1. 응용전자 회로
생체의공학과
박기택

2. Active Filter OpAmp를 사용하고 이득이 있는 filter Passive filter Active filter
LPF
HPF

3. Active Filter Lowpass Filter Highpass Filter Bandpass Filter
실제 필터
Lowpass Filter
Highpass Filter
pass band
stop band
𝑊 𝑐
𝑊 𝑐
𝑊 𝑐
Transmit band
Bandpass Filter
Bandstop Filter(Notch Filter)
𝑊 𝑐 :𝑐𝑢𝑡𝑜𝑓𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑊 𝑐
𝑊 𝐿
𝑊 𝑐
𝑊 𝐿

4. Active Filter V(s)=R(s) 𝑉(𝑗𝑤) 𝐼(𝑗𝑤) =𝑗𝑤𝐿 𝑉(𝑠) 𝐼(𝑠) =𝐿 ,+
V(s)=R(s)
𝑉 𝑡 =𝐿 𝑑𝑖(𝑡) 𝑑𝑡 i V -
𝑉 𝑠 =𝑠𝐿𝐼 𝑠
𝑉 𝑗𝑤 =𝑗𝑤𝐿𝐼 𝑗𝑤
𝑠=𝜎+𝑗𝑤 + i V
𝑉(𝑠) 𝐼(𝑠) =𝐿
𝑉(𝑗𝑤) 𝐼(𝑗𝑤) =𝑗𝑤𝐿
𝑖 𝑡 =𝐶 𝑑𝑉(𝑡) 𝑑𝑡 , -
𝐼 𝑠 =𝑠𝐶𝑉 𝑠
𝐼 𝑗𝑤 =𝑗𝑤𝐶𝑉 𝑗𝑤
𝑠=𝜎+𝑗𝑤
𝑉(𝑠) 𝐼(𝑠) = 1 𝑠𝑐
𝑉(𝑗𝑤) 𝐼(𝑗𝑤) = 1 𝑗𝑤𝑐 ,

5. Active Filter H Vi(t) Vo(t)
H(S) = Vo(S) Vi(S) = 𝑏 𝑚 𝑆 𝑚 + ⋯ 𝑏 0 𝑎 𝑛 𝑆 𝑛 + ⋯ + 𝑎 0
Transfer function = N(S) D(S)
𝐻 𝑠 = 𝐻 0 𝑠− 𝑍 1 𝑠− 𝑍 2 ⋯(𝑠− 𝑍 𝑚 ) 𝑠− 𝑃 1 𝑠− 𝑃 2 ⋯(𝑠− 𝑃 𝑛 )
𝑍 𝑖 :𝑧𝑒𝑟𝑜
𝑠= 𝑍 𝑖
𝐻 𝑠 =0
𝑃 𝑖 :𝑝𝑜𝑙𝑒
𝑠= 𝑃 𝑖
𝐻 𝑠 =∞

6. Pole – Zero Diagram 𝑗𝑤
𝑠=𝜎+𝑗𝑤 𝜎
Pole zero diagram을 그렸을때 나타나는 특성을 통해 전달함수의 특성을 알 수 있다.
*보통 통신에 사용 되는 전달함수는 𝜎가 0이다
Left half Plame
Right half Plame

7. Active Filter 𝐻 𝑠 = 𝐴 1 𝑠− 𝑃 1 + 𝐴 2 𝑠− 𝑃 2 +⋯+ 𝐴 𝑛 𝑠− 𝑃 𝑛
𝐻 𝑠 = 𝐴 1 𝑠− 𝑃 𝐴 2 𝑠− 𝑃 2 +⋯+ 𝐴 𝑛 𝑠− 𝑃 𝑛
CASE 1 𝑃 𝑘 = 𝜎 𝑘 (실근)
𝜎 𝑘 >0
Diverge
𝐿 −1 𝐴 𝑘 𝑠− 𝜎 𝑘 = 𝐴 𝑘 𝑒 𝜎 𝑘 𝑡 𝑢(𝑡)
𝜎 𝑘 <0
Die out
* BIBO(Bounded Input Bounded Output) Stability
𝜎 𝑘 <0

8. Active Filter
CASE 2
𝐿 −1 𝐴 𝑘 𝑠− (𝜎 𝑘 +𝑗 𝑤 𝑘 ) + 𝐴 𝑘 ′ 𝑠− (𝜎 𝑘 −𝑗 𝑤 𝑘 ) = 2𝐴 𝑘 𝑒 𝜎 𝑘 𝑡 cos⁡( 𝑤 𝑘 𝑡+ 𝐴 𝑘 )
𝜎 𝑘 =0
𝜎 𝑘 >0
Diverge
Osillation
𝜎 𝑘 <0
𝜎 𝑘 <0 : 모든 pole이 LHP에 있음
Natural Response “die out”
Transient Response “homogeneous”
Steady-stata Frequency Response
S=jw 인 H(s) , w축에서의 변화만 분석
Die out

9. Transfer Function Vi(jw) Vo(jw) H Vi(t) Vo(t)+ 3 - 2 V+ 7 V- 4
OUT 6
OS1 1
OS2 5 R1 C R2
Transfer Function
Vi(jw)
Vo(jw) H
Vi(t)
Vo(t)
𝑉 𝑖 𝑡 = 𝑉 𝑖𝑚 cos⁡(𝑤𝑡+ 𝜃 𝑖 )
H(jw)= 𝑉 𝑜 (jw) 𝑉 𝑖 (jw)
𝑉 𝑜 𝑡 = 𝑉 𝑜𝑚 cos⁡(𝑤𝑡+ 𝜃 𝑜 )
Lowpass filter
H jw = 𝑉 𝑜 (jw) 𝑉 𝑖 (jw) =− 𝑅 2 1+𝑗𝑤 𝑅 2 𝑐 𝑅 1 =− 𝑅 2 𝑅 𝑗𝑤 𝑅 2 𝑐 𝐻
3dB
log 1 𝑅 2 𝐶
log 𝑤

10. Transfer Function Highpass filter
20log 𝑅 2 𝑅 1
3dB
𝑅 𝑗𝑤𝑐
𝑤 𝑐 = 1 𝑅 1 𝐶 1
log 𝑤
Bandpass filter
𝑅 2 1+𝑗𝑤 𝑅 2 𝑐
H= 𝑉 𝑜 (𝑗𝑤) 𝑉 𝑖 (𝑗𝑤) =− 𝑅 2 1+𝑗𝑤 𝑅 2 𝑐 𝑅 𝑗𝑤𝑐 =− 𝑅 2 𝑅 1 𝑗𝑤 𝑅 2 𝐶 2 (1+𝑗𝑤 𝑅 1 𝐶 1 )(1+𝑗𝑤 𝑅 2 𝐶 2 )
20log 𝑅 2 𝑅 1
𝑅 𝑗𝑤𝑐
𝑤 𝐿 = 1 𝑅 1 𝐶 1
𝑤 𝐻 = 1 𝑅 2 𝐶 2
log 𝑤

11. First order Active Filter
Lowpass filter
H jw = 𝑉 𝑜 (jw) 𝑉 𝑖 (jw) =− 𝑅 2 1+𝑗𝑤 𝑅 2 𝑐 𝑅 1 =− 𝑅 2 𝑅 𝑗𝑤 𝑅 2 𝑐 = 𝐻 𝑜 1 1+𝑗 𝑤 𝑤 𝑜
𝐻 𝑜 =− 𝑅 2 𝑅 1
𝑤 𝑜 = 1 𝑅 2 𝑐
Dc gain(low frequency gain)
𝑤=𝑤 𝑜 = 1 𝑅 2 𝑐 일때 H= 𝐻 𝑜 1+𝑗 = 1 2 𝐻
H= 𝐻 𝑜 −3[𝑑𝐵]
3dB
-20dB/dec
𝑓 𝑜 = 𝑤 𝑜 2𝜋 = 1 2𝜋 𝑅 2 𝑐
log 1 𝑅 2 𝐶
log 𝑤
3dB cut off frequency

12. First order Active Filter
Highpass filter
H jw = 𝑉 𝑜 (jw) 𝑉 𝑖 (jw) = 𝐻 𝑜 𝑗 𝑤 𝑤 𝑜 1+𝑗 𝑤 𝑤 𝑜
𝐻 𝑜 =− 𝑅 2 𝑅 1
𝑤 𝑜 = 1 𝑅 1 𝑐
High frequency gain
𝑓 𝑜 = 𝑤 𝑜 2𝜋 = 1 2𝜋 𝑅 1 𝑐
3dB cut off frequency
20log 𝑅 2 𝑅 1
3dB
𝑤=𝑤 𝑜 = 1 𝑅 1 𝑐 1 일때
20db/dec
H= 𝐻 𝑜 −3[𝑑𝐵]
𝑤 𝑐 = 1 𝑅 1 𝐶 1
log 𝑤

13. First order Active Filter
Bandpass filter
H= 𝑉 𝑜 (𝑗𝑤) 𝑉 𝑖 (𝑗𝑤) =− 𝑅 2 1+𝑗𝑤 𝑅 2 𝑐 𝑅 𝑗𝑤𝑐 = 𝐻 𝑜 𝑗 𝑤 𝑤 𝐿 (1+𝑗 𝑤 𝑤 𝐿 )(1+𝑗 𝑤 𝑤 𝐻 )
𝐻 𝑜 =− 𝑅 2 𝑅 1 :𝑝𝑎𝑠𝑠𝑏𝑎𝑛𝑑 𝑔𝑎𝑖𝑛(𝑚𝑖𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑔𝑎𝑖𝑛)
𝑤 𝐿 = 1 𝑅 1 𝑐 1 :𝐿𝑜𝑤 3𝑑𝐵 𝑐𝑢𝑡𝑜𝑓𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑤 𝐻 = 1 𝑅 2 𝑐 2 :𝐻𝑖𝑔ℎ 3𝑑𝐵 𝑐𝑢𝑡𝑜𝑓𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑤=𝑤 𝐿 𝑜𝑟 𝑤= 𝑤 𝐻 일때
20log 𝑅 2 𝑅 1
H= 𝐻 𝑜 −3[𝑑𝐵]
𝑤 𝐿 = 1 𝑅 1 𝐶 1
𝑤 𝐻 = 1 𝑅 2 𝐶 2
log 𝑤

14. First order Active Filter
Phase - Shifter
𝑉 𝑝 = 1 𝑠𝑐 𝑅+ 1 𝑠𝑐 𝑉 𝑖 = 1 1+𝑠𝑅𝑐 𝑉 𝑖
𝑉 𝑜 =− 𝑉 𝑖 +2 𝑉 𝑝
= 1−𝑠𝑅𝑐 1+𝑠𝑅𝑐 𝑉 𝑖
20log 𝑅 2 𝑅 1
H= 𝑉 𝑜 (𝑗𝑤) 𝑉 𝑖 (𝑗𝑤) = 1−𝑗𝑤𝑅𝐶 1+𝑗𝑤𝑅𝐶 = −2tan −1 𝑤𝑅𝑐
log 𝑤 H
log 𝑤

15. Second order Active Filter
H s = 𝑁(𝑠) { 𝑠 𝑤 𝑜 } 2 +2𝜉 𝑠 𝑤 𝑜 +1
𝑁 𝑠 :𝑠의 다항식(m<=2)
𝑤 𝑜 : undamped natural frequency
𝜉 :damping ratio
𝑃 1.2 :(−𝜉 ± 𝜉 2 −1 ) 𝑤 𝑜
1. 𝜉>1 (서로 다른 실근 2개)
Over damped
Natural response는 2개의 지수함수로 이루어짐
lim 𝑡→∞ 𝐴 1 𝑒 − 𝑡 𝜏 𝐴 2 𝑒 − 𝑡 𝜏
2. 0<𝜉<1(𝑐𝑜𝑚𝑝𝑙𝑒𝑥 𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑒)
𝑃 1.2 : −𝜉 𝑤 𝑜 ±𝑗 𝑤 𝑜 1− 𝜉 2
Natural response : damped sine
2𝐴 𝑘 𝑒 − 𝜉𝑤 0 𝑡 cos⁡( 𝑤 0 1− 𝜉 𝐴 𝑘 )

16. Second order Active Filter
3. 𝜉=0 : undamped
Natural response
2𝐴 𝑘 cos⁡( 𝑤 0 𝑡+ 𝐴 𝑘 )
4. 𝜉<0
stable
Diverge (unstable)
𝜉<0
𝜉=0
𝜉<0
𝜉>1
𝜉>1
𝜉=0
𝜉<0

17. Second order Active Filter
H jw = 𝑁(𝑗𝑤) 1−{ 𝑤 𝑤 𝑜 } 2 + 𝑗 𝑤 𝑤 𝑜 𝑄
Damping Q
osilation
1. LPF
H jw = 𝐻 𝑂𝐿𝑝 ∗ 𝐻 𝐿𝑝 (𝑗𝑤)
𝐻 𝑂𝐿𝑝 :dc gain (low freq gain)
소자값에 따라 결정
Q값에 따라서 overshoot, undershoot 발생
= 𝐻 𝑂𝐿𝑝 1 1−{ 𝑤 𝑤 𝑜 } 2 + 𝑗 𝑤 𝑤 𝑜 𝑄
1. 𝑤 𝑤 𝑜 ≪1
3. 𝑤 𝑤 𝑜 =1
𝐻 𝑗𝑤 = 𝐻 𝑂𝐿𝑝
𝐻 𝐿𝑝 𝑗𝑤 =−𝑗𝑄
-40dB/dec
2. 𝑤 𝑤 𝑜 ≫1
𝐻= 𝐻 𝑂𝐿𝑝 +𝑄[𝑑𝐵]
log 𝑤
𝐻 𝑗𝑤 = 𝐻 𝑂𝐿𝑝 ∗ −1 { 𝑤 𝑤 𝑜 } 2
𝐻 𝐿𝑝 =−40 log 𝑤 𝑤 𝑜 [dB]