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important formula how to solve the first order differential circuits..
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First Order Differential Circuits
MADE BY-Aadish Bhagat (130110111003)Arjun Gupta (130110111021)Shubham Kurup (130110111027)Pragnesh Patel (130110111042)Denish Shah (140113111012)
GUIDED BY – Prof. Shankar K. Parmar
G. H. Patel College of Engineering & Technology
Subject :- Circuits and Networks (2130901)
First Order Differential Circuits• Consider a differential equation of nth order• • Where,• a0,a1,a2…. Are positive constants
• i=dependent variable• t=independent variable• v(t)=forcing function
• solution of a non homogeneous differential equation using integrating factor
• consider a first order non homogeneous equation as
• +pi=q, Where
• P is constant and q may be function of independent variable or constant
• Multiplying the equation by ept
• ept + ept pi = qept
• (i ept)=q ept
• now integrating the above equation
• ept= + k-------------(1)• equation 1 is generalized solution of the equation
• consider q is constant• i(t)=e-pt ept + k e-pt
• i(t) = + k e-pt
• at t=0 switch is getting closed
FIG
• after switching actionv = iR + Where p =and q = • i(t) = + K
• ABOUT INDUCTOR(L): 1)L behaves as open circuit if it is initially not charged
• 2)L behaves as current source if it is initially charged
• 3)L behaves as short circuit at steady state
Seven important steps to solve first order differential equation
• 1)Draw the circuit after switching action takes place.
• 2)Apply KVL or KCL to get first order differential equation.
• 3)Get generalized solution.• 4)Draw the circuit at t=0-.
• 5)Identify whether L or C is charged or not.• 6)Replace it at t=o+ and get the quantity for which
we have found generalized solution.• 7)Get particular solution.
• At t=0 switch is getting closed.• Steady state is attained previously.
QUESTION 30Ω 20Ω
𝟏𝟐𝑯10V
20V
𝒊 (𝒕 )
Answer• After switching
30Ω 20Ω
10V
20V
𝟏𝟐𝑯
𝒊 (𝒕 )
• ←first order differential equation
P Q
𝒊𝒍(𝟎−)
30Ω
20Ω
20V
10V
𝒕=𝟎−
• Putting the value of K in general solution𝐾=0.1
𝒕=𝟎+¿¿ 30Ω
20Ω
20V
10V
𝟑𝟓𝑨
Equation For Capacitor
After Switching
𝑖(𝑡)
𝑖(𝑡)
• V = iR + .•
Differentiate•
+ I = 0.•
Q = 0 , P = •
i(t) = k e
Note : Voltage across capacitor can not change instantly
T =
Vc(0-) = 0 V
i(0) = i(0) = =K i(t) =
T =
Vc (0-)=0v
• ABOUT Capacitor (C): 1)C behaves as short circuit if it is initially not charged
2)C behaves as voltage source if it is initially charged 3)C behaves as open circuit at steady state
Question:- find
5V
1Ω
2Ω
𝟏𝟐Ω
1Ω𝟏𝟐𝐅
𝑖𝑐
• After switching:-
0 2V -V32
1
02
1 )V -V(2
1
5-V
b a
b a a
dt
dVdt
dV
a
aNode A:-
5V
1Ω
2Ω
𝟏𝟐Ω
1Ω
𝟏𝟐𝐅
𝑖𝑐
ANSWER
Node B :-
7
5 4V V
5 4V V72
5 2V V]21
2
1[
0 )V -V(21
V
2
5 -V
a b
a b
a b
a bb b
tK
dt
dVdt
dVdt
dVdt
dV
a
a
a
a
e 726
26
90 (t)V
7
90 V
7
267
45 V
7
13
2
17
10 5 V]
7
83[
2
1
5]7
5 4V[ 2 -V3
2
1
a
a
a
a
a a
At t=
v312
3
5*23
)0-
( V c
5V
1Ω
𝟑𝟐Ω
-
+
𝒗𝒄𝟎−
At
e
e
e
t
t
dt
dV
t
K
a
7
26*
14
12
]7
26*7
26*
26
12[
2
1
2
1(t)i
7
26
26
12
26
90(t)V
26
903(0)V
3V (0)V
c
a
a
a
0t
3V
1Ω2Ω
1Ω
𝟏𝟐Ω
5V
THANK YOU