rl + rc circuit ( بسم الله الرحمن الرحيم )
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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
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