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Circuit With Resistance and Capacitance OnlyAuthor: E.E. Kimberly In Fig. 87 (a) is shown a circuit containing only resistance and an uncharged condenser. If this circuit is connected to a source of alternating voltage instead of direct voltage, the current that will flow to establish dynamic equilibrium must satisfy the condition that
The angle ϕ is the angle at which the circuit is closed after v = 0 when is positive.If the condenser is initially uncharged, there is no voltage across it and it acts with no impedance. The initial current then is . As a< result of initial current flow, the condenser acquires a charge which produces a counter voltage. That voltage is
The steadystate current is
where The transient current is
Therefore, When and
By substitution of this quantity for K, we obtain:
When , etc., there will be no transient component of current. Therefore, when the circuit is closed at the instant at which i89 would be a maximum, there is no transient component.When the condenser has an initial charge, the residual condenser voltage is This voltage may be either positive or negative in the circuit loop and is, therefore, added algebraically to the transient voltage. From equation (86),
The constant of integration K for this equation is
and (819) The current is that caused by the original charge Q and is positive if the flow is around the series loop in the positive direction. It is apparent that there will be no transient current when the condenser initially is negatively charged. Hence.When the condenser is initially charged, the general expression for the current is
Example 85.  A circuit containing a resistance R = 5 ohms and uncharged capacitance C = 200 microfarad is connected to a 60cycle voltage source of 70.0 volts at the instant when the voltage has reached 86.6 per cent of its maximum Value in a positive sense. Write the equation for the current.
Solution.  From equation (820),


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