FPGA Based Digital World - AC Analysis Basic Laws

Basic Laws

Phasor Ralatiohships for Circuit Elements

For resistor R, capcitor C and inductor L, the voltage-current relationships are listed below.

And it's also illustrated graphically.

LCR_Volt_Curr_Rel

[Example] The voltage v(t)=9cos(2t+30°)is applied to a 10mH inductor, calcuate the steady-sate current through the inductor.

Solution:

The source phasor is: V = 9*exp(30°), ω=2 rad/s

--> I=V/jωL = 9*exp(30°)/j(2*0.01)=9*exp(30°)/(0.2*exp(90°))=45*exp(-60°)A

--> i(t)=45*cos(2t-60°)

Basic Laws in Phasor Doamin

Ohm's law, Kirchhoff's laws of KVL and KCL are still valid in phasor domain.

Ohm's Law in Phasor Form

V=ZI

where V and I are phasor voltage and phasor current,

Z is the impedance, measured in ohms(Ω).

(Note: Z is not a phasor, and not corresponds to a sinusoid)

Impedance Z can be expressed in rectangular form as

Z=R+jX

where R is resistance and X is reactance, measured in ohms.

Admittance Y is the reciprocal of impedance, in the rectangular form as

Y=1/Z=I/V

Y expressed in rectangular form is

Y=G+jB

where G is conductance and B is suscptance.

The relationship between R,X and G, B is

G=R/(R*R+X*X), B=-X/(R*R+X*X)

KCL in Phasor Form

Kirchhoff's current law(KCL) applies to a node, and is based on the law of conservation of charge.

The sum of current phasors entering a node(or a closed boundary) is zero:

I1 + I2 +... + In = 0

KVL in Phasor Form

Kirchhoff's voltage law(KCL) applies to a loop , and is based on the law of conservation of energy.

The sum of all voltage phasors around a loop(or a closed path) is zero:

V1 + V2 +... + Vn = 0