Ch.8 Second-Order Circuits

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Ch.8 Second-Order Circuits

Part 1 – Source Free 2차회로

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2nd Order Circuit

• 2nd Order Circuit이란?

- 2차 미분방정식으로 표현될 수 있는 회로

- 에너지 저장이 가능한 소자(L 및 C)가 2개 포함된 회로

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Initial & Final Values

4 Pusan National University

Example 8.1

The switch in the following figure has been closed for a long time.

It is open at 𝑡 = 0. Find:

(a) 𝑖(0⁺^ሻ, 𝑣(0⁺ሻ (b) ^𝑑

𝑑𝑡𝑖(0⁺ሻ, ^𝑑

𝑑𝑡 𝑣(0⁺ሻ (c) 𝑖(∞ , ^ሻ 𝑣(∞ሻ

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Initial & Final Values

Example 8.2

In the following circuit, calculate:

(a) 𝑖_𝐿(0⁺ሻ, 𝑣_𝐶(0⁺ሻ, 𝑣_𝑅(0⁺ሻ (b) ^𝑑

𝑑𝑡𝑖_𝐿(0⁺ሻ, ^𝑑

𝑑𝑡𝑣_𝐶 0⁺ , ^𝑑

𝑑𝑡𝑣_𝑅(0⁺) (c) 𝑖_𝐿(∞ , ^ሻ 𝑣_𝐶(∞ , ሻ 𝑣_𝑅(∞ሻ

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Source Free Series RLC Circuit

6

• 전원이 없는 Series RLC 회로 - 초기값 : 𝐼₀, 𝑉₀(= ¹

𝐶 ׬_−∞⁰ 𝑖𝑑𝑡) - KVL 적용

Pusan National University

2 2

1 ^t 0 0

di d i R di i

Ri L idt

dt C ^ dt L dt LC

 



    

2차 미분방정식

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Source Free Series RLC Circuit

• 미분방정식의 해

2 2

2 2 2

0 2

2 2

1 2 1 2 0

0 1 0

1 0 or 2 0; characteristic equation

, 1 or , ; natural frequencies

2 2

where = ; damping facto 2

st st AR st A st st R

i Ae As e se e Ae s s

L LC L LC

s R s s s

L LC

R R

s s s s

L L LC

R L

 

  



 

          

 

      

 

         

1 2 1 2

0

1 1 2 2 1 2

r or neper frequency

1 ; resonant frequency (undamped natural frequency) ( ) ^{s t}, ( ) ^{s t} ( ) ^{s t} ^{s t}

LC

i t A e i t A e i t A e A e

 

      ^{경계조건 𝑖(0ሻ,} ^𝑑𝑖(0^𝑑𝑡^ሻ

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Source Free Series RLC Circuit

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• 𝛼와 𝜔₀에 따른 해의 특성

- Overdamped case (𝛼 > 𝜔₀)

⇒ 𝛼 > 𝜔₀ ⇒ 𝐶 > 4 Τ𝐿 𝑅² or 𝛼² > 𝜔₀²

⇒ 𝑠₁, 𝑠₂는 모두 음의 실수

⇒ 𝑖(𝑡ሻ = 𝐴₁𝑒^𝑠¹^𝑡 + 𝐴₂𝑒^𝑠²^𝑡는 점진적으로 감소

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Source Free Series RLC Circuit

- Critically Damped case (𝛼 = 𝜔₀)

⇒ 𝑠₁ = 𝑠₂ = −𝛼 = − ^𝑅

2𝐿

⇒ 𝑖(𝑡ሻ = 𝐴₁𝑒^𝑠¹^𝑡 + 𝐴₂𝑒^𝑠²^𝑡 = 𝐴₃𝑒^−𝛼𝑡

(경계조건은 2개이나 변수는 하나이므로 해가 될수 없음)

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Source Free Series RLC Circuit

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- Critically Damped case (𝛼 = 𝜔₀)

 

2

2 2

1

1 1

1 1 2 1 2

2 0 0

0 where

or

( ) ( )

t

t t t

d i di d di di

i i i

dt dt dt dt dt

d di

f f f i

dt dt

f A e

di di

i A e e e i A

dt dt

d e i A e i A t A i t A t A e

dt



  

    

 



   

          

   

 

    

      

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Source Free Series RLC Circuit

- Underdamped case (𝛼 < 𝜔₀)

 

 

2 2

1 2 0

( ) ( )

1 2 1 2

1 2

1 2 1 2

1 2

, ( )

( )

cos sin (cos sin )

( ) cos ( ) sin

( cos sin )

d d d d

d

j t j t t j t j t

t

d d d d

t

d d

t

d d

s s j

i t A e A e e A e A e

e A t j t A t j t

e A A t j A A t

e B t B t

      



    

   

 

     



       

   

     

   

 

Damped natural frequency

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Source Free Series RLC Circuit

Example 8.3

In the following figure, 𝑅 = 40𝛺, 𝐿 = 4H, and 𝐶 = Τ1 4F. Calculate the characteristic roots of the circuit. Is the natural response

overdamped, underdamped or critically damped?

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Source Free Series RLC Circuit

Example 8.4

Find ሻ𝑖(𝑡 in the following figure. Assume that the circuit has reached steady state at 𝑡 = 0⁻.

0 t ^

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Src-Free Parallel RLC Circuit

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• 전원이 없는 Parallel RLC 회로 - 초기값 : 𝐼₀(= ¹

𝐿 ׬_−∞⁰ 𝑣(𝑡ሻ𝑑𝑡), 𝑉₀ - KCL 적용

2 2

1 1 1

0 0

v t dv d v dv

vdt C v

R  L



^  dt   dt  RC dt  LC 

2차 미분방정식

2

cf. d i2 R di i 0 dt  L dt  LC 

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Src-Free Parallel RLC Circuit

• Characteristic Equation

2

2 2

1 2 0

0

1 1

0

1 1 1

, 2 2

1 1

where ,

2

s s

RC LC

s s RC RC LC

RC LC

  

 

  

 

          

 

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Src-Free Parallel RLC Circuit

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• Overdamped case (𝛼 > 𝜔₀)

• Critically Damped case (𝛼 = 𝜔₀)

• Underdamped case (𝛼 < 𝜔₀)

1 2

( ) ^{s t} ^{s t} v t  A e  A e

1 2

( ) ( ) ^t

v t  A  A t e^^

1 2

( ) ^t( cos _d sin _d ) v t  e^^ A  t  A  t

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Src-Free Parallel RLC Circuit

Example 8.5

In the following circuit, find 𝑣(𝑡 for 𝑡 > 0, assuming 𝑣(0ሻ = 5V, ሻ 𝑖(0ሻ = 0, 𝐿 = 1H and 𝐶 = 10mF. Consider these cases: 𝑅 = 1.923𝛺, 𝑅 = 5𝛺 and 𝑅 = 6.25𝛺.

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Src-Free Parallel RLC Circuit

Example 8.5

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Src-Free Parallel RLC Circuit

Example 8.6

Find 𝑣(𝑡 for 𝑡 > 0 in the following RLC circuit. ሻ