CIRCUIT ELEMENTS
AND
KIRCHHOFF’S LAWS
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2.1 Active Components
2.2 Passive Components
2.3 Node , Branch and Loop
2.4 Kirchhoff’s Laws
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2.1 Active Components
n
n
Ideal independent voltage source
Symbol and model
is
vs
vs
is = ∞
is
is
is
= 100 A
=0A
= -50 A
is = -∞
n
It maintains a prescribed voltage independent of the
current through this source.
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2.1 Active Components
n
n
Ideal independent current source
Symbol and model
is
n
vs
is
vs = ∞
vs = 100 V
vs = 0 V
vs = -50 V
vs = -∞
It maintains a prescribed current independent of the
voltage across this source.
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2.1 Active Components
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Ideal dependent voltage and current sources are
convenient for modeling transistors
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An ideal dependent (or controlled) voltage
source is an active element whose source
quantity is controlled by other voltage or
current.
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2.1 Active Components
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Symbol and model of dependent voltage source
vs = µ vx
voltagevoltage-controlled voltage source
(VCVS)
vs = γ ix
currentcurrent-controlled voltage source
(CCVS)
vx and ix are controlling parameter.
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2.1 Active Components
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Symbol and model of dependent current source
is = α vx
voltagevoltage-controlled current source
(VCCS)
is = β ix
currentcurrent-controlled current source
(CCCS)
vx and ix are controlling parameter.
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2.1 Active Components
i
+
vs
R
v
Rs → 0 , p = vi → ∞
-
is
Rs → ∞ , p = vi → ∞
Ideal sources do not exist in real world.
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2.2 Passive Components
(a) Resistance
n
Symbol and model
v = Ri
i = Gv
R: resistance , in ohms (Ω)
G: conductance , in siemens (S)
Ohm’s Law
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2.2 Passive Components
Georg Simon Ohm
1787-1854
Germany
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???
19XX-20XX
R.O.C. Taiwan
2.2 Passive Components
(b) Inductor
n
Symbol and model
v=L
di
dt
1 t
vdτ
L ∫−∞
1 0
1 t
= ∫ vdτ + ∫ vdτ
L −∞
L 0
1 t
= i0 + ∫ vdτ
L 0
i=
L: inductance , in henry (H)
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2.2 Passive Components
(c) Capacitor
n
Symbol and model
i=C
dv
dt
1 t
idτ
C ∫−∞
1 0
1 t
= ∫ idτ + ∫ idτ
C −∞
C 0
1 t
= v0 + ∫ idτ
C 0
v=
C: capacitance , in Farad (F)
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2.2 Passive Components
(d) Coupling Inductor
n
Symbol and model
i1
v1
 v1   L1
v  =  M
 2 
i2
L1
L2
v2
M  d  i1 
L2  dt i2 
 i1 
1
i  = L L − M 2
 2
1 2
 L2
−∞  − M

∫
t
− M   v1 
dτ
L1   v2 
M: mutual inductance , in henry (H)
L1: primary self inductance , in henry (H)
L2: secondary self inductance , in henry (H)
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2.2 Passive Components
(e) Ideal Transformer
n
Symbol and model
i1
v1
v2
=n
v1
i2
v2
i1
= −n
i2
n: turn ratio , positive real number
n
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It is convenient for modeling practical transformers
or coupling inductors.
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2.3 Node , Branch and Loop
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A branch represents a single two-terminal element.
n
A node is a point where two or more elements join.
n
A loop is a closed path in a circuit without passing
through any intermediate node more than once.
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2.3 Node , Branch and Loop
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Example
node : a , b , c , d , e , f
6 nodes
branch : V1 , V2 , R3 , R4 , R5 , R6 , R7 , R8
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8 branches
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2.3 Node , Branch and Loop
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Example
loop A : a => b => c => d => a
loop B : b => e => c => b
loop C : c => e => f => d => c
loop D : a =>b =>e =>f =>d => a
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2.4 Kirchhoff’s Laws
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Kirchhoff’s current law (KCL)
The algebraic sum of all the current leaving any
node in a circuit equals zero
, for any node
N : number of elements connected to this node
in : the nth element current leaving this node
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2.4 Kirchhoff’s Laws
n
The algebraic sum of all the current entering any
node in a circuit equals zero
, for any node
n
If
N = L∪M
, then
, for any node
L : set of n with in leaving this node
M : set of m with im entering this node
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2.4 Kirchhoff’s Laws
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KCL is based on the law of
conservation of charge.
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KCL can be applied to a super node
(closed boundary)
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2.4 Kirchhoff’s Laws
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Kirchhoff’s voltage law (KVL)
The algebraic sum of all the element
voltage drops around any loop in a
circuit equals zero.
, for any loop
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2.4 Kirchhoff’s Laws
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The algebraic sum of all the element
voltage rises around any loop in a
circuit equals zero.
, for any loop
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2.4 Kirchhoff’s Laws
n
If M = K + J , then
M = set of k with Vk being a voltage drop in this loop
J = set of j with Vj being a voltage rise in this loop
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KVL is based on the law of conservation of energy.
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2.4 Kirchhoff’s Laws
Gustav Robert Kirchhoff
1824-1887
Germany
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???
19XX-20XX
R.O.C. Taiwan
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2.4 Kirchhoff’s Laws
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Example 1
∑ i (leaving ) = 0 , KCL in time domain.
k
k
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Node b : (-i1) + i7 + i2 = 0
Node e : (-i7) + (-i4) + i5 = 0
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2.4 Kirchhoff’s Laws
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Example 1 (cont.)
Node b : i1 + (-i7) + (-i2) = 0
Node e : i7 + i4 + (-i5) = 0
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2.4 Kirchhoff’s Laws
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Example 1 (cont.)
Node b : i7 + i2 = i1
Node e : i5 = i7 + i4
Super Node
Super Node :i5+i2=i1+i4
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2.4 Kirchhoff’s Laws
n
Example 2
N
∑ i (t ) = 0 , for any node
n =1
n
Assume in (t ) = I n cos(ωt + φn )
= Re[ I n e j (ωt +φn ) ]
= Re[ I n e jφn e jωt ] , n=1,2,3,…,N
Re: real part
r
jφn
I
Define n = I n e = I n cos φn + jI n sin φn
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Then
r
in (t ) = Re[ I n e jωt ]
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2.4 Kirchhoff’s Laws
n
Example 2 (cont.)
r
N
∑ Re[ I
n
e jωt ] = 0
n =1
Q Re operator & ∑ operator
are commutative.
N uu
r
Re[(∑ I n )e jωt ] = 0
n =1
∴
uur
∑ I n = 0 , complex equation
N
n=1
KCL in phasor domain
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2.4 Kirchhoff’s Laws
n
Example 3
N
∑ i (t ) = 0
n =1
Let I n ( s ) @
Then
, for any node
n
∫
∞
0
in ( t ) e-st dt
∞ N
∫ ∑ i ( t ) dt=0
0
n
n=1
∫ operator and ∑ operator
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are commutative
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2.4 Kirchhoff’s Laws
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Example 3 (cont.)
∞
∞
∑ ∫ i (t ) e
n
0
n=1
-st
dt =0
∞
∑ I ( s )=0 , scalar equation
n
n=1
KCL in s domain
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2.4 Kirchhoff’s Laws
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Example 4
∑V (drop)=0
k
, KVL in time domain
K
loop
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a,b,c,d,e,f,a
V2 + V3 + V6 + (-V7)+ (-V8) + (-V4) = 0
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2.4 Kirchhoff’s Laws
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Example 4 (cont.)
loop
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a,b,c,d,e,f,a
(-V2) + (-V3) + (-V6) + V7+ V8 + V4 = 0
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2.4 Kirchhoff’s Laws
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Example 4 (cont.)
loop
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a,b,c,d,e,f,a
V2 + V3 + V6 = V7+ V8 + V4
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2.4 Kirchhoff’s Laws
Example 5
n
M
∑ v ( t ) ( drop )=0
k
, for any node
k=1
Assume
vk ( t )=Vk cos (ωt+φk )
∴ vk ( t )= Re Vk e jφk e jωt 
uur
Let Vk =Vk e jφk =Vk cos φk + jVk sin φ k
uur
Then vk (t ) = Re[Vk e jωt ]
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2.4 Kirchhoff’s Laws
Example 5 (cont.)
M
uur jωt

=0
Re
V
ke


Then ∑
k=1
n
 M uur  jωt 
⇒ Re  ∑Vk  e  =0
 k=1 

M uu
r
⇒ ∑Vk =0 , complex equation
k=1
KVL in phasor domain
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2.4 Kirchhoff’s Laws
Example 6
n
N
∑ v (t ) = 0
k
n =1
Define
∞ M
∫ ∑v
0
k =1
M
∑∫
k =1
∞
0
M
∑V
k =1
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K
k
∞
VK ( s ) = ∫ vk (t ) e − st dt
0
( t ) e − st d t = 0
v k ( t ) e − st d t = 0
( s ) = 0 , sca la r eq u tio n
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K V L in s d o m a in
2.4 Kirchhoff’s Laws
Example 7 (Chapter Problem 2.33)
n
Design an electric wiring system to control a
single appliance from two or more locations.
A three-way switch
：single pole double throw switch
(單極双投開關)
2
2
1
1
3
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2.4 Kirchhoff’s Laws
n
Example 7 (cont.)
A four-way switch
：double pole double throw switch
(双極双投開關)
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2.4 Kirchhoff’s Laws
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Example 7 (cont.)
(a) Two three-way switches can be connected
between two locations.
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2.4 Kirchhoff’s Laws
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Example 7 (cont.)
(b) One four-way switch is required for each
location in excess of two.
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Summary
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Objective 1：Understand ideal circuit components.
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Objective 2：Be able to state and use component
models.
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Objective 3：Be able to state and use KCL and KVL.
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Assignment : Chapter Problems
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Problem 2.3
2.9
2.26
2.29
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Due within one week.
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