 0)
=
+
+
+
+
=
( )=
()
=
( )
=
( )
=
=
EE 201
RLC transient – 19
+
+
=
This has exactly the same form as the series case, with inductor current
replacing capacitor voltage. The steady-state function will be iss = If,
and the transient function will have the general form:
()=
+
=
+
+
=
=
=
=
EE 201
=
damping factor - note the
difference from series case
resonant frequency
RLC transient – 20
We will obtain the exact same set of results for the parallel case:
>
( >
overdamped – both roots are real and negative.
)
()=
<
+
( <
underdamped – roots are complex conjugates.
)
()=
=
()=
EE 201
cos
( =
+
sin
+
=
critically damped – repeated root.
)
( +
+
)
+
(Virtually impossible to
have critical damping.)
RLC transient – 21