Formula Sheets: Sinusoidal Steady State Analysis | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

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Net w ork Theory: Sin usoidal Steady State Analysis
F orm ula Sheet for Electrical GA TE
Phasors
• Sin usoidal Signal : v(t) = V
m
cos(?t+?) .
• Phasor Represen tation : V = V
m
e
j?
= V
m
?? .
• RMS V alue :
V
rms
=
V
m
v
2
, I
rms
=
I
m
v
2
• Time Domain to Phasor : v(t) = V
m
cos(?t+?)? V = V
m
?? .
Imp edance and A dmittance
• 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
).
• P olar F orm : Z =|Z|?? , where |Z| =
v
R
2
+X
2
, ? = tan
-1
(
X
R
)
, X is reactance.
• A dmittance (Y ) : Y =
1
Z
.
Y
R
=
1
R
, Y
L
=
1
j?L
, Y
C
= j?C
Units: S .
• Series/P arallel :
Z
series
= Z
1
+Z
2
+...,
1
Z
parallel
=
1
Z
1
+
1
Z
2
+...
Y
parallel
= Y
1
+Y
2
+...,
1
Y
series
=
1
Y
1
+
1
Y
2
+...
A C Circuit Analysis
• Ohm’s La w in Phasor F orm : V = IZ .
• Kirc hhoff ’s La ws :
?
V
k
= 0 ( KVL, phasor v oltages),
?
I
k
= 0 ( K CL, phasor cur ren ts)
• V oltage Divider :
V
x
= V
total
·
Z
x
Z
1
+Z
2
+···+Z
n
• Curren t Divider :
I
x
= I
total
·
Z
eq
Z
x
, where
1
Z
eq
=
?
1
Z
i
( excluding Z
x
)
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Net w ork Theory: Sin usoidal Steady State Analysis
F orm ula Sheet for Electrical GA TE
Phasors
• Sin usoidal Signal : v(t) = V
m
cos(?t+?) .
• Phasor Represen tation : V = V
m
e
j?
= V
m
?? .
• RMS V alue :
V
rms
=
V
m
v
2
, I
rms
=
I
m
v
2
• Time Domain to Phasor : v(t) = V
m
cos(?t+?)? V = V
m
?? .
Imp edance and A dmittance
• 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
).
• P olar F orm : Z =|Z|?? , where |Z| =
v
R
2
+X
2
, ? = tan
-1
(
X
R
)
, X is reactance.
• A dmittance (Y ) : Y =
1
Z
.
Y
R
=
1
R
, Y
L
=
1
j?L
, Y
C
= j?C
Units: S .
• Series/P arallel :
Z
series
= Z
1
+Z
2
+...,
1
Z
parallel
=
1
Z
1
+
1
Z
2
+...
Y
parallel
= Y
1
+Y
2
+...,
1
Y
series
=
1
Y
1
+
1
Y
2
+...
A C Circuit Analysis
• Ohm’s La w in Phasor F orm : V = IZ .
• Kirc hhoff ’s La ws :
?
V
k
= 0 ( KVL, phasor v oltages),
?
I
k
= 0 ( K CL, phasor cur ren ts)
• V oltage Divider :
V
x
= V
total
·
Z
x
Z
1
+Z
2
+···+Z
n
• Curren t Divider :
I
x
= I
total
·
Z
eq
Z
x
, where
1
Z
eq
=
?
1
Z
i
( excluding Z
x
)
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P o w er i n A C Circuits
• Complex P o w er : S = VI
*
= P +jQ .
P = V
rms
I
rms
cos?realpower,W
Q = V
rms
I
rms
sin? ( reactiv e p o w er, -reactive)
|S| = V
rms
I
rms
=
v
P
2
+Q
2
( apparen t p o w er, )
• P o w er F actor : cos? , where ? is the p hase difference b et w een V and I .
• P o w er in Imp edance : P = I
2
rms
R , Q = I
2
rms
X .
F requency Domain Analysis
• T ransfer F unction : H(j?) =
V out(j?)
V
in
(j?)
or
I out(j?)
I
in
(j?)
.
• Gain : |H(j?)| (in dB: 20log
10
|H(j?)| ).
• Phase : ?H(j?) = tan
-1
(
Im(H)
Re(H)
)
.
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Imp edance (? ), A dmittance (S ), P o w er (W ,
, -reactive ).
• Alw a ys use RMS v alues for p o w er calculations unless sp ecified otherwise.
• Ensure phase angles are correctly accoun ted for in phasor calculations.
• F or complex circuits, use no dal or mesh analysis in the phasor domain.
• V erify conjugate of curren t (I
*
) in complex p o w er form ula.
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