Model 2: A Simple Electrical Circuit (25 Marks) The complex impedance of the linear LCR circuit shown in Figure 1 is given by Z(w) = R + wLj - j/(wC) where L, C, and R are physical constants for the resistance, inductance, and capacitance of the system, j = ∑(-1) is the imaginary unit, and w is the angular frequency at which the circuit is being excited. Figure 1: Simple LCR circuit a) Obtain an expression for the impedance of the circuit, i.e., |Z(w)|, the magnitude of Z(w). [6] b) Express Z(w) in polar form and represent it on the Gauss plane. In order to do this, you will need to identify the real and imaginary components of Z(w) and calculate its magnitude and argument. [6] c) Use differential calculus to find and characterise (maximum, minimum, or inflection) any real-valued stationary points of |Z(w)| as a function of w. Assume that L, C, and R are strictly positive real numbers. [10] Hint: Look out for terms that will vanish at the critical point when considering the second derivative. For ease of notation, represent |Z(w)| as a function z(w).