a) Produce a differential equation for the behaviour of the circuit linking the charge Q in C to some function F(t). Hence produce a Laplace transfer function linking the charge in C to the injected charge.

a) Produce a differential equation for the behaviour of the circuit linking the charge Q in C to some function F(t). Hence produce a Laplace transfer function linking the charge in C to the injected charge.
b) The inductance L was placed into this circuit to reduce the pick-up of ‘noise’ from nearly circuits. Noise has not been a problem in practice so, in an attempt to reduce the incidence of failure, the manufacturer is considering replacing the inductor with a wire link. What would the differential equation be in this case?
2. Comment on the similarities and differences between the equations for Scenario A part b) and Scenario B part a) above. To what extent are these scenarios analogous?
Scenario B
An electronics company is suffering a large number of circuit board failures in the field. All boards are tested before leaving the factory so the company knows that all boards shipped to site were sound. All failures happened at, or within seconds of, the first power up after installation. On inspection, all failures were found to be of the same microprocessor chip. The company suspects that the failures are being caused by static electric discharges to the microprocessor’s supply and drain (+ve and –ve power) terminals during installation. In its powered-off state these pins of the microprocessor behave as a fixed resistor R. Most of the circuitry around these pins is inactive until the system is powered up but there is one loop consisting of an inductor L and capacitor C linking the supply and drain that the company thinks may be vulnerable to static discharge. This loop can be considered to be simply R, L and C connected in series as shown below. Note that in this situation there is no supply voltage.
Static discharge events involve the movement of small quantities of charge from very high voltage sources and can be modelled to sufficient accuracy by assuming that a time-varying additional charge Q [equal to some function F(t)] is directly injected into C.