Overview Review Capacitor Inductor Test & next time

Circuit Analysis - Lecture 4

Capacitors and Inductors

2024/2025
Overview Review Capacitor Inductor Test & next time

Overview

1 Review

2 Capacitor

3 Inductor

4 Test & next time

Overview Review Capacitor Inductor Test & next time

Review

Lecture 1:
Circuit model, circuit variables, elements, element equations
(EEQ)
Lecture 2:
Structured analysis approach: node voltage method
Lecture 3:
Source transformations, superposition, Thévenin, Norton
Maximum power transfer
Lecture 4:
Capacitor and Inductor
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Capacitor

Capacitor:
2 conductors separated by non conductor (dielectric)
E.g. plates separated by air, glass, paper, ...

Equal but oposite charge on plates

Overview Review Capacitor Inductor Test & next time

Circuit Elements: Capacitor

Remember voltage = energy required to move unit charge
from A to B (plate 1 to plate 2 in capacitor):
dwAB
VAB =
dq
If Q small, it takes small effort, if Q is large, takes much
effort/releases much energy
Increasing Q means storing energy in Capacitor
Decreasing Q means extracting energy from Capacitor
Ratio between charge and voltage is constant:
Q
Capacity = C =
V
C depends on area of plates, dielectric material, distance
between plates
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Capacitor

To build up charge we need a current:

dq(t)
i(t) =
dt
With:
Q
C=
V
We get:
dq dv
Q = CV ⇒ =C = i(t)
dt dt

This is the element equation of the Capacitor

Passive: cannot deliver more energy than is stored
N.B. You only need to know the EEQ, not this derivation!
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Capacitor

i Capacity
+ Symbol: C
v Dimension: Farad (F)
C − 1 F = 1 As/V = 1 s/Ω

or:
Element Equation (EEQ):
1
Rt
i(t) = C dv (t) v (t) = v (t0 ) + C i(τ )d(τ )
dt t0
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Capacitor

v(t)
1
i
+ 0 1 2 t
v
C − i(t) 1

0 1 2 t
−1
or:
Element Equation (EEQ):
1
Rt
i(t) = C dv (t) v (t) = v (t0 ) + C i(τ )d(τ )
dt t0
Overview Review Capacitor Inductor Test & next time

Energy and power in a capacitor

Rt
i w (t) − w (t0 ) = C dvdτ(τ ) v (τ )dτ
t0
C +
vs +− p,w v vR(t)
− =C vdv
v (t0 )

= 21 C v 2 (t) − v 2 (t0 )

p(t) = i(t)v (t) for w (t0 ) = 0 and v (t0 ) = 0:

= C dvdt(t) v (t)
w (t) = 21 Cv 2 (t)
Overview Review Capacitor Inductor Test & next time

Capacitors in parallel

i1 i2 i = i1 + i2
+
i v = C1 dv dv
dt + C2 dt

C1 C2 −
= (C1 + C2 ) dv
dt

Cv = C1 + C2

Capacity ∼ area of plates

Parallel ⇒ bigger area
Overview Review Capacitor Inductor Test & next time

Capacitors in series
+
i 1
Rt
v1 (t)= v1 (t0 ) + i(τ )dτ
+ C1
t0
C1 v1 v2 (t)= v2 (t0 ) + 1
Rt
i(τ )dτ
− C2
i v v (t)= v1 (t) + v2 (t)
t0

+ 
1 1
 Rt
C2 v2 = v1 (t0 ) + v2 (t0 ) + C1 + C2
t0
i(τ )dτ
− 
1 1
 Rt
= v (t0 ) + C1 + C2 i(τ )dτ
− t0

1 1 1
Cv = C1 + C2

Capacity ∼ 1/distance between plates; Series ⇒ distance bigger

Overview Review Capacitor Inductor Test & next time

Circuit Elements: Inductor

Inductor:
Wire, usually wound around as coil around core
Core is of magnetic material

Energy stored in magnetic field

Resists changes in current (magnetic field)
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Inductor

Increasing flux means storing energy in Inductor

Decreasing flux means extracting energy from Inductor
Ratio between magnetic flux and current is constant:
ϕ
Inductance = L =
I
Inductance L determines how much magnetic flux is generated
by current I .
L depends on number of windings, magnetic material of core
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Inductor

Faraday’s law of induction:

dϕ
ϵ=−
dt
ϵ is electromotive force (dynamo) = voltage
This is about energy generated in a coil (dynamo)
We consider energy stored in coil, so:
dϕ
v =+
dt
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Inductor

To build up flux we need voltage (Faraday/Lenz):

dϕ(t)
v (t) =
dt
With:
ϕ
L=
I
We get:
dϕ di
ϕ = LI ⇒ = L = v (t)
dt dt

This is the element equation of the Inductor

Passive: cannot deliver more energy than is stored
N.B. You only need to know the EEQ, not this derivation!
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Inductor

i Inductor
+ Symbol: L
v Dimension: Henry (H)
L − 1 H = 1 Vs/A = 1 Ωs

or:
Element Equation (EEQ):
1
Rt
v (t) = L di(t) i(t) = i(t0 ) + L v (τ )d(τ )
dt t0
Overview Review Capacitor Inductor Test & next time

Circuit Elements: Inductor

i(t)
1
i
+ 0 1 2 t
v
L − v(t)1

0 1 2 t
−1
or:
Element Equation (EEQ):
1
Rt
v (t) = L di(t) i(t) = i(t0 ) + L v (τ )d(τ )
dt t0
Overview Review Capacitor Inductor Test & next time

Energy and power in an inductor

Rt
w (t) − w (t0 ) = i(τ )L di(τ )
dτ dτ
i t0

L +
is p,w v i(t)
−
R
=L idi
i(t0 )

= 12 L i 2 (t) − i 2 (t0 )

p(t) = i(t)v (t)

for w (t0 ) = 0 and i(t0 ) = 0:

= i(t)L di(t)
dt
w (t) = 21 Li 2 (t)
Overview Review Capacitor Inductor Test & next time

Inductors in parallel

1
Rt
i1 (t)= i1 (t0 ) + L1 v (τ )dτ
t0
Rt
i1 i2 i2 (t)= i2 (t0 ) + 1
L2 v (τ )dτ
+ t0
i v
L1 L2 − i(t)= i1 (t) + i2 (t)
  Rt
1 1
= i1 (t0 ) + i2 (t0 ) + L1 + L2 v (τ )dτ
t0
  Rt
1 1
= i(t0 ) + L1 + L2 v (τ )dτ
t0

1 1 1
Lv = L1 + L2
Overview Review Capacitor Inductor Test & next time

Inductors in series

+
i di
v1 (t)= L1 dt
+ di
L1 v1 v2 (t)= L2 dt
−
i v v (t)= v1 (t) + v2 (t)
+ di di
L2 v2 = L1 dt + L2 dt
− di
= (L1 + L2 ) dt
−
Lv = L1 + L2
Overview Review Capacitor Inductor Test & next time

Test and next time:

Test on source trafo, superpos, Thevenin, Norton, max power,

Capacitors and Inductors
4 October, 13.45-15.15

Next time:
Responses of circuits with resistors and one capacitor or
inductor (1st order circuits).