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) ($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

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) ($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

Delmar's Standard Textbook Of Electricity

7th Edition

ISBN:9781337900348

Author:Stephen L. Herman

Publisher:Stephen L. Herman

Chapter21: Resistive-capacitive Series Circuits

Section: Chapter Questions

Problem 6RQ: A 15-F AC capacitor is connected in series with a 50 resistor. The capacitor has a voltage rating...

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