EE101EE101EE101EE101Circuit TheoryCircuit TheoryCircuit TheoryCircuit Theory

TRANSIENTS CIRCUIT ANALYSISCapacitors and InductorsCapacitors and Inductors

Engr. Gerard F. Manuel

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Transients Circuit Transients Circuit AnalysisAnalysisAnalysisAnalysis

- In transients analysis, two storage elements are added In transients analysis, two storage elements are added to the resistance. These are capacitors and inductors. Both have the ability to store energy, which can retrieve at some later time.

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1 1 C ito (1)C ito (1)1 1 Capacitors (1)Capacitors (1)A capacitor is a passive element designed p p gto store energy in its electric field.

• A capacitor consists of two conducting plates

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separated by an insulator (or dielectric).

1 1 C ito (2)C ito (2)1 1 Capacitors (2)Capacitors (2)Factors governing Capacitance:acto s go e g Capac ta ce

Area of the metal platesSpacing between the platesSpacing between the platesTypes of dielectric material

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1 1 C ito C ito (3)(3)1 1 Capacitors Capacitors (3)(3)Capacitance C is the ratio of the charge q on one Capac ta ce C s t e at o o t e c a ge q o o eplate of a capacitor to the voltage difference vbetween the two plates, measured in farads (F).

vCq =dAC ε

=and

• Where ε is the permittivity of the dielectric material

d

e e ε s t e pe tt ty o t e d e ect c ate abetween the plates, A is the surface area of each plate, d is the distance between the plates.

• Unit: F pF (10–12) nF (10–9) and μF (10–6)

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• Unit: F, pF (10 12), nF (10 9), and μF (10 6)

1 1 C ito C ito (4)(4)1 1 Capacitors Capacitors (4)(4)If i is flowing into the +ve If i is flowing into the +ve terminal of C◦ Charging => i is +ve◦ Discharging => i is –ve◦ Discharging => i is ve

• The current-voltage relationship of capacitor according to above convention is

d 1tdvdCi = )(1

00

tvtdiC

vt

t+= ∫and

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1 1 C ito C ito (5)(5)1 1 Capacitors Capacitors (5)(5)The energy w stored in The energy, w, stored in the capacitor is

1 2

21 vCw =

• A capacitor is – an open circuit to dc (dv/dt = 0). – its voltage cannot change abruptly.

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C ito (6)C ito (6)Capacitors (6)Capacitors (6)Example 1Example 1

The current through a 100-μF capacitor is

i(t) = 50 sin(120 πt) mA.

Calculate the voltage across it at t =1 ms and t = 5 ms.

T k (0) 0 Take v(0) =0.

Answer:

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v(1ms) = 93.14mV

v(5ms) = 1.7361V

1 1 C ito C ito (7)(7)1 1 Capacitors Capacitors (7)(7)Example 2

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

Calculate the voltage across it at t = 2 ms

Answer:

v(2ms) = 100 mV

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v(2ms) = 100 mV

2 2 Series and Parallel Series and Parallel C ito (1)C ito (1)Capacitors (1)Capacitors (1)

The equivalent capacitance of N parallel-The equivalent capacitance of N parallelconnected capacitors is the sum of the individual capacitances.

Neq CCCC +++= ...21

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2 2 Series and ParallelSeries and ParallelC ito (2)C ito (2)Capacitors (2)Capacitors (2)The equivalent capacitance of N series-connectedThe equivalent capacitance of N series connectedcapacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

1111

Neq CCCC1...111

21

+++=

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2 2 Series and Parallel Series and Parallel C ito (3)C ito (3)Capacitors (3)Capacitors (3)Example 3Example 3Find the equivalent capacitance seen at the terminals of the circuit in the circuit shown below:

Answer:

Ceq = 40μF

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2 2 Series and Parallel Series and Parallel C ito (4)C ito (4)Capacitors (4)Capacitors (4)Example 4a p eFind the voltage across each of the capacitors in the circuit shown below:

Answer:

v1 = 30V

v2 = 30V

v = 10Vv3 = 10V

v4 = 20V

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3 3 I d to (1)I d to (1)3 3 Inductors (1)Inductors (1)An inductor is a passive element designed An inductor is a passive element designed to store energy in its magnetic field.

• An inductor consists of a coil of conducting wire.

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3 3 I d to I d to (2)(2)3 3 Inductors Inductors (2)(2)Factors governing Capacitance:

Number of turns in the coilPermeability of the magnetic circuit.Cross-sectional area of the magnetic circuitCross sectional area of the magnetic circuitLength of the inductor or spacing between the turns

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3 3 I d to I d to (3)(3)3 3 Inductors Inductors (3)(3)Inductance is the property whereby an inductor Inductance is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H).

didLv = ANL μ

=2

andtd l

L

• The unit of inductors is Henry (H), mH (10–3) and μH (10–6)

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and μH (10 ).

3 3 I d to I d to (4)(4)3 3 Inductors Inductors (4)(4)The current-voltage relationship of an inductor:The current voltage relationship of an inductor:

)()(10titdtvi

t+= ∫

The power stored by an inductor:

)()( 00

titdtvL

it

+∫• The power stored by an inductor:

21 iLw =2

iLw =

• An inductor acts like a short circuit to dc (di/dt = 0)

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• An inductor acts like a short circuit to dc (di/dt = 0) and its current cannot change abruptly.

3 3 I d to I d to (5)(5)3 3 Inductors Inductors (5)(5)Example 5The terminal voltage of a 2-H inductor is

v = 10(1-t) V v 10(1 t) V

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

A i(0) 2 A Answer:Assume i(0) = 2 A. i(4s) = -18V

w(4s) = 320J

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3 3 I d to (5)I d to (5)3 3 Inductors (5)Inductors (5)Example 6a p e 6

Determine vc, iL, and the energy stored in the capacitor and inductor in the circuit of circuit shown capacitor and inductor in the circuit of circuit shown below under dc conditions at steady state.

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4 4 Series and ParallelSeries and ParallelI d to (1)I d to (1)Inductors (1)Inductors (1)The equivalent inductance of series-connectedThe equivalent inductance of series connectedinductors is the sum of the individual inductances.

Neq LLLL +++= ...21

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4 4 Series and Parallel Series and Parallel I d to (2)I d to (2)Inductors (2)Inductors (2)

The equivalent capacitance of parallel inductors is The equivalent capacitance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.

Neq LLLL1...111

21

+++=

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4 4 Series and ParallelSeries and ParallelC ito (3)C ito (3)Capacitors (3)Capacitors (3)Example 7Example 7Calculate the equivalent inductance for the inductive ladder network in the circuit h b lshown below:

Answer:

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Leq = 25mH

4 4 Series and Parallel Series and Parallel C ito (4)C ito (4)Capacitors (4)Capacitors (4)

Current and voltage relationship for R, L, C

+

+

+

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