Formula Sheets: Transient Analysis in AC & DC Circuits | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

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Net w ork Theory: T ransien t Analysis in A C & DC
Circuits F orm ula Sheet for Electrical GA TE
General T ransien t Analysis
• T ransien t Resp onse : Resp onse of a circuit to a sudden c hange (e.g., switc hing) un til
it reac hes steady state.
• Comp onen ts : T ransien t (natural) + Steady-state (forced) resp onse.
• Initial Conditions : V alues of curren ts and v oltages at t=0
+
.
R-C Circuits (DC)
• Time Constan t : t =RC (s ), where R is resistance (? ), C is capacitance (F ).
• Capacitor V oltage (Charging) :
V
C
(t)=V
f
+(V
i
-V
f
)e
-t/t
where V
f
is final v oltage, V
i
is initial v oltage.
• Capacitor Curren t :
I
C
(t)=C
dV
C
dt
=
V
i
-V
f
R
e
-t/t
• Differen tial Equation :
RC
dV
C
dt
+V
C
=V
s
where V
s
is the source v oltage.
R-L Circuits (DC)
• Time Constan t : t =
L
R
(s ), where L is inductance (H ).
• Inductor Curren t :
I
L
(t)=I
f
+(I
i
-I
f
)e
-t/t
where I
f
is final curren t, I
i
is initial curren t.
• Inductor V oltage :
V
L
(t)=L
dI
L
dt
=(V
i
-V
f
)e
-t/t
• Differen tial Equation :
L
dI
L
dt
+RI
L
=V
s
1
Page 2


Net w ork Theory: T ransien t Analysis in A C & DC
Circuits F orm ula Sheet for Electrical GA TE
General T ransien t Analysis
• T ransien t Resp onse : Resp onse of a circuit to a sudden c hange (e.g., switc hing) un til
it reac hes steady state.
• Comp onen ts : T ransien t (natural) + Steady-state (forced) resp onse.
• Initial Conditions : V alues of curren ts and v oltages at t=0
+
.
R-C Circuits (DC)
• Time Constan t : t =RC (s ), where R is resistance (? ), C is capacitance (F ).
• Capacitor V oltage (Charging) :
V
C
(t)=V
f
+(V
i
-V
f
)e
-t/t
where V
f
is final v oltage, V
i
is initial v oltage.
• Capacitor Curren t :
I
C
(t)=C
dV
C
dt
=
V
i
-V
f
R
e
-t/t
• Differen tial Equation :
RC
dV
C
dt
+V
C
=V
s
where V
s
is the source v oltage.
R-L Circuits (DC)
• Time Constan t : t =
L
R
(s ), where L is inductance (H ).
• Inductor Curren t :
I
L
(t)=I
f
+(I
i
-I
f
)e
-t/t
where I
f
is final curren t, I
i
is initial curren t.
• Inductor V oltage :
V
L
(t)=L
dI
L
dt
=(V
i
-V
f
)e
-t/t
• Differen tial Equation :
L
dI
L
dt
+RI
L
=V
s
1
R-L-C Circuits (DC)
• Differen tial Equation :
LC
d
2
V
C
dt
2
+RC
dV
C
dt
+V
C
=V
s
• Characteristic Equation :
s
2
+
R
L
s+
1
LC
=0
• Damping F actor : a =
R
2L
, Natural F requency : ?
0
=
1
v
LC
.
• Resp onse T yp es :
– Ov erdamp ed (a>?
0
): s
1
,s
2
real and distinct.
V
C
(t)=A
1
e
s
1
t
+A
2
e
s
2
t
+V
f
– Critically Damp ed (a =?
0
): s
1
=s
2
.
V
C
(t)=(A
1
+A
2
t)e
-at
+V
f
– Underdamp ed (a<?
0
): s=-a±j?
d
, where ?
d
=
v
?
2
0
-a
2
.
V
C
(t)=e
-at
(A
1
cos?
d
t+A
2
sin?
d
t)+V
f
A C T ransien t Analysis
• Steady-State A C Resp onse : Use phasors, imp edance Z =R+jX (? ).
• T ransien t in A C : Com bine DC transien t solution with steady-state A C solution
(phasor form).
• Imp edance :
Z
L
=j?L, Z
C
=
1
j?C
• A C Source : V
s
(t)=V
m
cos(?t+?) or I
s
(t)=I
m
cos(?t+?) .
• T otal Resp onse : V(t)=V
transien t
(t)+V
steady-state
(t) .
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Resistance (? ), Inductance (H ), Capacitance
(F ).
• Initial conditions (V
C
(0
+
) , I
L
(0
+
) ) are crit ical for transien t analysis.
• F or A C, con v ert to phasor domain for steady-state; solv e transien t separately .
• In R-L-C, determine damping t yp e using a and ?
0
.
• Use Laplace transforms for complex circuits: s -domain imp edanceZ
L
(s)=sL ,Z
C
(s)=
1
sC
.
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