Formula Sheets: R-L & R-C Transients | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

 Page 1


Basic Electrical T ec hnology: R-L & R-C T ransien ts
F orm ula Sheet for Electrical GA TE
R-L Circuit T ransien ts
• Time Constan t :
t =
L
R
where t is the time constan t (s ), L is inductance (H ), R is resistance (? ).
• Step Resp onse (DC Source Applied at t=0 ) :
i(t)=
V
R
(
1-e
-
t
t
)
(Inductor curren t)
v
L
(t)= Ve
-
t
t
(Inductor v oltage)
where V is the DC source v oltage (V ).
• Initial Condition Resp onse (Source Remo v ed, Initial Curren t I
0
) :
i(t)= I
0
e
-
t
t
v
L
(t)=-L
di(t)
dt
=-I
0
Re
-
t
t
• Stored Energy in Inductor :
W
L
=
1
2
LI
2
where W
L
is energy (J ), I is steady-state curren t.
R-C Circuit T ransien ts
• Time Constan t :
t = RC
where C is capacitance (F ).
• Step Resp onse (DC Source Applied at t=0 ) :
v
C
(t)= V
(
1-e
-
t
t
)
(Capacitor v oltage)
i(t)=
V
R
e
-
t
t
(Capacitor curren t)
• Initial Condition Resp onse (Source Remo v ed, Initial V oltage V
0
) :
v
C
(t)= V
0
e
-
t
t
i(t)=-
V
0
R
e
-
t
t
• Stored Energy in Capacitor :
W
C
=
1
2
CV
2
where W
C
is energy (J ), V is steady-state v oltage.
1
Page 2


Basic Electrical T ec hnology: R-L & R-C T ransien ts
F orm ula Sheet for Electrical GA TE
R-L Circuit T ransien ts
• Time Constan t :
t =
L
R
where t is the time constan t (s ), L is inductance (H ), R is resistance (? ).
• Step Resp onse (DC Source Applied at t=0 ) :
i(t)=
V
R
(
1-e
-
t
t
)
(Inductor curren t)
v
L
(t)= Ve
-
t
t
(Inductor v oltage)
where V is the DC source v oltage (V ).
• Initial Condition Resp onse (Source Remo v ed, Initial Curren t I
0
) :
i(t)= I
0
e
-
t
t
v
L
(t)=-L
di(t)
dt
=-I
0
Re
-
t
t
• Stored Energy in Inductor :
W
L
=
1
2
LI
2
where W
L
is energy (J ), I is steady-state curren t.
R-C Circuit T ransien ts
• Time Constan t :
t = RC
where C is capacitance (F ).
• Step Resp onse (DC Source Applied at t=0 ) :
v
C
(t)= V
(
1-e
-
t
t
)
(Capacitor v oltage)
i(t)=
V
R
e
-
t
t
(Capacitor curren t)
• Initial Condition Resp onse (Source Remo v ed, Initial V oltage V
0
) :
v
C
(t)= V
0
e
-
t
t
i(t)=-
V
0
R
e
-
t
t
• Stored Energy in Capacitor :
W
C
=
1
2
CV
2
where W
C
is energy (J ), V is steady-state v oltage.
1
General T ransien t Analysis
• General Solution (First-Order Differen tial Equation) :
x(t)= x(8)+[x(0
+
)-x(8)]e
-
t
t
where x(t) is the v ariable (curren t or v oltage), x(8) is steady-state v alue, x(0
+
) is
initial v alue p ost-switc hing.
• Time to Reac h Steady State : Appro ximately 5t for 99% of final v alue.
t˜5t
P o w er and Energy Dissipation
– P o w er Dissipated in Resistor :
P = I
2
R =
V
2
R
where P is p o w er (W ), I is RMS curren t, V is RMS v oltage across resistor.
– Energy Dissipated in Resistor (T ransien t) :
W
R
=
?
8
0
P(t)dt
Key Notes
* Initial Conditions : Inductor curren t and capacitor v oltage cannot c hange in-
stan taneously .
* Steady-State : R-L: i=
V
R
, v
L
=0 ; R-C: v
C
= V , i=0 .
* GA TE F o cus : Solv e for i(t) , v(t) , time constan ts; handle step and initial condi-
tion resp onses.
* Units : Use SI units for GA TE (e.g., V , A , ? , H , F , s ).
* Switc hing Analysis : Determine t = 0
+
conditions carefully based on circuit
configuration.
2