Electrical Engineering – RLC Circuits Switching
EE Modul 3: RLC Circuits Switching
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Chapter Content
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RC Circuit (1)

A first-order circuit is characterized by a first-order differential
equation.
By KCL
iR  iC  0
Ohms law
v
dv
C
0
R
dt
Capacitor law
 Apply Kirchhoff’s laws to purely resistive circuit results in algebraic
equations.
 Apply the laws to RC and RL circuits produces differential equations.
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RC Circuit (2)

The natural response of a circuit refers to the behavior (in terms
of voltages and currents) of the circuit itself, with no external
sources of excitation.
Time constant
RC
Decays more slowly
Decays faster
•
•
The time constant  of a circuit is the time required for the response
to decay by a factor of 1/e or 36.8% of its initial value.
v decays faster for small  and slower for large .
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RC Circuit (3)
The key to working with a
source-free RC circuit is finding:
v(t )  V0 e t /
where
RC
1. The initial voltage v(0) = V0
across the capacitor.
2. The time constant  = R·C.
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RC Circuit (4)
Refer to the circuit below, determine vC, vx, and io for t ≥ 0.
Assume that vC(0) = 30 V.
Answer:
vC = 30e–0.25t V
vx = 10e–0.25t
io = –2.5e–0.25t A
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RC Circuit (5)
The switch in circuit below is opened at t = 0, find
v(t) for t ≥ 0.
Answer:
V(t) = 8e–2t V
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Chapter Content
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Summary / Zusammenfassung
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RL Circuit (1)

A first-order RL circuit consists of a inductor L (or its
equivalent) and a resistor (or its equivalent)
By KVL
vL  vR  0
di
L
 iR  0
dt
Inductors law
Ohms law
di
R
  dt
i
L
Michael E.Auer
24.10.2012
i (t )  I 0 e
Rt / L
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RL Circuit (2)
i (t )  I 0 e
where
t / 
L

R
1. The initial voltage i(0) = I0 through
the inductor.
2. The time constant  = L/R.
•
The general form is very similar to a RC source-free circuit.
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RL Circuit (3)
Comparison between a RL and RC circuit
A RL source-free circuit
i (t )  I 0 e
Michael E.Auer
t / 
A RC source-free circuit
L
where  
R
24.10.2012
v(t )  V0 e t /
where   RC
EE03
Electrical Engineering – RLC Circuits Switching
Source-Free RL Circuit (4)
For the circuit, find i(t) for t > 0.
Answer:
i(t) = 2e–2t A
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Chapter Content
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Summary / Zusammenfassung
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Unit-Step Function (1)

Michael E.Auer
The unit step function u(t) is 0 for negative values of t and 1 for
positive values of t.
 0,
u(t )  
1,
t0
t0
 0,
u (t  to )  
1,
t  to
t  to
 0,
u (t  to )  
1,
t   to
t   to
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Unit-Step Function (2)
Represent an abrupt change for:
1.
voltage source.
2.
current source:
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Chapter Content
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Summary / Zusammenfassung
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
The Step-Response of a RC Circuit (1)

The step response of a circuit is its behavior when the excitation
is the step function, which may be a voltage or a current source.
• Initial condition:
v(-0) = v(+0) = V0
• Applying KCL,
C
or
dv v  Vs u (t )

0
dt
R
v  Vs
dv

u (t )
dt
RC
• Where u(t) is the unit-step function
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
The Step-Response of a RC Circuit (2)

Integrating both sides and considering the initial conditions, the
solution of the equation is:
V0
v(t )  
t / 
V
(
V
V
)
e


0
s
 s
Final value at
t -> ∞
Michael E.Auer
24.10.2012
Initial value at
t=0
t0
t 0
Source-free
Response
EE03
Electrical Engineering – RLC Circuits Switching
The Step-Response of a RC Circuit (3)
1. The initial capacitor voltage v(0).
2. The final capacitor voltage v() — DC voltage
across C.
3. The time constant .
t /
v (t)  v ()  [v (0)  v ()]e
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
The Step-Response of a RC Circuit (4)
Find v(t) for t > 0 in the circuit in below. Assume the switch has
been open for a long time and is closed at t = 0.
Calculate v(t) at t = 0.5.
Answer:
v(t )  15e 2t  5 and v(0.5) = 0.5182V
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Chapter Content
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Summary / Zusammenfassung
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
The Step-response of a RL Circuit (1)

The step response of a circuit is its behavior when the excitation
is the step function, which may be a voltage or a current source.
•
Initial current
i(-0) = i(+0) = Io
•
Final inductor current
i(∞) = Vs/R
•
Time constant  = L/R
Vs t
Vs
i (t )   ( I o  )e u (t )
R
R
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
The Step-response of a RL Circuit (2)
1. The initial inductor current i(0) at t = +0.
2. The final inductor current i().
3. The time constant .
i (t )  i ( )  [i (0 )  i ( )] e
Michael E.Auer
24.10.2012
 t /
EE03
Electrical Engineering – RLC Circuits Switching
The Step-response of a RL Circuit (3)
The switch in the circuit shown below has been closed for a long
time. It opens at t = 0.
Find
i(t) for t > 0.
Answer:
i (t )  2  e 10t
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Chapter Content
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Summary / Zusammenfassung
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Delay Circuit
  R1  R2   C
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Photo Flash
R1 >> R2
τ1 >> τ 2
R2 = Flash lamp
High current pulse
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Automobile Ignition Circuit
Example:
R = 4Ω; Vs = 12V; L = 6mH
Cut-off time toff = 1µs
V
i  s  3A
R
Michael E.Auer
voff  L 
i
3A
 6mH 
 18kV
t
1s
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Relay Circuit
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Chapter Content
Source-Free RC Circuit / Quellenfreier RC-Kreis
Source-Free RL Circuit / Quellenfreier RL-Kreis
Singularity Functions / Sprungfunktionen
Step-Response RC Circuit / Sprungantwort RC-Kreis
Step-Response RL Circuit / Sprungantwort RL-Kreis
Applications / Anwendungen
Summary / Zusammenfassung
Michael E.Auer
24.10.2012
EE03
Electrical Engineering – RLC Circuits Switching
Summary
•
•
•
•
•
•
•
•
Michael E.Auer
When analyzing RC and RL circuits, one must always keep in mind that the capacitor is an
open circuit to steady-state dc conditions while the inductor is a short circuit to steady-state
dc conditions.
Circuits comprising a resistor and a single energy-storage elemet (capacitor, inductor) are
called first-order circuits, because its behavour is described by a first-order differential
equation.
The transition from one state to another follows an exp function.
The time constant τ is the time required for a response to decay to 1/e of its initial value. For
RC circuits, τ = R·C and for RL circuits, τ = R / L.
The unit step function is one of the singularity functions.
The steady-state response is the behavior of the circuit after an independent source has
been applied for a long time.The transient response is the component of the complete
response that dies out with time.
The total or complete response of the circuit consists of the steady-state response and the
transient response.
The step response is the response of the circuit to a sudden application of a dc current or
voltage . Finding the step response of a first-order circuit requires the initial value x(+0), the
final value x(∞) and the time constant τ (x stands for i or v). With this three items, we obtain
the step response as
Instantaneous value = Final + [ Initial – Final ] · exp (-(t - t0) / τ )
24.10.2012
EE03