Question 4. a) State what are the through- and across-variables of an electrical system? [4 marks] b) Show through equations and sketches how the capacitor element of an electrical system can be modelled mathematically to derive its transfer function. [5 marks] c) Show how the series resistor-capacitor (RC) lumped element electrical system is modelled, by sketching the circuit and deriving the constitutive equation and transfer function relating output voltage (across the capacitor) to input voltage. [10 marks] Question 4 a) For an electrical system, the through-variable is the current i, while the across-variable is the potential or voltage difference VAB across an element/system. b) Sketch capacitor symbol; capacitance C Potential difference v related to charge Q and C by v=Q/C Since current i=dQ/dt, Q = integral of i.dt, so v=1/C(integral of i.dt) or i = Cdv/dt Show Laplace transform (this is analogous to the mechanical model for a spring element) c) Sketch of resistor and capacitor in series with a voltage source. The directions of v_r and v_c polarity should be indicated. Applying Kirchoff's Voltage Law (KVL), v_i=v_r+v_c (where v_c is the output voltage in this case) Since v=iR and i=C(dv_c/dt) we get v_i = RC(dv_c/dt) + v_c Develop Laplace form of this and show time constant T=RC