Part 1: Circuit's damping coefficient and damping type 1a) For R = 19, L = 1H,C = 10F find the circuit's damping coefficient from the characteristic equation and assign to variable Ze1. 1b) What is the damping type? Define variable Damping1 and store the string "over", "under", or "critical" based on an if...elseif statement for the response type. All the response string characters should be lower case. Ex: if Ze1 < 1 Damping1 "under" % Use elseif for the second condition and else for the last condition for the other two possible damping types. end Part 2: Circuit's impulse response 2a) Use residue function to find the impulse response coefficients. Store the result elements in variables A, B, C, and D as defined above. A and C are the coefficients of the first and second exponentials respectively. B and D are the coefficients of the first and second exponentials multiplying by t, respectively. 2b) Define a time interval starting at t0 = 0 to tend = 60 seconds with a time step of ts = 1 s. Find the impulse response for this time interval and store the result in variable h1. 2c) Plot the impulse response for the given time interval. 2d) Find the impulse response for t = 10 s and t = 30 s and assign to variables h1_10sec and h1_30sec respectively. Find the values either from the plot or the h1 array. Part 3: Circuit's step response 3a) Determine the circuit's step response using conv function and the circuit's impulse response h1. Find the step response for the above time interval and store the result in variable ystep1. 3b) Plot the step response for the given time interval. 3c) Find the step response for t = 10 s and t = 30 s and assign to variables ystep1_10sec and ystep1_30sec respectively. Find the values either from the plot or the ystep1 array. A second order circuit's impulse and step response The figure below shows a series second order RLC circuit. The system input is a step voltage source vin(t) = u(t) and the output is the voltage across resistor R. + Vin(t) C m L i(t). RVout (t) Series RLC As shown in the section regarding second order LCCDES impulse response, the Kirchhoff's voltage law equation can be written as di ½ i(t)dt + L + Ri = vix(t) = u(t). di Differentiating results: + di² Ldt LC d²i Rdi 1 + i= =(t). The circuit's characteristic equation can be written as: = 0. R + = + s+ L LC A system's impulse response can be found using the residue function. Ex: for the LCCDE b₂ d²v + b₁ dy b₁1/4 + bo= αι +ao, dx di [R, P] = residue ([a1 a0], [b2 b1 b0]) Vectors [a1 a0] and [b2 b1 b0] are the input (right hand side) and output (left hand side) coefficients respectively. Vector R contains the residues and vector P contains the roots (poles) of the characteristic equation. If the impulse response has the form h(t) = Aet + Cet then A = R(1), B = P(1), C = R(2), and D = P(2). Convolution of the impulse response with any given input including step function can provide the system's output. y(t) = x(t) * h(t). The unit step function in MATLAB can be defined as heaviside (1: length).