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An excerpt from 'electrical engineering by c.t. Pan'. It covers the topics of capacitors, inductors, and mutual induction, including their symbols, component models, energy storage, series and parallel combinations, and mutual induction between two coils.
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6.1 The Capacitor 6.1 The Capacitor
6.1 The Capacitor 6.1 The Capacitor Circuit symbol and component model. Circuit symbol and component model.
, ( )
t q C vC q(t)= iC d q : charge C : capacitance , in F(Farad)
@ ⋅ (^) ∫ τ τ
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6.1 The Capacitor 6.1 The Capacitor
2 2
( ) ( ) ( )
( ) 1 0 2 2
c c c c
c c c C C C
t c
If v > 0 and i > 0 , or v < 0 and i < 0 the capacitor is being charged. If v i < 0 , the capacitor is discharging. dv t p(t)= v t i t v t C dt Energy in a capacitor
w p d C v t q C −∞^ τ^ τ
⋅
=
= (^) ∫ = = ≥
6.1 The Capacitor 6.1 The Capacitor
0
( ) ( )
1 ( ) ( ) ( )
lim ( ) ( ) , (0 ) (0 )
C
t C C (^) t C
C C C C C
b v t is a continuous function if there is only finite strength energy sources inside the circuit.
v t v t i d C v t v t v v i.e. v (t
ε
ε
ε τ τ
ε
− →
= +
= =
Q ∫
) can not change abruptly for finite i (t C )
a When v C is constant , then i = 0.c ie equivalent to open circuit
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6.1 The Capacitor 6.1 The Capacitor
(c) An ideal capacitor does not dissipate energy. It stores
energy in the electrical field.
(d) A nonideal capacitor has a leakage resistance
ESR : equivalent series resistance
6.2 The Inductor6.2 The Inductor
λ
t L iL v (^) L d λ : flux linkage , in web - turns L : inductance , in H (Henry)
λ @ ⋅ λ = (^) ∫ τ τ
Circuit symbol and component model. Circuit symbol and component model.
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i l N
A
6.2 The Inductor6.2 The Inductor
2
2
, f l u x l i n k a g e
B A A N i l N A N^ i l L A N i l
φ^ μ
λ φ μ
λ μ
= =
= =
∴ = =
6.2 The Inductor6.2 The Inductor The unit of inductance is the henry (H) , named in honorThe unit of inductance is the henry (H) , named in honor of the American inventor Joseph Henry (1797of the American inventor Joseph Henry (1797--1878) .1878). 1H = 1 volt-1H = 1 volt-second / amperesecond / ampere (a) when(a) when^ iiLL is constant , thenis constant , then^ vvLL=0 .=. i.e.i.e. , equivalent to short circuit, equivalent to short circuit (b)(b) iiLL(t) is a continuous function if there is only finite(t) is a continuous function if there is only finite strength source inside the circuit .strength source inside the circuit.
( ) ( ) (^) ( ) ( )
0
1
lim , 0 0
.. can not change abruptly for finite.
t L L (^) t L
L L L L L L
i t i t v dt L i t i t or i i i e i t v t
τ
+∈
− ∈→
∈ = +
∈ = =
∫
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6.2 The Inductor6.2 The Inductor
(c) An ideal inductor does not dissipate energy .(c) An ideal inductor does not dissipate energy. It stores energy in the magnetic field .It stores energy in the magnetic field.
(d) A nonideal inductor contains winding resistance and(d) A nonideal inductor contains winding resistance and parasitic capacitance .parasitic capacitance.
6.2 The Inductor6.2 The Inductor
2
2
12 2 1 5 5 10 1 1 10 50 2 (^1 2 2 ) 2
L
C L
C
L
I I A
V I V
W J
W J
= = =
= =
= × × =
= × × =
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6.3 Series- 6.3 Series-Parallel Combinations ofParallel Combinations of Capacitance and InductanceCapacitance and Inductance N capacitors in parallel N capacitors in parallel
i i 1 i 2 iN v c 1 c 2 L cN
1 2
1 2
1 1 2 (0) (0)
N
N N k k^ eq eq N k
i i i i
c dv^ c dv^ c dv dt dt dt c dv^ C dv dt dt C c c c v v
=
= + + +
= + + +
= ^ = ∴ = + + + =
∑
Q L
L
(^1 2) L 1 2
N N
6.3 Series- 6.3 Series-Parallel Combinations ofParallel Combinations of Capacitance and InductanceCapacitance and Inductance N capacitors in series N capacitors in series
L
L
0
0
0
0
1 1
0
1 2 0 1 0 2 0 0
1
(^1) ( )
1
1 1 1 1
t k k k t N t N k k k k^ o t t eq t
eq N N
v t (^) c i d v t
v i d v t c
i d v t C
C C C C v t v t v t v t
τ τ
τ τ
τ τ
= =
= +
∴ = ^ + = +
∴ = + + +
= + + +
Q
L
L
i 1 = i 2 = L=iN
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6.3 Series- 6.3 Series-Parallel Combinations ofParallel Combinations of Capacitance and InductanceCapacitance and Inductance N inductors in series N inductors in series
L
1 2
1 2
1 1 2 (0) 1 (0) 2 (0) (0)
N
N N k eq k eq N N
v v v v
L di^ L di^ L di dt dt dt di di L L dt dt L L L L i i i i
=
= + + +
= + + +
= ^ =
∴ = + + + = = =
∑
Q L
L
L L
1 2 1 2
N N
6.3 Series- 6.3 Series-Parallel Combinations ofParallel Combinations of Capacitance and InductanceCapacitance and Inductance N inductors in parallel N inductors in parallel
L
0 0 0
0
0
1 0 1 2 0 2 0
1 1 0 0
1 2 0 1 0 2 0 0
1 1 1
1
1
1 1 1 1
t t t t t N Nt N (^) t N k (^) k t k k t eqt
eq N N
i i t (^) L vd i t (^) L vd i t (^) L vd
L vd^ i t vd i t L
L L L L i t i t i t i t
τ τ τ
τ
τ
= =
= + + + + + +
= + = +
∴ = + + +
= + + +
L
L
L
1 2 1 2
N N
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1 2
1 1
Assume uniform magnetic field intensity HAssume uniform magnetic field intensity H 1 1 1 1
1 1
1 2 1 1 2 1 1 1 2 2 1 12 2 1 2 1 2 1 1 1
,
;
,
H N^ I^ B H N^ I l l B d A A N^ I l N A N^ I^ N A N^ N^ I l l L A N^ M A N^ N I l I l
μ^ μ
φ μ
λ φ μ^ λ φ μ λ μ λ μ
= ∴ = =
= ⋅ =
= = = =
= =
@ @
6.4 Mutual Inductance6.4 Mutual Inductance
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Dot convention for mutually coupled inductors: Dot convention for mutually coupled inductors: When the reference direction for a current enters theWhen the reference direction for a current enters the dotted terminal of a coil , the polarity of the induceddotted terminal of a coil , the polarity of the induced voltage in the other coil is positive at its correspondingvoltage in the other coil is positive at its corresponding dotted terminal.dotted terminal.
6.4 Mutual Inductance6.4 Mutual Inductance
i i 11 ↗↗ ,, vv 11 >0 ,>0 , φ↗φ↗ , ( i, ( i 22 =0 )=0 ) Another dot of coil 2 should be placed in terminal c.Another dot of coil 2 should be placed in terminal c.
V 1 >0 V^2
6.4 Mutual Inductance6.4 Mutual Inductance
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Then the induced voltage at coil two will increase Then the induced voltage at coil two will increase and so will iand so will i 22 .. This will violate the conservation of energy.This will violate the conservation of energy.
V 1 >0 V 2
6.4 Mutual Inductance6.4 Mutual Inductance
The procedure for determining dot markings The procedure for determining dot markings
Step1Step1 Assign current direction references for the coils.Assign current direction references for the coils.
Step2Step2 Arbitrarily select one terminal of one coil andArbitrarily select one terminal of one coil and mark it with a dot.mark it with a dot.
Step3Step3 Use the right-Use the right-hand rule to determine the directionhand rule to determine the direction of the magnetic flux due to the current of the otheof the magnetic flux due to the current of the otherr coil.coil.
6.4 Mutual Inductance6.4 Mutual Inductance
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Step4 Step4 If this flux direction has the same direction as thatIf this flux direction has the same direction as that of the first dot terminal current , then the secondof the first dot terminal current , then the second dot is placed at the terminal where the seconddot is placed at the terminal where the second current enters. Otherwise, the second dot shouldcurrent enters. Otherwise, the second dot should be placed at the terminal where the second currentbe placed at the terminal where the second current leaves.leaves.
6.4 Mutual Inductance 6.4 Mutual Inductance
Step 1Step 1 : Assign: Assign ii 11 ,, ii 22 ,, ii 33 directions.directions.
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6.4 Mutual Inductance 6.4 Mutual Inductance
For coils 2 and 3 For coils 2 and 3
6.4 Mutual Inductance 6.4 Mutual Inductance
Step 3Step 3 :: Check the relative flux directions and determine theCheck the relative flux directions and determine the dot position at the other coildot position at the other coil For coil 1 and 2For coil 1 and 2 andand are in the same directionare in the same direction
L 1 L 2
M 12
φ (^) 1 φ 2
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6.4 Mutual Inductance 6.4 Mutual Inductance
For coil 1 and 3 For coil 1 and 3
andand are in opposite directionare in opposite direction
L 1 L 3
6.4 Mutual Inductance 6.4 Mutual Inductance
For coil 2 and 3 For coil 2 and 3 φ 2 (^) andand φ 3 are in opposite directionare in opposite direction
