Lecture Electric circuit theory: First-order crcuits - Nguyễn Công Phương

Nguyễn Công Phương

Electric Circuit Theory
First-Order Circuits


Contents
I. Basic Elements Of Electrical Circuits
II. Basic Laws
III. Electrical Circuit Analysis
IV. Circuit Theorems
V. Active Circuits
VI. Capacitor And Inductor
VII. First-Order Circuits
VIII.Second Order Circuits
IX. Sinusoidal Steady State Analysis
X. AC Power Analysis
XI. Three-phase Circuits
XII. Magnetically Coupled Circuits
XIII.Frequency Response
XIV.The Laplace Transform
XV. Two-port Networks
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2


First-Order Circuits
1.
2.
3.

4.
5.
6.
7.
8.

Introduction to Transient Analysis
Initial Conditions
The Source-free RC Circuit
The Source-free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
The Classical Method
First-order Op Amp Circuits

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3


Introduction to Transient Analysis (1)
– +

– +
10 V

5Ω

10 V

t=0

0.1 H

0.1 H

i
2A

i

Any change in an
electrical circuit,
which brings about
a change in energy
distribution,
will result in a
transient-state.

5Ω

Steady-state

Transient-state

Steady-state t

0
12 V

v
Steady-state

Transient-state

Steady-state t

0
– +

– +
10 V

5Ω

12 V

t=0

0.1 mF
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6Ω
0.1 mF

+
v
–
4

Introduction to Transient Analysis (2)
vL(t)
Inductors in
DC circuits

Transient
-state

Old steady-state

0
Short-circuit

Capacitors in
DC circuits

New steady-state

t
Not short-circuit

Short-circuit

iC(t)
Transient
-state

Old steady-state

New steady-state

0
Open-circuit

t
Not open-circuit

Open-circuit

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5


Introduction to Transient Analysis (3)

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6


First-Order Circuits
1.
2.
3.
4.
5.
6.
7.

8.

Introduction to Transient Analysis
Initial Conditions
The Source-free RC Circuit
The Source-free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
The Classical Method
First-order Op Amp Circuits

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7


20 V

Initial Conditions (1)

–+
–+
10 V

5Ω
t=0
i

0.1 H
i A

Steady-state/Initial condition 2

4
2A

Steady-state/Initial condition 1
5Ω
–+
10 V

t=0
i

0
0–

0.1 H

t

Prior to switching

0+
After switching

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8

Initial Conditions (2)
• 1st switching rule/law: the current (magnetic flux) in an
inductor just after switching is equal to the current (flux) in the
same inductor just prior to switching
iL(0+) = iL(0–)
λ(0+) = λ(0–)
• 2nd switching rule/law: the voltage (electric charge) in a
capacitor just after switching is equal to the voltage (electric
charge) in the same capacitor just prior to switching
vC(0+) = vC(0–)
q(0+) = q(0–)
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9


Initial Conditions (3)

Ex. 1

The switch has been at A for a long time,
and it moves to B at t = 0; find I0?
−

i (0 ) = 0 A
i (0+ ) = i (0− )

→ i(0+ ) = 0 A → I 0 = 0 A

–+

10 V

A
5Ω
t=0
B

i

0.1 H

Ex. 2

The switch has been at A for a long time,
and it moves to B at t = 0; find I0?
20
i (0 ) =
= 4A
5

–+
–+

−

i (0+ ) = i (0− )

20 V

→ i(0+ ) = 4 A → I 0 = 4 A

10 V

A

5Ω
t=0

B

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i

0.1 H

10


Initial Conditions (4)

Ex. 3

The switch has been at A for a long time,
and it moves to B at t = 0; find V0?
−

v (0 ) = 0 V
+

−

v (0 ) = v(0 )

→ v (0 ) = 0 V → V0 = 0 V
+

–+
10 V

A
5Ω
t=0
B
0.1 mF

+
V0
–

Ex. 4

The switch has been at A for a long time,
and it moves to B at t = 0; find V0?
v (0− ) = 20 V
v (0+ ) = v(0− )

20 V
–+
–+

→ v (0+ ) = 20 V → V0 = 20 V

10 V

A
5Ω
t=0
B

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0.1 mF

+
V0
–
11


First-Order Circuits
1.
2.
3.
4.
5.
6.
7.
8.

Introduction to Transient Analysis
Initial Conditions
The Source-free RC Circuit
The Source-free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
The Classical Method
First-order Op Amp Circuits

First-Order Circuits - sites.google.com/site/ncpdhbkhn

12


The Source-free RC Circuit (1)

iR + iC = 0
dv v
→C + =0
dt R
dv
v
→
+
=0
dt RC
dv
1
→
=−

dt
v
RC
t
→ ln v = −
+ ln A
RC
v
t
→ ln = −
A
RC
−

t
RC

→ v (t ) = Ae
v (0) = v t = 0 = V0

E

–+

iR
R

R
t=0

C

+

v

iC
C

–

v (0) = v t = 0 = V0

v

→ v (t ) = V0 e

−

= V0 e

t
RC

−

t

τ

V0

0.368V0
0

τ

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t
13


The Source-free RC Circuit (2)
Ex. 1

+

R1 = 6 Ω; R2 = 12 Ω; vC(0) = 10 V;
C = 0.01F; find vC ?

R1

vC
C

R2

–

+

R12
C

τ = R12C = 4 × 0.01 = 0.04s
vC = vC (0)e

−

t

τ

= 10e

t
−
0.04

= 10e −25t V

vC
–

6 × 12
R12 =
= 4Ω
6 + 12

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14


The Source-free RC Circuit (3)
Ex. 2

t=0

+

R2

R1

–
E

+

V0

E

t>0

vC
C

–

+

R2

V0 = 24 V

R1

–

–

V0 = E = 24 V

R1

–

t<0

+

R2

+

E = 24 V; R1 = 8 Ω; R2 = 12 Ω; C = 0.01F;
the switch has been closed for a long time,

and it is opened at t = 0; find vC for t ≥ 0?

τ = ( R1 + R2 )C = (8 + 12) × 0.01 = 0.2s
vC = V0 e

−

t

τ

= 24e

−

t
0.2

= 24e −5t V

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15


First-Order Circuits
1.
2.
3.
4.

5.
6.
7.
8.

Introduction to Transient Analysis
Initial Conditions
The Source-free RC Circuit
The Source-free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
The Classical Method
First-order Op Amp Circuits

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16


The Source-free RL Circuit (1)
vR + vL = 0
di
→ Ri + L = 0
dt
di
R
→ = − dt
i
L
R

→ ln i = − t + ln A
L
i
R
→ ln = − t
A
L

E

–+

i

R

+

t=0

vR

L

–

R

–

L vL
+

i (0) = i t = 0 = I 0

i

I0

R
− t
L

R
− t
→ i(t ) = Ae
→ i(t ) = I 0 e L 0.368I0
i (0) = i t =0 = I 0

= I0e

−

t

τ

0

τ

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t
17


The Source-free RL Circuit (2)
Ex.

t=0
R1

R3

+

L

R2

–

E = 24 V; R1 = 5 Ω; R2 = 4 Ω; R3 = 12 Ω;
L = 0.01H; the switch has been closed for
a long time, and it is opened at t = 0;
find iL for t ≥ 0?

iL

E

t<0

t>0
R1

R3

+

R2

R3

I0

L

R2

–

E

I0 = 0.75 A

4 × 12
24
Req = 5 +

= 8Ω → i1 =
= 3A
4 + 12
8
R2
4
→ I 0 = i1
=3
= 0.75 A
R2 + R3
4 + 12

τ=

L
0.01
=
= 0.000625s
R23 4 + 12

iL (t ) = I 0 e

−

t

τ

= 0.75e −1600 t A

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18


First-Order Circuits
1.
2.
3.
4.
5.
6.
7.
8.

Introduction to Transient Analysis
Initial Conditions
The Source-free RC Circuit
The Source-free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
The Classical Method
First-order Op Amp Circuits

First-Order Circuits - sites.google.com/site/ncpdhbkhn

19


Step Response of an RC Circuit (1)

+

E1

−

V0 = v(0 ) = v(0 ) = E1
Ri + v = E2

– +

E2

– +

→ ln(v − E2 ) V

v (t )
0

→

t=0
i

dv
v
E2
dv
+

=
→ RC + v = E2 →
dt RC RC
dt
dv
v − E2
→
=−
dt
RC

R
+
v
C –

dv
dt
=−
v − E2
RC

t
=−
RC

t

0

v − E2
t
→ ln
=−
V0 − E2
RC

→ v (t ) = E2 + (V0 − E2 )e

−

t
RC

t
−
v − E2
→
= e RC
V0 − E2

−

t

= E2 + ( E1 − E2 )e τ ,

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t >0

20


Step Response of an RC Circuit (2)
v (t ) = E2 + (V0 − E2 )e

−

t
RC

−

t

= E2 + ( E1 − E2 )e ,
τ

E1

t>0
E2

– +
– +

R
t=0
i

Forced response/steady-state response

+
v
C –

Natural response/transient-state response

v
E2

v

v
V0 = E1

E2
V0 = E1
0

E2
t

0

t

0

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t
21


First-Order Circuits
1.
2.
3.
4.
5.
6.
7.
8.

Introduction to Transient Analysis
Initial Conditions
The Source-free RC Circuit
The Source-free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
The Classical Method
First-order Op Amp Circuits

First-Order Circuits - sites.google.com/site/ncpdhbkhn

22


Step Response of an RL Circuit (1)

E
I 0 = i(0 ) = i (0 ) = 1
R
di
di E2 − Ri
Ri + L = E2
→ =
dt
dt
L
di
dt
di
R
→
=
→
= dt
E2
E2 − Ri L
−i L
R
E2
t
t
−i
R
E
R
 2 

R
=− t
→ ln  − i  = − t → ln
E2
L
L 0
 R
 I0
− I0
R
+

−

E1

E2

–+
–+

R
t=0
L
i

E2
−i
R
− t

→ R
=e L
E2
− I0
R

E2 
E2  − RL t E2  E1 E2  −τt
→ i(t ) =
+  I0 −  e =
+ − e ,
R 
R 
R R R 
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t >0
23


Step Response of an RL Circuit (2)
E 
E
i(t ) = 2 +  I 0 − 2
R 
R

E1

R

 − Lt
e


E2  E1 E2  −τt
=
+  − e ,
R R R 

t>0

E2

–+
–+

R
t=0
L
i

Forced response/steady-state response
Natural response/transient-state response

i
E2
R

i

v
I0 =

E2
R
E1
I0 =
R

0

t

0

t

E2
R
0

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E1
R

24

t

First-Order Circuits
1.
2.
3.
4.
5.
6.
7.
8.

Introduction to Transient Analysis
Initial Conditions
The Source-free RC Circuit
The Source-free RL Circuit
Step Response of an RC Circuit
Step Response of an RL Circuit
The Classical Method
First-order Op Amp Circuits

First-Order Circuits - sites.google.com/site/ncpdhbkhn

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