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Capacitor. 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.

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capacitor
Capacitor

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
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

series

parallel

Capacitors in

V=V1+V2+V3

q=q1=q2=q3

V=V1=V2=V3

q=q1+q2+q3

inductor
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
Sinusoidal waves
  • Why sinusoids: fundamental waves, ex. A square can be constructed using sinusoids with different frequencies (Fourier transform).
  • x(t)=Acos(wt+f)
  • f=1/T cycles/s, 1/s, or Hz
  • w=2pf rad/s
  • f =2p (Dt / T) rad

=360 (Dt / T) deg.

average and rms quantities in ac circuit
Average and RMS quantities in AC Circuit

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.

complex number review

b

a

Complex number review

Euler’s indentity

phasor
Phasor

How can an ac quantity be represented by a complex number?

Acos(wt+q)=Re(Aej(wt+q))=Re(Aejwtejq )

Since Re and ejwtalways exist, for simplicity

Acos(wt+q)  Aejq=Aq Phasor representation

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

v(t) = Acos(wt+q)

and a frequency-domain (or phasor) form

V(jw) = Aejq=Aq

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

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

ac i v relationship for r l and c
AC i-V relationship for R, L, and C

Source vS(t) =Asinwt

Resistive Load

vR and iR are in phase

Phasor representation: vS(t) =Asinwt = Acos(wt-90°)= A -90°=VS(jw)

IS(jw) =(A / R)-90°

Impendence: complex number of resistance Z=VS(jw)/ IS(jw)=R

Generalized Ohm’s law VS(jw) = Z IS(jw)

Everything we learnt before applies for phasors with generalized ohm’s law

capacitor load
Capacitor Load

ICE

VC(jw)= A -90°

Notice the impedance of a capacitance decreases with increasing frequency

inductive load
Inductive Load

ELI

Phasor: VL(jw)=A -90°

IL(jw)=(A/wL) -180°

ZL=jwL

Opposite to ZC, ZL increases with frequency

ac circuit analysis
AC circuit analysis
  • Effective impedance: example
  • Procedure to solve a problem
    • Identify the sinusoidal and note the excitation frequency.
    • Covert the source(s) to phasor form
    • Represent each circuit element by its impedance
    • Solve the resulting phasor circuit using previous learnt analysis tools
    • Convert the (phasor form) answer to its time domain equivalent.
ex 4 21 p188
Ex. 4.21 P188

R1=100 W, R2=75 W, C= 1mF, L=0.5 H, vS(t)=15cos(1500t) V.

Determine i1(t) and i2(t).

Step 1: vS(t)=15cos(1500t), w=1500 rad/s.

Step 2: VS(jw)=15 0

Step 3: ZR1=R1, ZR2=R2, ZC=1/jwC, ZL=jwL

Step 4: mesh equation