capacitors inductors

8/3/2019 Capacitors Inductors

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Circuit Theory

Capacitors and Inductors



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Capacitors and Inductors

1. Capacitors

2. Series and Parallel Capacitors

3. Inductors

4. Series and Parallel Inductors



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1 Capacitors (1)

• A capacitor is a passive element designed

to store energy in its electric field.

• A capacitor consists of two conducting platesseparated by an insulator (or dielectric).



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1 Capacitors (2)

• Capacitance C is the ratio of the charge q on oneplate of a capacitor to the voltage difference v between the two plates, measured in farads (F).

• Where is the permittivity of the dielectric materialbetween the plates, A is the surface area of eachplate, d is the distance between the plates.

• Unit: F, pF (10–12), nF (10–9), and

vC q d

AC

and

F (10–6)



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Capacitors (4)

• The energy, w , stored inthe capacitor is

• A capacitor is

– an open circuit to dc (dv/dt = 0).

– its voltage cannot change abruptly.

2

21 vC w



Example

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Example

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Example

The current through a 100-F capacitor is

i(t) = 50 sin(120 t) mA.

Calculate the voltage across it at t =1 ms and

t = 5 ms.

Take v (0) =0.

Answer:

v(1ms) = 93.14mV

v(5ms) = 1.7361V



Example

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Example

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Example

An initially uncharged 1-mF capacitor has thecurrent shown below across it.

Calculate the voltage across it at t = 2 ms and

t = 5 ms.

Answer:

v(2ms) = 100 mV

v(5ms) = 500 mV



Example – DC Conditions

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Example

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Capacitors (2)

• The equivalent capacitance of N series-connected capacitors is the reciprocal of the sum of thereciprocals of the individual capacitances.

N eq C C C C

1...

111

21

svC C

C v

21

2

1 sv

C C

C v

21

1

2

• Voltage Divider Rule



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Inductors (2)

• Inductance is the property whereby an inductorexhibits opposition to the change of currentflowing through it, measured in henrys (H).

• is the inductor core permeability• N: Number of turns

• L: Coil length

• A: Cross sectional area

• Unit: Henry (H).

t d

id Lv

l

A N L

2

and



Example

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Example

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Inductors (4)

ExampleThe terminal voltage of a 2-Hinductor is

v = 10(1-t) V

Find the current flowing through it att = 4 s and the energy stored in itwithin 0 < t < 4 s.

Assume i(0) = 2 A. Answer:

i(4s) = -18V

w(4s) = 320J



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Series and Parallel Inductors (1)

• The equivalent inductance of series-connected inductors is the sum of the individualinductances.

N eq L L L L ...21

sv L L

Lv

21

1

1 sv

L L

Lv

21

2

2

• Voltage Divider Rule



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Series and Parallel Inductors (2)

• The equivalent capacitance of parallel inductorsis the reciprocal of the sum of the reciprocals of the individual inductances.

N eq L L L L

1...

111

21

si L L

Li

21

2

1 si

L L

Li

21

1

2

•Current Divider Rule



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Series and Parallel-Inductor (3)

Calculate the equivalent inductance for theinductive ladder network in the circuitshown below:

Answer:

Leq = 25mH



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capacitors inductors

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