fundamentals of engineering supplied-reference handbook - Ventech!

AC CIRCUITS
For a sinusoidal voltage or current of frequency f (Hz) and
period T (seconds),
f = 1/T = ω/(2π), where
ω = the angular frequency in radians/s.
Average Value
For a periodic waveform (either voltage or current) with
period T,
X ave = 1
T
( T ) x(
t)
∫
0
dt
The average value of a full-wave rectified sinusoid is
Xave = (2Xmax)/π
and half this for half-wave rectification, where
Xmax = the peak amplitude of the waveform.
Effective or RMS Values
For a periodic waveform with period T, the rms or effective
value is
12
⎡ T
2 ⎤
Xeff = Xrms = ⎢( 1 T) ∫ x ( t) dt⎥
⎣ 0 ⎦
For a sinusoidal waveform and full-wave rectified sine
wave,
X = X = X 2
eff
rms max
For a half-wave rectified sine wave,
Xeff = Xrms = Xmax/2
For a periodic signal,
∞
2 2
rms dc n
n= 1
X = X + ∑ X where
Xdc is the dc component of x(t)
Xn is the rms value of the n th harmonic
Sine-Cosine Relations
cos (ωt) = sin (ωt + π/2) = – sin (ωt – π/2)
sin (ωt) = cos (ωt – π/2) = – cos (ωt + π/2)
Phasor Transforms of Sinusoids
Ρ [Vmax cos (ωt + φ)] = Vrms ∠ φ = V
Ρ [Imax cos (ωt + θ)] = Irms ∠ θ = I
For a circuit element, the impedance is defined as the ratio
of phasor voltage to phasor current.
Z = V I
For a Resistor, ZR = R
1
For a Capacitor, ZC = = jXC
jωC 169
ELECTRICAL AND COMPUTER ENGINEERING (continued)
For an Inductor,
ZL = jωL = jXL, where
XC and XL are the capacitive and inductive reactances
respectively defined as
1
X C = − and X L = ωL
ωC
Impedances in series combine additively while those in
parallel combine according to the reciprocal rule just as in
the case of resistors.
ALGEBRA OF COMPLEX NUMBERS
Complex numbers may be designated in rectangular form or
polar form. In rectangular form, a complex number is
written in terms of its real and imaginary components.
z = a + jb, where
a = the real component,
b = the imaginary component, and
j = − 1
In polar form
z = c ∠ θ, where
c =
2
2
a + b ,
θ = tan –1 (b/a),
a = c cos θ, and
b = c sin θ.
Complex numbers are added and subtracted in rectangular
form. If
z1 = a1 + jb1 = c1 (cos θ1 + jsin θ1)
= c1 ∠ θ1 and
z2 = a2 + jb2 = c2 (cos θ2 + jsin θ2)
= c2 ∠ θ2, then
z1 + z2 = (a1 + a2) + j (b1 + b2) and
z1 – z2 = (a1 – a2) + j (b1 – b2)
While complex numbers can be multiplied or divided in
rectangular form, it is more convenient to perform these
operations in polar form.
z1 × z2 = (c1 × c2) ∠ θ1 + θ2
= (c1 /c2) ∠ θ1 – θ2
z1/z2
The complex conjugate of a complex number z1 = (a1 + jb1)
is defined as z1* = (a1 – jb1). The product of a complex
number and its complex conjugate is z1z1* = a1 2 + b1 2 .