Active Learning Assignment

Sub: Circuits and Networks (2130901 )Topic: Time Domain response of second order linear circuitGuided By: Prof. Sweta Khakhakhar Branch: Electrical Engineering Batch : B1Prepared By: (1) Abhishek Choksi 140120109005

(2) Himal Desai 140120109008 (3) Harsh Dedakiya 140120109012

Contents:

• Discharging of a capacitor through an inductor• Source free series RLC circuit• Source free parallel RLC circuit• Types of Response• Second order linear network with constant input

Discharging of a capacitor through an inductor• Consider LC network as shown

in fig.• Initially the switch is at point A.

So the voltage source Vs is connected across capacitor. Due to this a capacitor is fully charged to Vs volts.

• At time t = 0 the switch is moved to position ‘B’. An equivalent network is shown in fig.

Equation in Term of Capacitor Voltage:• The capacitor current can be expressed in term of voltage

across capacitor as, So, ---------- (1)• Now as shown in fig. = ---------- (2)• Differentiating both side with respect to ‘t’ we get, ---------- (3)

• The voltage across inductor is given by, So, ---------- (4)• But from fig = ---------- (5)• Putting Equation (5) in equation (3) we get, ---------- (6)• This is second order differential equation in term of

capacitor voltage

Equation in Term of Inductor Current

• Consider Equation (5) • Differentiating both the side with respect to time we get, ---------- (7)• Putting Equation (2) in Equation (7) we get, ---------- (8)• This is the Second order differential equation in term of

inductor current.

Source Free Series RLC Circuit• Consider a source free series RLC circuit as shown in

fig.

Equation in Term of • Applying KVL to the given network,

VR + VL + VC = 0 ---------- (1)Now,VR = Voltage across resistor = IR R ---------- (2)VL = Voltage across inductor = ---------- (3)VC = Voltage across capacitor = ---------- (4)• Putting these value in Equation (1) we get, IR R + + = 0 ---------- (5)• Now referring fig (1) IR = IL = IC . We want equation in term

of IL . So replacing IR and IC by IL in Equation (5) we get,

IL R + + = 0 ---------- (6)• To eliminate integration sign, differentiate with respect to

time ’t’ R• Rearranging the equation,

• To make coefficient of equal to 1, divide both the side by 1, ---------- (7)• This is a second order differential equation of source free

RLC circuit in term of .

Equation in Term of • Consider Equation 3, we have to express every current in

term of . So put IR = IC , IL = IC . The last term in the equation 5 indicates, voltage across capacitor, .

---------- (8)• Now we want equation in term of . The capacitor current IC

can be expressed in term of capacitor voltage as, • Putting this value in the equation 8 we get, RC

RC • Rearranging the equation, • To obtain the coefficient of , equal to 1, divide both the side

by LC, ----------(9)• This is the second order differential equation, for source

free RLC circuit in term of capacitor voltage .

Source free Parallel RLC Circuit

• Consider source free parallel RLC circuit as shown in fig

Equation in Term of

• As shown in fig. R, L and C are connected in parallel, so voltage across each element is same.

VR = VL = VC ---------- (1)• Now applying KCL we get,

iR + iL + iC = 0• Expressing every current in term of corresponding voltage

we get, ---------- (2)• But we want every voltage in term of . Now from the

equation (1) VR = VC and VL = VC • Putting these values in equation (2) we get,

• To eliminate integration sign, take derivatives with respect

to time, • Rearranging the terms, +• To make coefficient of equal to 1, divide both side by C + ---------- (3)• This is the second order differential equation, for source

free parallel RLC circuit in term of capacitor voltage .

Equation in Term of iL • Consider Equation (2) it is, • Now the second term indicate the equation of that means, iL =

• Thus we can write, + iL + = 0• From the equation (1), ) VR = VL and VL = VC + iL + = 0 ---------- (4)

• But the voltage across inductor is, VL = L ---------- (5)• Putting Equation (5) in Equation (4) we get, • Rearranging the terms, • To make the coefficient of first term equal to 1, divide both

side by CL, ---------- (6)• This is the second order differential equation, for parallel

RLC circuit in term of inductance current

Type of Response• The output of source free second order linear system is an

oscillatory waveform.• The work ‘damping’ indicates, decrease in the peak

amplitude of oscillation. It is due to the effect of energy that is absorbed by elements of system.

• Depending on the damping, the response of the system is classified as follows:

1. Critical damping response2. Under damping response 3. Over damping response 4. Undamped response

Critical Damping Response • If the damping factor is

sufficient to prevent oscillations then a system response is called as critical response.

• Consider that, the current is exponentially increasing as shown in the fig. Here the response is not oscillating. So it is critically damped response.

Under Damping Response

• If the amount of damping is less than the response is called as under damped response.

• The oscillation are present but eventually decays out as shown in fig.

Over Damped Response

• In this case, amount of damping is large. Any small change in the circuit parameter, will prevent the damping.

Undamped Response

• If there is no damping then it is called as undamped response. Here the oscillation are called as sustained oscillations. It is shown in fig.

Second Order Linear Circuit with Constant Input• When some independent sources are present in RLC circuit

then its differential equation is,

• Compared to the equation of the source free circuits, this equation are extra term f(t).

• Here the term f(t) depend on the source input and the derivative of input.

• If f(t) is not constant then the solution becomes complicated, so such differential equations are solved using laplace transform.

.

• If f(t) is having some constant value that means, f(t) = F and the solution can be easily obtained

• Consider the general solution of differential equation,

• Here the roots are,

• And the general solution is given by,• In equation 3 is the general solution when f(t) = 0.• The value of X(F) is independent of the root and it is

given by,

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