INDUCTANCES, CAPACITANCES 

 AND RESISTANCES 



Series L, C and R connected to a constant direct voltage generator 



The behaviour of this relatively simple-looking circuit (Figure 5.1) is quite 

 complicated, but well repays study; it is the electrical analogue of mechanical 

 devices such as the electromagnetic penwriter, and an understanding of it is 

 of great help in grasping how such devices perform. 



Close at 



t=0 



R 

 -VvW- 



6 



\ 



/ 



Figure 5.1 



On closing the switch we have 



E^v^ + vc + Vl 



= iR + 



h\' 



dt-{-L 



d/ 

 dt 



but / = d^/d/, therefore 



^-d7^ + c + ^d/2 



Let us consider the voltage across the capacitance, Vq. This is qfC, so 



RC 



dvc 

 dt 



^Vc + LC 



d/2 



There are three types of solution to this equation, depending on the amount 

 of resistance in the circuit. When the resistance is small, the circuit is 'lightly 

 damped' and the response is oscillatory, the oscillations dying away slowly 

 when R is very small, and more quickly as R is raised. This oscillatory res- 

 ponse is called 'ringing', by analogy with the striking of a bell. On further 

 increasing R we reach at length a point at which the oscillations die away, 

 as it were, before they have started, and the response of the circuit is said to 

 be 'critically damped'. If R be made greater still, the response is of the 

 'heavily damped' variety, extremely sluggish. 



72 



