Dr. C.Y. Evrenosoglu
EE220
EE220
EE220
RC & RL Circuits
RC & RL Circuits
Dr. E
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order circuits
order circuits
•Circuits that contain only one of the energy storage
elements are represented by a firstorder differential
equation. That is why these circuits are called firstorder
circuits
–Only one inductor and no capacitors.
–Only one capacitor and no inductors.
•Thévenin’s & Norton’s equivalent circuits are used to
simplify the analysis with one energy storage element
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order circuits
order circuits
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order RC circuit
order RC circuit
(
)
(
)
()
()
()
()
()
()
RC
V
RC
tv
dt
tdv
Vtv
dt
tdv
CR
0tv
dt
tdv
CRV
0tvtiRV
=+⇒
=+⋅⇒
=+⋅+−⇒
=
+
⋅
+
−
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order RL circuit
order RL circuit
(
)
()
()
()
()
()
()
()
()
I
L
R
ti
L
R
dt
tdi
ti
dt
tdi
R
L
I
ti
R
dt
tdi
L
I
ti
R
tv
I
tiiI
R
=+⇒
+=⇒
+=⇒
+=⇒
+
=
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order differential equation
order differential equation
•Complete Response of the differential equation =
Transient Response (tis small) + Steady State Response (t→ ∞)
(Natural Response) (Forced Response)
(
)
(
)
RC
V
RC
tv
dt
tdv
=+
If input voltage V is constant (DC), the answer will be:
()()()
RC
t
eV0vVtv
−
−+=
Transient (Natural) Response:The part of the
response which disappears after long enough time
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order differential equation
order differential equation
(
)
(
)
RC
V
RC
tv
dt
tdv
=+
If input voltage V is constant (DC), the answer will be:
()()()
RC
t
eV0vVtv
−
−+=
Steady State (Forced) Response:The part of the
response which stays after long enough time passes.
Time constant = τ
Dr. C.Y. Evrenosoglu
EE220
Example
Example
•R = 1 ,
C = 1 F
, initial capacitor voltage, v(0) = 0. 5 V DC is applied to an
RC circuit. The capacitor voltage?
()()()
()()
()
t
11t
RC
t
e15
e505tv
eV0vVtv
−
×
−
−
−=
−+=
−+=
0
1
2
3
4
5
6
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time  t [s]
Voltage  [V]
Voltage across a capacitor in a series RC circuit with 5 V input
Dr. C.Y. Evrenosoglu
EE220
Example
Example
•R = 1 ,
C = 1 mF
, initial capacitor voltage, v(0) = 0. 5 V DC is applied to an
RC circuit. The capacitor voltage?
()()()
()()
()
t10
m11
t
RC
t
3
e15
e505tv
eV0vVtv
−
×
−
−
−=
−+=
−+=
0
1
2
3
4
5
6
x 10
3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time  t [s]
Voltage  [V]
Voltage across a capacitor in a series RC circuit with 5 V input
Dr. C.Y. Evrenosoglu
EE220
Steady
Steady


State
State
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time  t [s]
Voltage  [V]
Voltage across a capacitor in a series RC circuit with 5 V input
Transient (Natural) response
SteadyState (Forced) Response
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order RL circuit
order RL circuit
(
)
()
()
()
()
()
()
()
()
I
L
R
ti
L
R
dt
tdi
ti
dt
tdi
R
L
I
ti
R
dt
tdi
L
I
ti
R
tv
I
tiiI
R
=+⇒
+=⇒
+=⇒
+=⇒
+
=
Dr. C.Y. Evrenosoglu
EE220
1
1
st
st
order differential equation for RL circuit
order differential equation for RL circuit
•Complete Response of the differential equation =
Transient Response (t is small) + Steady State Response (t
→ ∞)
(Natural Response) (Forced Response)
If input current Iis constant (DC), the answer will be:
Transient (Natural) Response:The part of the
response which disappears after long enough time
(
)
()
I
L
R
ti
L
R
dt
tdi
=+
()()()
t
L
R
eI0iIti
−
−+=
Dr. C.Y. Evrenosoglu
EE220
()()()
τ
t
eI0iIti
−
−+=
1
1
st
st
order differential equation for RL circuit
order differential equation for RL circuit
If input current Iis constant (DC), the answer will be:
Steady State (Forced) Response:The part of the
response which stays after long enough time passes.
Time constant
τ= L/ R
(
)
()
I
L
R
ti
L
R
dt
tdi
=+
Dr. C.Y. Evrenosoglu
EE220
REMEMBER: Initial Conditions
REMEMBER: Initial Conditions
•DC circuits
–Independent voltage and current sources are DC (constant) –They
don’t change with time.
•The circuit includes at least one capacitor or one inductor. (Ifmultiple
capacitors/inductors find the equivalent.)
•The change in capacitor voltage or in inductor current is NOT
instantaneous. It is continuous.
–We denote the time immediately before the switch opens/closes ast0
–We denote the time immediately after the switch opens/closes as t0+
–The capacitor voltage or inductor current have the same values right
before and right after the switch closes.
•A
capacitor
in a DC circuit behaves like an open circuitin steady state
.
•An
inductor
in a DC circuit behaves like a short circuitin steady state
.
•Draw
the circuit for before the switch operation
and after the switch
operation
!
Dr. C.Y. Evrenosoglu
EE220
Example 8.3
Example 8.3
–
–
1
1
•What is the value of the capacitor voltage
50 msafter the switch
opens?
Dr. C.Y. Evrenosoglu
EE220
Example 8.3
Example 8.3
–
–
2
2
•Find the inductor current after the switch closes. How long willit
take for the inductor current
to reach 2 mA?
Dr. C.Y. Evrenosoglu
EE220
Example 8.3
Example 8.3
–
–
3
3
•The switch has been open for a long time and the circuit has
reached steady state before the switch closes at time t = 0. Find the
capacitor voltage
for t ≥0.
t < 0t > 0
Dr. C.Y. Evrenosoglu
EE220
Example 8.3
Example 8.3
–
–
4
4
•The switch has been open for a long time and the circuit has
reached steady state before the switch closes at time t = 0. Find the
inductor current
for t ≥0.
t < 0
t > 0
Dr. C.Y. Evrenosoglu
EE220
Example 8.3
Example 8.3
–
–
5
5
•The circuit is at steady state before the switch opens. Find the
current i(t)for t > 0.
t > 0
Dr. C.Y. Evrenosoglu
EE220
Example 8.3
Example 8.3
–
–
6
6
•Find the capacitor voltage after the switch opens. What is the value
of the capacitor voltage 50 msafter the switch opens?
Dr. C.Y. Evrenosoglu
EE220
Sequential Switching Example (Section 8.4)
Sequential Switching Example (Section 8.4)
•The circuits is at steady state before the switch changes state at t= 0.
t < 0
t< 0
0 < t< 1 ms
t= 1 ms
Dr. C.Y. Evrenosoglu
EE220
Stability of 1
Stability of 1
st
st
order circuits
order circuits
•If the transient response (natural response) vanishes in time
and steadystate response is within limits (not increasing in
time; either settles down a value or oscillates within bounds),
then the circuit is called stable
.
•If the circuit is stable, then the steadystate (forced) response
depends on the input
to the circuit.
•If the circuit is unstable, then the forced response will
NOT
depend on the input
and the forced response cannot be called
steadystate. Since there is no settling.
•Recall the time constant τinsolutions of the RC and RL
circuits with DC input:
()()()
RC
t
eV0vVtv
−
−+=
()()()
R/L
t
eI0iIti
−
−+=
τ
Dr. C.Y. Evrenosoglu
EE220
Stability of 1
Stability of 1
st
st
order circuits
order circuits
•Recall the time constant τinsolutions of the RC and RL
circuits with DC input:
•If the R(usually TheveninR) is larger than 0, the circuit is
stable.
•The negative TheveninRcan appear in circuits with
dependent sources
and OPAMPs
. It may not mean a practical
resistance.
•If there is no dependent source or OPAMPsconnected to the
capacitor/inductor, the circuit is guaranteed to be STABLE as
long as the TheveninR is larger than 0.
()()()
RC
t
eV0vVtv
−
−+=
()()()
R/L
t
eI0iIti
−
−+=
τ
Dr. C.Y. Evrenosoglu
EE220
Example 8.5
Example 8.5
–
–
1
1
•The first order switch is at steady state before the switch closes at t= 0. Find the
capacitor voltage, v(t), for t> 0.
t < 0
t> 0 –Thévenin’s Equivalent
Dr. C.Y. Evrenosoglu
EE220
Example 8.5
Example 8.5
–
–
1
1
•The first order switch is at steady state before the switch closes at t= 0. Find the
capacitor voltage, v(t), for t> 0.
t < 0
t> 0 –Thévenin’s Equivalent
Dr. C.Y. Evrenosoglu
EE220
Example 8.5
Example 8.5
–
–
2
2
•What restrictions must be placed on the gain of the dependent source to ensure
that the circuit is stable? Design the circuit to have a time constant of +20 ms.
t> 0 –Thévenin’s Equivalent
Dr. C.Y. Evrenosoglu
EE220
OPAMP & RC Circuits
OPAMP & RC Circuits
(
)
(
)
(
)
()
()()
()
()()
()()
tydx
RC
1
dt
tyd
RCtx
dt
tyd
C
R
tx
dt
ty0d
C
R
0tx
t
0
=
−
⇒
−=⇒
−=⇒
−
=
−
∫
ττ
(
)
(
)
()
()
dt
tdx
RCty
R
ty
dt
tdx
C
−=⇒
−=
Integrator
Differentiator
Dr. C.Y. Evrenosoglu
EE220
Unit Step Source
Unit Step Source
•Unit step function
>
<
=−
0
0
0
1
0
)(
tt
tt
ttu
Dr. C.Y. Evrenosoglu
EE220
Example 8.7
Example 8.7
–
–
2
2
•Find the capacitor voltage for t> 0 when v(0) = 0 and the
current source, is
(t) = (10sin(2t))u(t) A.
(
)
(
)
()
()
0)2sin(20
2
)(
>=+
=+
tt
tv
dt
tdv
CR
tiR
CR
tv
dt
tdv
N
sN
N
>>
v = dsolve('Dv+v/2=20*sin(2*t)','v(0)=0');ezplot(v,[0,2])
Dr. C.Y. Evrenosoglu
EE220
Example 8.7
Example 8.7
–
–
2
2
Vttetv
t
)2cos(
17
160
)2sin(
17
40
17
160
)(
2/
−+=
−
>> t = 0:0.1:12;
>> v = (160/17*exp(t/2)+40/17*sin(2*t)160/17*cos(2*t));
>> plot(t,v,'r'); xlabel('timet [s]'); ylabel('v(t)'); grid
0
2
4
6
8
10
12
10
5
0
5
10
15
time  t [s]
v(t)
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