Formula Sheets: A.C. Analysis | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

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Net w ork T heory: A.C. Analysis F orm ula Sheet for
Electrical GA TE
Phasors and Imp edance
• Phasor Represen tation : A C v oltage or curren t, V(t) = V
m
cos(?t+?) .
V = V
m
e
j?
= V
m
?? ( phasor form)
• Imp edance (Z ) :
Z
R
= R, Z
L
= j?L, Z
C
=
1
j?C
where R is resistance (? ), L is inductance (H ), C is capacitance (F ), ? is angular
frequency (rads
-1
).
• T otal Imp edance : F or series, Z = Z
1
+Z
2
+... ; for parallel,
1
Z
=
1
Z
1
+
1
Z
2
+... .
• P olar F orm : Z =|Z|?? , where|Z| =
v
R
2
+X
2
, ? = tan
-1
(
X
R
)
, X is reactance.
A C P o w er
• Instan taneous P o w er : p(t) = v(t)i(t) .
• Complex P o w er : S = VI
*
= P +jQ , where V and I are RMS phasors.
P =|S|cos? ( real p o w er, W)
Q =|S|sin? ( reactiv e p o w er,-reactive)
|S| =
v
P
2
+Q
2
( apparen t p o w er, )
• P o w er F actor : cos? , where ? is the p hase angle b et w een V and I .
• RMS V alues :
V
rms
=
V
m
v
2
, I
rms
=
I
m
v
2
Resonance
• Series Resonance :
?
0
=
1
v
LC
, f
0
=
1
2p
v
LC
where ?
0
is resonan t angular frequency (rads
-1
), f
0
is resonan t frequency (Hz ).
• Qualit y F actor (Q ) :
Q =
?
0
L
R
=
1
R
v
L
C
• Bandwidth : BW =
?
0
Q
=
R
L
(rads
-1
).
• P arallel Resonance : Imp edance is maxim um at ?
0
, same resonan t frequency as
series.
Z
max
˜
L
RC
( at resonance)
1
Page 2


Net w ork T heory: A.C. Analysis F orm ula Sheet for
Electrical GA TE
Phasors and Imp edance
• Phasor Represen tation : A C v oltage or curren t, V(t) = V
m
cos(?t+?) .
V = V
m
e
j?
= V
m
?? ( phasor form)
• Imp edance (Z ) :
Z
R
= R, Z
L
= j?L, Z
C
=
1
j?C
where R is resistance (? ), L is inductance (H ), C is capacitance (F ), ? is angular
frequency (rads
-1
).
• T otal Imp edance : F or series, Z = Z
1
+Z
2
+... ; for parallel,
1
Z
=
1
Z
1
+
1
Z
2
+... .
• P olar F orm : Z =|Z|?? , where|Z| =
v
R
2
+X
2
, ? = tan
-1
(
X
R
)
, X is reactance.
A C P o w er
• Instan taneous P o w er : p(t) = v(t)i(t) .
• Complex P o w er : S = VI
*
= P +jQ , where V and I are RMS phasors.
P =|S|cos? ( real p o w er, W)
Q =|S|sin? ( reactiv e p o w er,-reactive)
|S| =
v
P
2
+Q
2
( apparen t p o w er, )
• P o w er F actor : cos? , where ? is the p hase angle b et w een V and I .
• RMS V alues :
V
rms
=
V
m
v
2
, I
rms
=
I
m
v
2
Resonance
• Series Resonance :
?
0
=
1
v
LC
, f
0
=
1
2p
v
LC
where ?
0
is resonan t angular frequency (rads
-1
), f
0
is resonan t frequency (Hz ).
• Qualit y F actor (Q ) :
Q =
?
0
L
R
=
1
R
v
L
C
• Bandwidth : BW =
?
0
Q
=
R
L
(rads
-1
).
• P arallel Resonance : Imp edance is maxim um at ?
0
, same resonan t frequency as
series.
Z
max
˜
L
RC
( at resonance)
1
F requency Resp onse
• T ransfer F unction : H(s) =
V out(s)
V
in
(s)
or
I out(s)
I
in
(s)
in Laplace domain (s = j? for A C).
• Gain (Magnitude) : |H(j?)| (in dB: 20log
10
|H(j?)| ).
• Phase : ?H(j?) = tan
-1
(
Im(H)
Re(H)
)
.
• Cutoff F requency : F requency where|H(j?)| =
1
v
2
of maxim um ( for filters).
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Imp edance (? ), P o w er (W , , -reactive ),
F requency (Hz or rads
-1
).
• Use phasor notation for A C analysis; con v ert to time domain if needed.
• F or resonance, c hec k if series or parallel circuit; Q determines sharpness.
• F or p o w er calculations, use RMS v alues unless sp ecified otherwise.
• V erify phase angles for p o w er factor and ensure conjugate for curren t in S = VI
*
.
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