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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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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



Capacitors in






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



Complex number review

Euler’s indentity


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


VC(jw)= A -90°

Notice the impedance of a capacitance decreases with increasing frequency

inductive load
Inductive Load


Phasor: VL(jw)=A -90°

IL(jw)=(A/wL) -180°


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