RESISTANCE AND CAPACITANCE IN SERIES 



whole of the e.m.f. appears across the resistance, causing a current EfR to 

 flow through it to charge the capacitance, and the capacitance voltage begins 

 to rise at a rate dv/dt = ijC = EjCR. However, as soon as the voltage 

 across the capacitance rises, that across the resistance must fall, since their 

 sum must equal E: therefore the current falls, so that the capacitance voltage 

 may continue to rise, but more and more slowly. The final state of affairs, 

 which theoretically takes an infinite time to reach, is that the capacitance 

 voltage just reaches the generator voltage as the current falls to zero, when 

 circuit action ceases. 



This circuit is one of an extremely important class possessing an 'exponen- 

 tial response' which occurs frequently in electronics, and it is important to 

 be thoroughly at home with it. 



An important notion associated with the exponential response is that of 

 'time constant', which is measured in seconds and is a measure of the rapidity 

 with which the circuit responds to the application of E. Clearly we cannot 

 use the time taken for circuit action to cease, since this is theoretically infinite. 

 Time constant is defined simply as 



T = RC (seconds, ohms, farads, or more usefully seconds, megohms 



microfarads) 



Its meaning may be seen in two ways : 



(1) From the verbal description. The capacitance charges at an initial 

 rate EfCR. If this rate could be maintained, the time taken to reach the 

 charging voltage E would be EKEjCR) = CR. Thus the time constant is the 

 time it would take the circuit to reach equilibrium if the initial capacitance 

 charging rate could be maintained. 



(2) From the mathematical description. The capacitance voltage is 

 £(1 — e-^'^^). When t reaches T = CR, the time constant, then Vg = 

 £■(1 — e-^) — 0-632 E, that is, the time constant is the time taken by the 

 capacitance voltage to reach 63-2 per cent of its final value. 



The two pieces of analysis which follow are important in connection with 

 multi-stage capacitance coupled amplifiers, such as are often employed in 

 electrophysiology ; their importance will become evident later. 



Resistance and capacitance in series with a constant-voltage generator whose 

 output is of the form e — Ee~^'^^. 



Here we are feeding our series circuit with a voltage waveform similar to 

 Vji in Figure 3.9 (Figure 3.10). This time we have 



Ee-'l^^ = Vr-\-Vc 



= iR+ dv 



Jo 



29 



