The charge on a capacitor g(t) in a circuit with a resistor, a capacitor and an inductor connected in series driven by a given time-dependent voltage e(t) is governed by the second-order ODE d'e) + 2d'(0) + 2q(4) = v(t). (1) A schematic of the RLC circuit can be found in the course notes. In Questions 1 - 6 you will caleulate Laplace transforms / inverse Laplace transforms of fune- tions. These calculations will be useful in determining the response of (1) for an initial value problem specified in Question 7. You MUST use the method of solution as per the instructions found in each of the below questions. 1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on MyUni to find the Laplace transform of r(t) = av (t) + Bey(), where (t)=6(t-2) and va(t) = u(t - 3). Here, a and 3 are constants, and 6(t) and u(t) are the Dirac delta function and unit step function. 2 2. Complete the square of the denominator and use the a-shifting theorem to find the inverse Laplace transform of the function F(e) = + 2s + 2 Where applicable, you may use entries (1) through to (15) and the basie general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23). 5 3. Une partial fraetion decomposition, linearity and a-ahifting theorem to find the invere Laplace transform of the function H() s(o² +2» + 2)* Where applicable, you may use entries (1) through to (15) and the basic general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23). 4 4. Use the convolution theorem and your answer to Question 2 to verify the inverse Laplace transform of the funetion H(s) found in Question 3. i.e. Find the inverse Laplace transform of the function s(8² + 2s + 2)
The charge on a capacitor g(t) in a circuit with a resistor, a capacitor and an inductor connected in series driven by a given time-dependent voltage e(t) is governed by the second-order ODE d'e) + 2d'(0) + 2q(4) = v(t). (1) A schematic of the RLC circuit can be found in the course notes. In Questions 1 - 6 you will caleulate Laplace transforms / inverse Laplace transforms of fune- tions. These calculations will be useful in determining the response of (1) for an initial value problem specified in Question 7. You MUST use the method of solution as per the instructions found in each of the below questions. 1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on MyUni to find the Laplace transform of r(t) = av (t) + Bey(), where (t)=6(t-2) and va(t) = u(t - 3). Here, a and 3 are constants, and 6(t) and u(t) are the Dirac delta function and unit step function. 2 2. Complete the square of the denominator and use the a-shifting theorem to find the inverse Laplace transform of the function F(e) = + 2s + 2 Where applicable, you may use entries (1) through to (15) and the basie general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23). 5 3. Une partial fraetion decomposition, linearity and a-ahifting theorem to find the invere Laplace transform of the function H() s(o² +2» + 2)* Where applicable, you may use entries (1) through to (15) and the basic general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23). 4 4. Use the convolution theorem and your answer to Question 2 to verify the inverse Laplace transform of the funetion H(s) found in Question 3. i.e. Find the inverse Laplace transform of the function s(8² + 2s + 2)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The charge on a capacitor g(t) in a circuit with a resistor, a capacitor and an inductor connected
in series driven by a given time-dependent voltage v(t) is governed by the second-order ODE
"(t) + 20(t) +2q(t) = v(t).
(1)
A schematic of the RLC circuit can be found in the course notes.
.
In Questions 1 - 6 you will calculate Laplace transforms / inverse Laplace transforms of fune-
tions. These calculations will be useful in determining the response of (1) for an initial value
problem specified in Question 7.
You MUST use the method of solution as per the instructions found in each of the below
questions.
1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on
MyUni to find the Laplace transform of
v(t) = av (t) + Buy (t),
where n (t) = 6(t – 2) and ty(t) = u(t – 3). Here, a and 8 are constants, and 6(t) and u(t)
are the Dirac delta function and unit step function.
2 2. Complete the square of the denominator and use the s-shifting theorem to find the inverse
Laplace transform of the function
F(e) = 7 +2 +2
Where applicable, you may use entries (1) through to (15) and the basic general formulae
from the table of Laplace transforms on MyUni. You must NOT use any entries (16)
through to (23).
5 3. Use partial fraction decomponition, linearity and s-shifting theorem to find the inverse
Laplace transform of the function
1
H(s) =
s(s + 28 + 2)
Where applicable, you may use entries (1) through to (15) and the basic general formulae
from the table of Laplace transforms on MyUni. You must NOT use any entries (16)
through to (23).
4 4. Use the convolution theorem and your answer to Question 2 to verify the inverse Laplace
transform of the function H(s) found in Question 3. i.e. Find the inverse Laplace transform
of the function
H(^) = »(² + 2» + 2) -F(0).
1
s(s² + 2s + 2)
H(s) •
You MUST evaluate the resulting integral by hand, and clearly show all your working.
1 5. Use the t-shifting theorem and your answer to Question 2 to find the inverse Laplace trans-
form of the function
e-2
K(s) =+28+2°
1 6. Use the t-shifting theorem and your answer to Question 3 to find the inverse Laplace trans-
form of the function
J(s) =
s(s +2s + 2)'
4 7. Use Laplace transforms and the results from Questions 1 - 6 to find the response of (1)
subject to the initial conditions
q(0) =0 and d'(0) = 1,
with the time-dependent voltage
v(t) = ab(t – 2) + Bu(t – 3).
3 8. Plot the response, q(t), found in Question 7, on the interval 0<t< 8, for values of a = 1
and B = 2, and attach it to your assignment. Describe the behaviour of q(t) as t + oo in
the plot. Justify your description by examining the long-time behaviour of the individual
terms in the solution for q(t).
Transcribed Image Text:Answer of Q:2
F FIS)- as2
> Flt)= ē*sintAns
Answer of ONo:3
%3D
XSint
int>
ē cost - ē's Ans
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