Capacitor

d

A

V

qC

:capacitora of dimension physical theon dependsonly eCapacitanc

A circuit element that stores electric energy and electric charges

A capacitor always consists of two separated metals, one stores +q, and the other stores –q. A common

capacitor is made of two parallel metal plates.

Capacitance is defined as: C=q/V (F); Farad=Colomb/volt

Once the geometry of a capacitor is determined, the capacitance (C) is fixed (constant) and is independent of voltage V. If the voltage is increased, the charge will increase to keep q/V constant

Application: sensor (touch screen, key board), flasher, defibrillator, rectifier, random access memory RAM, etc.

Capacitor: cont.

• Because of insulating dielectric materials between the plates, i=0 in DC circuit, i.e. the braches with Cs can be replaced with open circuit.

• However, there are charges on the plates, and thus voltage across the capacitor according to q=Cv.

• i-v relationship:

i = dq/dt = C dv/dt

• Solving differential equation needs an initial condition

• Energy stored in a capacitor: WC =1/2 CvC(t)2

Capacitors in

V=V1=V2=V3

q=q1+q2+q3

321321 CCC

V

qqq

V

qCeq

parallel series

V=V1+V2+V3

q=q1=q2=q3

321

321

111

1

CCC

q

VVV

q

V

Ceq

Inductor

i-v relationship: vL(t)= LdiL/dt

L: inductance, henry (H)Energy stored in inductors

WL = ½ LiL2(t)

In DC circuit, can be replaced with short circuit

Sinusoidal waves

• Why sinusoids: fundamental waves, ex. A square can be constructed using sinusoids with different frequencies (Fourier transform).

• x(t)=Acos(t+)• f=1/T cycles/s, 1/s, or Hz =2f rad/s 2t / rad

=360 t / deg.

Average and RMS quantities in AC Circuit

01

0

T

dttxT

tx

It is convenient to use root-mean-square or rms quantities to indicate relative strength of ac signals rather than the magnitude of the ac signal.

rmsrmsavermsrms VIPV

VI

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,2

T

rms dttxT

x0

21

Complex number review

A

Ae

jA

ba

bj

ba

abajba

j

sincos

2222

22

Euler’s indentity

ab

11

2

1

2

1

2

1

11212121

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21

21

21 ,

A

Ae

A

A

c

c

AAeAAcc

AeAcAeAc

j

j

jj

Phasor

How can an ac quantity be represented by a complex number?Acos(t+)=Re(Aej(t+))=Re(Aejtej )

Since Re and ejt always exist, for simplicity

Acos(t+) AejPhasor representation

Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form

v(t) = Acos(t+)

and a frequency-domain (or phasor) formV(j) = Aej

In text book, bold uppercase quantity indicate phasor voltage or currents

Note the specific frequency of the sinusoidal signal, since this is not explicit apparent in the phasor expression