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Chapter 1: Basic Concepts
Basic Circuit Elements and Laws
ELE1110B Basic Circuit Theory
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ELE 1110B Lecture 01 - 2
Topics To Cover
Concepts
Charge, Current, Voltage, Power, Energy
Basic circuit elements and device laws
Resistor, Capacitor, Inductor
Circuit concepts and definitions
Nodes, branches, and loops Basic circuit laws
Kirchhoffs laws
Basic circuit analysis Reference
Alexander and Sadiku, Chapters 1, 2 and 6.
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ELE 1110B Lecture 01 - 3
Introduction an electric circuit
A circuit = the path of flow for charge carriers
To communicate from one point to another
To transfer energy from one point to another
Charge carriers are
electrons in a conductor
ions in an battery
Battery
(source)
Light bulb(device)
Switch
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ELE 1110B Lecture 01 - 4
An electric circuit
A circuit consists of source (energy) and/or drain
(device) elements
A circuit has constant flow of charges when it isclosed
The polarity of an electric circuit segment defines the
flow direction
Battery
(source)
Light bulb
(device)
Switch
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ELE 1110B Lecture 01 - 5
Charge and Current
Charge: coulombs (C)
1 C = 6.24 x 1018 electrons
Single electron has 1.6 x 10-19 C Conservation of Charges:
charges can neither be created nor destroyed, onlytransferred
The algebraic sum of the electric charges in a closed systemdoes not change with time
Electric current
dc: direct current, remains constant with time
ac: alternating current, varies (sinusoidally) with time
dt
dqi= =
t
todiq )(
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ELE 1110B Lecture 01 - 6
Voltage / potential difference It is the energy required to move one unit charge through an element
Polarity of a circuit segment indicates the flow direction of charges
a
b
+
-
Vab (= -Vba)
+ve charges(e.g. ions in
battery)
-vecharges
(electrons)
+I
-I
Passive sign convention
Powerp = vi
+ve: consumption
-ve: generation
dq
dEVab =
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ELE 1110B Lecture 01 - 7
Power supply and absorption
v
i i
i
+
-v+
-v+
-
v
i
+
- v
+
-
i
Independent voltage source
(const. or time-varying voltage)
Independent voltage source
(constant voltage)
Independent current source
Dependent voltage source Dependent current source
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ELE 1110B Lecture 01 - 8
Ideal Sources Ideal Independent Sources
Voltage source
An active element (which provides energy) provides a specific voltage
with its zero internal resistance. It is completely independent of other circuit variables, e.g. current through
it.
Current source
An active element provides a specific current with its infinite resistance.
It is completely independent of other circuit variables, e.g.voltage acrossthe source.
Ideal Dependent (Controlled) Sources An active element in which the source quantity is controlled by another
voltage or current. Voltage-controlled voltage source (VCVS), Current-controlled voltage
source (CCVS), Voltage-controlled current source (VCCS), Current-controlled current source (CCCS)
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ELE 1110B Lecture 01 - 9
Energy and Power
Energy
Average power
More convenient to measure than instantaneous power, e.g. bywattmeter
)()()( titvdt
dq
dq
dE
dt
dEtp ===
== t
t
t
tdivdptE
00
)()()()(
+
=
Tt
taverage
o
o dttpTp )(
1
time-varying, so called instantaneous power
instantaneous
voltage
instantaneous
current
v(t)
i(t)
p(t)
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ELE 1110B Lecture 01 - 10
Example 1.1 Example: Calculate the power supplied or absorbed by each element:
By passive sign convention,
p1 = 20(-5) = -100W (supplied power)
p2 = 12(5) = 60W (absorbed power)
p3 = 8(6) = 48W (absorbed power)
p4 = 8(-0.2I) = 8(-0.2 x 5) = -8W (supplied power)
p1 + p2 + p3 + p4 = 0 (Energy Conservation Law: p = 0)
20 V p1
p2
8 Vp3 p4
I = 5A
6A
0.2I+
-
12 V+ -
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ELE 1110B Lecture 01 - 11
Ohms Law
Short circuit: R=0
Open circuit: R=
Circuit Element resistor
+
-
v
i
R
vRivip
22 ===
GiGvvip
2
2===
resistance
conductance
GR
1=
iRv=
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ELE 1110B Lecture 01 - 12
Resistors in series
=
=++=N
n
nNeq RRRRR1
21 ...
=
=
=
=
=
=
N
nn
N
n
n
N
n
n
Ri
iR
vv
1
1
1
vRR
Rv
21
11
+= v
RR
Rv
21
22
+=Note:
Known as
voltage divider
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ELE 1110B Lecture 01 - 13
Resistors in parallel
Neq RRRR1...111
21
+++=
=
=
=
=
=
=
N
n n
N
n n
N
n
n
Rv
R
v
ii
1
1
1
1
i
RR
Ri
21
21
+
=
iRR
Ri
21
12
+= Known as
current divider
Note:
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ELE 1110B Lecture 01 - 14
Example 1.2
Calculate the equivalent resistance Rab in the circuit
a
b
Rab
10 1
632
c d
b b
a
b
Rab 32
c d
b b
10a
b
11.2
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ELE 1110B Lecture 01 - 15
Example 1.3 Calculate the equivalent resistance Rab in the circuit
a
b
Rab
10 1 1
54
6
3
12
c d
b b
a
b
Rab
10 1
632
c d
b b
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ELE 1110B Lecture 01 - 16
-to-Y transformationa
b c
a
b c
Rc Rb
Ra
R1
R2 R3
cba
ba
cba
ac
cba
cb
RRR
RRR
RRR
RRR
RRR
RRR
++=
++=
++=
3
2
1
3
133221
2
133221
1
133221
R
RRRRRRR
R
RRRRRRR
R
RRRRRRR
c
b
a
++=
++=
++=
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ELE 1110B Lecture 01 - 17
Circuit Element capacitor
Open circuit to dc signal
An ideal capacitor can store (in its electric field) and deliver energy without
dissipation, but real capacitor has a leakage resistance in parallel combined with theideal capacitive part.
When a voltage source v is applied to a capacitor, the amount of charge stored
q = Cv
For parallel-plate capacitor,
For electrolyte capacitors, High capacitance
Can endure high voltage
Polarity is prearranged and not allowed to be interchanged
d
AC
=
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ELE 1110B Lecture 01 - 18
Capacitor device laws
i C
v+ -
dt
dvvCivp ==
dtdvC
dtdqi ==
1. v const. i = 0 open circuit
2. v cannot change instantaneously because i = is needed
C
QvCtE
tEdiv
tEdptE
o
t
t
o
t
t
o
o
22
1)(
)()()(
)()()(
22 ==
+=
+=
)(1
)( ot
ttvid
Ctv
o +=
if E(to) = 0
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ELE 1110B Lecture 01 - 19
Capacitors in series
Neq CCCC
1...
111
21
+++=
)(1
)(1
)(1
)()(
11
1
1
oeq
t
teq
N
n on
t
t
N
n n
N
n
t
t on
n
N
n
n
tvidC
tvidC
tvidC
tvtv
o
o
o
+=
+
=
+=
=
==
=
=
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ELE 1110B Lecture 01 - 20
Capacitors in parallel
=
=++=N
n
nNeq CCCCC1
21 ...
dt
dvC
dt
dvC
dtdvC
tii
eq
N
n
n
N
n
n
N
n
n
=
=
=
=
=
=
=
1
1
1
)(
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ELE 1110B Lecture 01 - 21
Circuit Element inductor
Short circuit to dc signal If current is allowed to pass through an inductor, it is found that
The proportional constantL is called inductance and
An ideal inductor can store (in its magnetic field) and deliver energywithout dissipation, but a real inductor has a winding resistance in seriescombined with the ideal inductive part.
air-core iron-core
l
ANL
2
=
dtdiLv=
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ELE 1110B Lecture 01 - 22
Inductor device laws
i
dt
diLivp
==
dt
di
Lv=
1. i const.(dc current) v = 0 short circuit
2. i cannot change by a finite amount in zero time because v = is needed
)()(
1
)( o
t
t tidvLti o +=
If i(-) = 0
2
21)(
)()(
)()(
iLtE
Lidi
div
dptE
t
t
t
=
=
=
=
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ELE 1110B Lecture 01 - 23
Inductors in series
=
=++=N
n
nNeq LLLLL1
21 ...
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ELE 1110B Lecture 01 - 24
Inductors in parallel
Neq LLLL1...111
21
+++=
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ELE 1110B Lecture 01 - 25
Example 1.4
i(t)
vR(t)
vC(t)
vL(t)
i(t)
i(t)
i(t)
+
-
+
-
+
-
RtitvR = )()(
)()(
1
)( ot
tC tvdiCtv o +=
dt
tdiLtvL
)()( =
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ELE 1110B Lecture 01 - 26
Example 1.5
i(t)
vR(t)
vC(t)
vL(t)
i(t)
i(t)
i(t)
+
-
+
-
+
-
RtitvR = )()(
)()(
1
)( o
t
tC tvdiCtv o +=
dt
tdiLtvL
)()( =
impulse
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ELE 1110B Lecture 01 - 27
Symbol and units
joule (J)EEnergy
watt (W)pPower
volt (V)V, vVoltage
ampere (A)I, iCurrent
coulomb (C)QCharge
henry (H)LInductance
farad (F)CCapacitance
siemens (S)
or mho ( )
GConductance
ohm ()RResistance
UnitSymbol
prefixes:
1012 tera (T)109 giga (G)
106 mega (M)
103 kilo (k)
10-2
centi (c)10-3 milli (m)
10-6 micro ()
10-9 nano (n)
10
-12
pico (p)10-15 femto (f)
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ELE 1110B Lecture 01 - 28
Circuit concept and definition
Node a point where two or more circuit elements join (e.g a, b, c, d, e, f, g)
Essential node a node where three or more circuit elements join (e.g. b, c, e, g)
Branch connects between two nodes (i.e. consists of one circuit element)
Essential Branch connects between two essential nodes
v1
v2
R1
R2 R3
R4
R5
R6
R7 I
a
c
f
d
b
e
g
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ELE 1110B Lecture 01 - 29
Circuit concept and definition - Loop
Path
a trace of adjoining elements withno element included more than once
Loop
a closed path with no node passed
more than once(there are 8+ loops) Mesh
a loop that does not enclose any other loop
4 meshes
v1
v2
R1
R2 R3
R4
R5
R6
R7 I
a
c
f
d
b
e
g
V1 - R1 - R5 - R6 - R4 - V2V1 - R1 - I - R 4 - V2V1 - R1 - R7 - R4 - V2I - R5 - R6V1 - R1 - R5 - R3 - R2
V2 - R2 - R3 - R6 - R4R5 - R7 - R6R7 - I
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ELE 1110B Lecture 01 - 30
Kirchhoffs Current Law (KCL)
i1i2
i3
i4
iN
0=N
ni
Note: Assume one direction to be +ve, and in theabove case the direction of entering the node
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ELE 1110B Lecture 01 - 31
Kirchhoffs Voltage Law (KVL)
0=N
nv
Note: Assume one direction in the loop to be +ve, and
in the above case the anti-clockwise direction
vn
+
-
v1+
-
v2 +-v3 +-
v4+
-
v5+
-
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ELE 1110B Lecture 01 - 32
Example 1.6
500V
+ V-5i
io
+
Vo
-
205i
ab
c
Loop A
By KVL around loop A,
By KCL at node b,
Put (2) into (1):
)1...(1004
0)20()5(500
=+
=++
o
o
ii
ii
)2...(6
05
=
=+
ii
iii
o
o
VV
VV
Ai
Ai
ii
o
o
480)20(24
20)5(4
24
4
100)6(4
==
==
=
=
=+
Total power generated
Total power dissipated
W
PP iV
11600
)4)(5(480)4(500
5500
=
+=
+=
W
PP
11600
)24(480)4(20
205
=
+=+=
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ELE 1110B Lecture 01 - 33
50
Example 1.7(a)
500V
2.2
20 30
46 9
io
a
b c
d 500V
2.2
46 9
a
b c
d
R2 R3
R1
=++
=
=++=
=++
=
15503020
)50(30
10503020
)50(20
6503020
)30(20
3
2
1
R
R
R
Find io
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ELE 1110B Lecture 01 - 34
Example 1.7(b)
500V
2.2
46 9
a
b c
d
A
VV
i
VR
V
V
R
V
VV
o 350
126)336(9
9
276)336(
46
46
336)500(24//562.8
24//56
32
3
3
2
2
1
=
=
=+
=
=
+
=
=+
=
6
10 15
500V
2.2 + 6 = 8.2
10 + 46
= 5615 + 9
= 24
+
-V2 V3
+
-
V1
+
-