INDUCTANCES, CAPACITANCES AND RESISTANCES 



On solving the equation it turns out that everything depends on the 

 quantities 1/LC and R^I4L^. When R^I4L-<^llLC, the solution is approxi- 

 mately 



To see how this looks, let R^jAL^ = 0-\jLC. Then the equation becomes 



t 

 (LCy 



When R^I4L^ =-- IjLC the circuit is critically damped and the solution is 



Vc=E{\-e-^l^^'^^ncos,,^^„^ 



Vn = E 



-tKLO^'^l _j_ 



(LC) 



1/2 



When R^I4L^ ^ 1 /LC, the circuit is heavily damped and the general solution 

 is a somewhat cumbersome expression. However, in the case of R^I4L^ = 

 10/LC the solution is 



Vc = E{\ - 1-025 e-0162«LC)i/^ ^ Q.Q25 e-6162WiC)i'*) 



These three responses are plotted in Graph 22, which shows the three routes 

 by which the capacitance voltage eventually becomes equal to E. Graph 22 

 also shows, though not rigorously, that the critically damped circuit settles 

 into the final state soonest: this is a point of great importance. 



A physical description of what is happening may not be out of place. In 

 the lightly damped case, upon closing the switch, current tends to flow to 

 charge the capacitance, but its growth is retarded by the back e.m.f. across 

 the inductance, which opposes E. The initial slope of the current curve is 

 zero. As charging proceeds a magnetic field is built up around the induc- 

 tances. As V(j approaches E, the current would fall off were it not for the 

 e.m.f. across the inductance, which now aids E, charging the capacitance to a 

 voltage greater than E. Energy is being transferred from the magnetic field, 

 which is collapsing, to the capacitance in the form of additional stored charge. 



When the energy of the magnetic field is exhausted the capacitance voltage 

 begins to push current round the circuit the other way, against the action of 

 the generator, building up a magnetic field round the inductance once more 

 and giving up its own energy to do so. 



Thus, the oscillations represent the transfer of energy back and forth be- 

 tween the inductance and the capacitance. If R were not present this process 

 could continue indefinitely. The effect of R is to absorb a fraction of the 

 energy at each cycle, converting it irreversibly into heat, so that the oscilla- 

 tions die away. As R is increased they die away more and more quickly, 

 until in the critically damped case they have been suppressed altogether. If 

 R be made larger still, the circuit begins to turn into a simple R-C low-pass 

 filter, because the effect of R swamps that of L. Hence, increasing R is 

 increasing the time constant, and the response becomes increasingly sluggish. 



Capacitive and inductive reactance 



If series R, L, and C are connected to a constant alternating current 

 generator of output / = /sin cut (Figure 5.2), then y^ is in phase with /, v^ leads 



73 



