If we differentiate this equation with respect to time and substitute I = dq dt a second-order differential equation L d²I dt² dI 1 + R + dt 7 Problem: Now suppose an RLC circuit with a 25 100 F capacitor is driven by the voltage E(t) 27 T1, T2 = = I= dE dt resistor, a 0.09t² V. 1 100 we obtain H inductor, and a d² I dI i. Using differential operator notation, dt² dt differential equation associated with this circuit in terms of current I, differential operator D, and time t. = D²I and = DI, write the ii. Find the roots of the auxiliary polynomial of the corresponding homogeneous equation of I. Enter the roots as a list separated by commas. iii. Find the general solution of the corresponding homogeneous equation (complementary solution) for I. Use A and B for the arbitrary constants. Ih(t) iv. Find a particular solution for I. Where needed, round off all your values to at least five decimal places. Ip(t) v. Find the general solution for I in terms of t and arbitrary contastands A and B. I(t) =
If we differentiate this equation with respect to time and substitute I = dq dt a second-order differential equation L d²I dt² dI 1 + R + dt 7 Problem: Now suppose an RLC circuit with a 25 100 F capacitor is driven by the voltage E(t) 27 T1, T2 = = I= dE dt resistor, a 0.09t² V. 1 100 we obtain H inductor, and a d² I dI i. Using differential operator notation, dt² dt differential equation associated with this circuit in terms of current I, differential operator D, and time t. = D²I and = DI, write the ii. Find the roots of the auxiliary polynomial of the corresponding homogeneous equation of I. Enter the roots as a list separated by commas. iii. Find the general solution of the corresponding homogeneous equation (complementary solution) for I. Use A and B for the arbitrary constants. Ih(t) iv. Find a particular solution for I. Where needed, round off all your values to at least five decimal places. Ip(t) v. Find the general solution for I in terms of t and arbitrary contastands A and B. I(t) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Theory: Consider an RLC circuit shown below consisting of an inductor with an
inductance of I henry (H), a resistor with a resistance of R ohms (2), and a capacitor
with a capacitance of C farads (F) driven by a voltage of E(t) volts (V).
=
Given the voltage drop across the resistor is ER = RI, across the inductor is
L(dI / dt), and across the capacitor is Ec
9
EL
Kirchhoff's Law gives
C
E|
L
T1, T2 =
dI
dt
L
In(t)
R
If we differentiate this equation with respect to time and substitute I =
dq
dt
a second-order differential equation
=
с
d²I
dt²
=
1
+ RI + 69
7
Problem: Now suppose an RLC circuit with a
25
dI
+ R +
dt
=
1
E(t)
с
-I =
100
-F capacitor is driven by the voltage E(t) = 0.09t² V.
27
dE
dt
resistor, a
d²I
dI
i. Using differential operator notation,
dt²
dt
differential equation associated with this circuit in terms of current I, differential
operator D, and time t.
D²I and
1
100
ii. Find the roots of the auxiliary polynomial of the corresponding homogeneous
equation of I.
Enter the roots as a list separated by commas.
we obtain
H inductor, and a
iii. Find the general solution of the corresponding homogeneous equation
(complementary solution) for I.
Use A and B for the arbitrary constants.
= DI, write the
iv. Find a particular solution for I.
Where needed, round off all your values to at least five decimal places.
Ip(t)
v. Find the general solution for I in terms of t and arbitrary contastands A and B.
I(t)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Write the given second order ODE
VIEWStep 2: Write the corresponding second order ODE in terms of D,I and t
VIEWStep 3: Determine the roots of the auxiliary polynomial equation of the corresponding homogeneous equation
VIEWStep 4: Write the complementary function of the corresponding homogeneous equation
VIEWStep 5: Determine a particular solution
VIEWStep 6: Write the required general solution for I(t)
VIEWSolution
VIEW
Step by step
Solved in 7 steps with 7 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
