الرحمن الله بسمالرحيم

RL + RC CIRCUIT

A circuit containing a series combination of a resistor and a capacitor is called an RC circuit.

Maximum current of the circuit, I0 = [When, t = 0, Maximum current flows]Maximum Charge on Capacitor, Q = CƐ

RC circuit. Charging case

Expression of Charge q(t), voltage VC and current I during charging phase of an RC circuit: The voltage across a capacitor cannot

change instantaneously.By applying KVL, We get,

I = Putting this value of I and after rearranging, we get,

This is the equation of charge stored in a capacitor.

Equation of instantaneous current can be obtained by differentiating the equation of charge,

Voltage across the capacitor is, VC = q(t) / C

Graph: Time vs Charge (or voltage) Graph: Current vs Time

time constant =RC represents the time interval during which the current decreases to 1/e of its initial value; that is, after a time interval t, the current decreases

i = 0.368 Ii

By applying KVL in opposite direction, we get,

Now, I = . Again, when t=0 then q = Q

RC circuit Discharging case

This is the equation of charge remaining in the capacitor. The equation of current can be obtained by differentiating this equation.

VL = – L and VR = iL RBy applying KVL we get .. E – VR – VL = 0 or E – iL R – L = 0

Let, – iL = x then, = –

x + = 0 ,, = – dt

RL circuit

By integrating within the limit (x0 to x) and (0 to t),

ln = – t , x = x0 e –Rt/L

When t = 0, current iL = 0 thus, x = x0 = When t = t, current = iL thus, x = – iL

Now, – iL = e –Rt/L

, iL = ( 1 – e –Rt/L )

VL = E e –t/τ

VR = E (1 – e –t/τ )

The current in the circuit , iL = ( 1 – e –Rt/L )

time constant = τ =

Physically, τ is the time it takes the current in the circuit to reach ( 1 – e –1 ) = 0.637 or 63.7% of its final value .

By applying KVL we get .. VR – VL = 0 or iL R – L = 0

At no E

= – dt

Ln i= – dt + const At , t=0, i= const=ln Ln

VL = -E e –t/τ

VR = E e –t/τ

The current in the circuit , iL = e –Rt/L

Volt across L

Volt across R

E

E=0