Transfer (Gain) Function for AC Circuit

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Transfer (Gain) Function for AC Circuit John Matz 12/6/2020 George Mason University Math 401: Mathematics Through 3D Printing These two objects are graphs of the real an imaginary parts of the transfer function for a second order AC circuit. The transfer function (Vo/Vi) is 5s/(s^2+8s+5) where s = sqrt(-1)*(frequency), and in these examples the frequency is Pi rad/s. This is the answer to problem 14.3 from the fifth edition of Fundamentals of Electric Circuits by Charles K. Alexander and Matthew N. O. Sadiku, a picture of which can be seen in image #4, above. While most circuits are modeled by differential equations, it is possible to convert these models into the phasor domain, which is easier to work with. By restricting inputs and outputs to sinusoids the differential equation becomes a polynomial. I believe this is similar to a Laplace Transform, but my experience is more in applications than the theory. This part of the aforementioned textbook and the accompanying problems were very difficult for me to understand and solve, so I was hoping by modeling one of the solutions in Mathematica I would gain a better intuitive understanding for these very unintuitive concepts. I think I have accomplished that here. The large spikes and holes where the surface blows up correspond to the poles and zeros of the transfer function. These are defined in Fundamentals of Electric Circuits as, "The transfer function H(omega) of a circuit is the frequency-dependent ratio of a phasor output Y(omega) (an element of voltage or current) to a phasor input X(omega) (source voltage or current)", and, "A zero, as a root of the numerator polynomial, is a value that results in a zero value of the function. A pole, as a root of the denominator polynomial, is a value for which the function is infinite." These are critical for understanding how a given circuit responds to a given frequency. Attached are two STL files, one for the real part of the surface and one for the imaginary part. Also included is my Mathematica code, which was based on code given to me by my instructor in Mathematics Through 3D printing.