CHAPTER 5

Transient and Steady State

Response

(Second-Order Circuits)

Contents

Natural response of series RLC circuit

Natural response of parallel RLC circuit

Step response of series RLC circuit

Step response of parallel RLC circuit

What is second order?

• Circuits containing

two storage

elements.

• Second-order

circuit may have

two storage

elements of

different type or

the same type

Initial and final values

• Combination of R, L and C

• Find v(0), i(0), dv(0)/dt, di(0)/dt, i(∞) & v(∞)

• t(0-) the time just before switching event

• t(0+) the time just after switching event

• Assume the switching event take place at t=0

• Voltage polarity across capacitor

• Current direction across inductor

• Capacitor voltage always continuous v(0+) = v(0-)

• Inductor current always continuous i(0+)=i(0-)

Example

The switch in the figure shown has been closed for a long

time. It is open at t=0, Find:

(a) i(0+), v(0+)

(b) di(0+)/dt, dv(0+)/dt

(c) i(∞) , v(∞)

12 V

0.25 H4 Ω

0.1 F2 Ω

i

+V-

t=0

Exercise

The switch in the figure shown was open for a long time but

closed at t=0. Determine

(a) i(0+), v(0+)

(b) di(0+)/dt, dv(0+)/dt

(c) i(∞) , v(∞)

24 V

0.4 H

1/20 F2 Ω

i

+V-

t=0

10 Ω

The Source-Free Series RLC

• Applying KVL around the loop

𝑅𝑖 + 𝐿𝑑𝑖

𝑑𝑡+1

𝑐 −∞

𝑡

𝑖 𝑑𝑡 = 0

• Differentiate with respect to t

𝑑2𝑖

𝑑2+𝑅

𝐿

𝑑𝑖

𝑑𝑡+𝑖

𝐿𝐶= 0

• Finally,

𝑠2 +𝑅

𝐿𝑠 +

1

𝐿𝐶= 0

The Source-Free Series RLC

• Roots equation

𝑠1 = −𝑅

2𝐿+

𝑅

2𝐿

2

−1

𝐿𝐶

𝑠2 = −𝑅

2𝐿−

𝑅

2𝐿

2

−1

𝐿𝐶

or

𝑠1 = −𝛼 + 𝛼2 − 𝜔02

𝑠2 = −𝛼 − 𝛼2 − 𝜔02

where

𝛼 =𝑅

2𝐿, 𝜔0 =

1

𝐿𝐶

• 𝛼 (Np/s)

• 𝜔0 (rad/s)

The Source-Free Series RLC

Three type of solution

• If α > ω0 overdamped case

• If α = ω0 critically damped case

• If α < ω0 underdamped case

The Source-Free Series RLC

Overdamped case (α>ω0)

• Both roots S1 and S2 are negative and real

• The response is 𝑖 𝑡 = 𝐴1𝑒

𝑠1𝑡 + 𝐴2𝑒𝑠2𝑡

The Source-Free Series RLC

Critically damped case (α= ω0)

• Roots

𝑠1 = 𝑠2 = −𝛼 = −𝑅

2𝐿• The response is

𝑖 𝑡 = (𝐴2+𝐴1𝑡)𝑒−𝛼𝑡

The Source-Free Series RLC

Underdamped case(α<ω0)

• Roots

𝑠1 = −𝛼 + − 𝜔02 − 𝛼2 = −𝛼 +j𝜔𝑑

𝑠2 = −𝛼 − − 𝜔02 − 𝛼2 = −𝛼-j𝜔𝑑

where 𝜔𝑑 = 𝜔02 − 𝛼2

• The response is 𝑖 𝑡 = 𝑒−𝛼𝑡(𝐵1 cos𝜔𝑑𝑡 + 𝐵2 sin𝜔𝑑𝑡)

Example

Find i(t) for t > 0

+

v(t)

-

Exercise

Find i(t) in the circuit below. Assume that the

circuit has reached steady state at t=0-

Source Free Parallel RLC Circuits

• Initial inductor current and

initial voltage capacitor

𝑖 0 = 𝐼0 =1

𝐿 ∞

0

𝑣 𝑡 𝑑𝑡

𝑣 0 = 𝑉0• Applying KCL

𝑣

𝑅+1

𝐿 −∞

𝑡

𝑣𝑑𝑡 + 𝐶𝑑𝑣

𝑑𝑡= 0

Source Free Parallel RLC Circuits

• Derivatives with respect t and diving by C

𝑑2𝑣

𝑑𝑡2+1

𝑅𝐶

𝑑𝑣

𝑑𝑡+1

𝐿𝐶𝑣 = 0

or 𝑠2 +1

𝑅𝐶𝑠 +

1

𝐿𝐶

• Roots of the characteristics equation are

𝑠1,2 = −1

2𝑅𝐶±

1

2𝑅𝐶

2

−1

𝐿𝐶

Source Free Parallel RLC Circuits

or 𝑠1,2 = −𝛼 ± 𝛼2 −𝜔02

where 𝛼 =1

2𝑅𝐶, 𝜔0 =

1

𝐿𝐶

• 𝛼 (Np/s)

• 𝜔0 (rad/s)

The Source-Free Parallel RLC

Three type of solution

• If α > ω0 overdamped case

• If α = ω0 critically damped case

• If α < ω0 underdamped case

The Source-Free Parallel RLC

Overdamped case (α>ω0)

• Both roots S1 and S2 are negative and real

• The response is

𝑣 𝑡 = 𝐴1𝑒𝑠1𝑡 + 𝐴2𝑒

𝑠2𝑡

The Source-Free Parallel RLC

Critically damped case (α= ω0)

• The roots are real and equal so the response is

𝑣 𝑡 = (𝐴1+𝐴2𝑡)𝑒−𝛼𝑡

The Source-Free Parallel RLC

Underdamped case(α<ω0)

• Roots

𝑠1,2 = −𝛼 ± j𝜔𝑑

where 𝜔𝑑 = 𝜔02 − 𝛼2

• The response is

𝑣 𝑡 = 𝑒−𝛼𝑡(𝐴1 cos𝜔𝑑𝑡 + 𝐴2 sin𝜔𝑑𝑡)

Example

Find v(t) for t>0 in the RLC circuit shown

below

Step Response of a Series RLC Circuit

• Applying KVL around the

loop for t>0

𝐿𝑑𝑖

𝑑𝑡+ 𝑅𝑖 + 𝑣 = 𝑉𝑠

but 𝑖 = 𝐶𝑑𝑣

𝑑𝑡

substitute i in equation above

𝑑2𝑣

𝑑𝑡2+𝑅

𝐿

𝑑𝑣

𝑑𝑡+𝑣

𝐿𝐶=𝑉𝑠𝐿𝐶

Step Response of a Series RLC Circuit

• There is two components in the equation (i) transient

response 𝑣𝑡 𝑡 (ii) steady-state response 𝑣𝑠𝑠 𝑡

𝑣 𝑡 = 𝑣𝑡 𝑡 + 𝑣𝑠𝑠 𝑡

• The transient response 𝑣𝑡 𝑡 is similar as discussed in

source-free circuit.

• The final value of the capacitor voltage is the same as

the source voltage Vs

𝑣𝑠𝑠 𝑡 = 𝑣 ∞ = 𝑉𝑠

Step Response of a Series RLC Circuit

• The complete response solution are:-

𝑣 𝑡 = 𝑉𝑠 + 𝐴1𝑒𝑠1𝑡 + 𝐴2𝑒

𝑠2𝑡 (Overdamped)

𝑣 𝑡 = 𝑉𝑠 + (𝐴1+𝐴2𝑡)𝑒−𝛼𝑡 (Critically damped)

𝑣 𝑡 = 𝑉𝑠 + (𝐴1𝑐𝑜𝑠𝜔𝑑𝑡 + 𝐴2𝑠𝑖𝑛𝜔𝑑𝑡)𝑒−𝛼𝑡 (Underdamped)

Example

For the circuit shown in figure below, find

v(t) and i(t) for t>0.

Given R = 5 Ω, C = 0.25 F

Step Response of a Parallel RLC Circuit

• Applying KCL at the top

node for t > 0,𝑣

𝑅+ 𝑖 + 𝐶

𝑑𝑣

𝑑𝑡= 𝐼𝑠

but 𝑣 = 𝐿𝑑𝑖

𝑑𝑡

substitute vin equation above

and dividing by LC:

𝑑2𝑖

𝑑𝑡2+1

𝑅𝐶

𝑑𝑖

𝑑𝑡+𝑖

𝐿𝐶=𝐼𝑠𝐿𝐶

Step Response of a Parallel RLC Circuit

• There is two components in the equation (i) transient

response 𝑖𝑡 𝑡 (ii) steady-state response 𝑖𝑠𝑠 𝑡

𝑖 𝑡 = 𝑖𝑡 𝑡 + 𝑖𝑠𝑠 𝑡

• The transient response 𝑖𝑡 𝑡 is similar as discussed in

source-free circuit.

• The final value of the current through the inductor is the

same as the source current Is

Step Response of a Parallel RLC Circuit

• The complete response solution are:-

𝑖 𝑡 = 𝐼𝑠 + 𝐴1𝑒𝑠1𝑡 + 𝐴2𝑒

𝑠2𝑡 (Overdamped)

𝑖 𝑡 = 𝐼𝑠 + (𝐴1+𝐴2𝑡)𝑒−𝛼𝑡 (Critically damped)

𝑖 𝑡 = 𝐼𝑠 + (𝐴1𝑐𝑜𝑠𝜔𝑑𝑡 + 𝐴2𝑠𝑖𝑛𝜔𝑑𝑡)𝑒−𝛼𝑡 (Underdamped)

Example

Find i(t) and v(t) for t > 0

END