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Delmar's Standard Textbook Of Electricity
7th Edition
ISBN: 9781337900348
Author: Stephen L. Herman
Publisher: Cengage Learning
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
Transcribed Image Text:Consider the homogeneous RLC circuit (no voltage source) shown in the diagram below. Before
the switch is closed, the capacitor has an initial charge go and the circuit has an initial current go.
R
w
i(t)
q(t)
C
н
After the switches closes, current flows through the circuit and the capacitor begins to discharge.
The equation that describes the total voltage in the loop comes from Kirchoff's voltage law:
di(t)
L
+ Ri(t) + (t) = 0,
dt
(1)
where i(t) and q(t) are the current and capacitor charge as a function of time, L is the inductance,
R is the resistance, and C is the capacitance. Using the fact that the current equals the rate of
change of the capacitor charge, and dividing by L, we can write the following homogeneous (no
input source) differential equation for the charge on the capacitor:
ä(t)+2ag(t)+wg(t) = 0,
(2)
where
R
a
2L
and w₁ = C
LC
The solution to this second order linear differential equation can be written as:
where
81=
q(t) = Ae³¹- Bel
82 =
(3)
(4)
(5)

Transcribed Image Text:($1+20)90 +90
(82 +20)90 +90
and B
81-82
81-82
(6)
1. (6 marks) Show that the solution for q(t) from equation (4) satisfies the homogenous differ-
ential equation for the RLC circuit with no voltage source from equation (2).
2. (5 marks) Write a MATLAB function to evaluate q(t), ġ(t), and ä(t) from time t = 0 tot - T
in steps of AT. The equation for q(t) describes the charge on the capacitor as a function of
time, q(t) describes the current in the circuit, and ä(t) describes the rate of change of the
current.
Note that s₁ and s₂ can be complex numbers, depending on the values of a and up. MATLAB
automatically handles the computation of complex numbers.
A function template can be found in the file HomogeneousRLC.m, which takes as input the
resistance, inductance, and capacitance of the circuit, the initial charge on the capacitor, the
initial current in the circuit, the time step, and the final time.
3. Using the MATLAB function that you wrote in the previous question, calculate the voltage
drop across each component in the RLC circuit, and the energy stored in each of the inductor
and capacitor. Use a value of zero for the initial current go, and chose the initial charge on
the capacitor go so that the initial voltage on the capacitor is 1V. The time scale will need
to be chosen carefully so that the entire output is clearly visible.
You may need to look up some simple formulas to calculate these voltages and energies, but
the important pieces of information - the charge on the capacitor, the current, and the rate
of change of the current - are all provided by the function from the previous question.
(a) (10 marks) Choose realistic values of R, L, and C such that a> wp and make a graph
that shows the voltage drop across the resistor, inductor, and capacitor, and the sum of
all three, as a function time. Make another graph that shows the energy stored in each
of the inductor and capacitor, and the total energy stored, as a function of time.
(b) (10 marks) Choose values of R, L, and C such that a <wo and make another pair of
graphs as before that show the voltages and energies in the RLC circuit as a function of
time. Label the R, L, C, and go values on each graph.
4. (4 marks) A special case occurs when a wo, which results in 81-82. Equation (4) cannot
be evaluated directly if s₁ = 82 since q(t). If L'Hospital's rule is applied as s₁→ 82 then
where
q(t) que+(ago + ġo) teat
R
81-82-0 and at
2L
(7)
(8)
Modify the MATLAB function that you wrote to check for the condition a = wo (or equiva-
lently, $1 =82) and calculate q(t), q(t), and ä(t) accordingly.
Submit the MATLAB code for your function as a .m file.
Expert Solution
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Delmar's Standard Textbook Of Electricity
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ISBN:9781337900348
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