1. 6장 : 주파수 응답과 시스템 Frequency Response and System
김효진
한소희
정희형
추연웅
이보미

2. Frequency Response & system
𝑉 𝐿 = 𝑍 𝐿 𝑍 𝑇 + 𝑍 𝐿 = Hㆍ 𝑉 𝑆
정현파 전원(Vs)

3. 예)
𝑉 𝑇 = 𝑍 2 𝑍 1 + 𝑍 2 Vs
Z1 = R1 + ZC1 = R jwc1
Z2 = R2 // ZC2 = 𝑅 2 X 1 j𝑊 𝐶 2 𝑅 j𝑊 𝐶 2 = 𝑅 2 1+𝑗𝑤 𝑅 2 𝐶 2
ZT = Z1 + Z2

4. 예)
𝑉 𝑇 = 1 𝑗𝑤𝐶 𝑅+ 1 𝑗𝑤𝑐 𝑉 𝑠 = 1 1+𝑗𝑤𝑅𝐶 𝑉 𝑆
ZT = 𝑅 1 𝑗𝑤𝐶 𝑅+ 1 𝑗𝑤𝑐 = 𝑅 1+𝑗𝑤𝑅𝐶
𝑉 𝐿 = 𝑅 𝐿 𝑍 𝐿 + 𝑅 𝐿 𝑉 𝑇 = 𝑅 𝐿 𝑅 1+𝑗𝑤𝑅𝐶 + 𝑅 𝐿 X 1 1+𝑗𝑤𝑅𝐶 𝑉 𝑆
= 𝑅 𝐿 𝑅+ 𝑅 𝐿 (1+𝑗𝑤 𝑅 𝑅 𝐿 𝑅+ 𝑅 𝐿 𝐶) 𝑉 𝑆
= 𝑅 𝐿 𝑅+ 𝑅 𝐿 (1+𝑗𝑤𝑅 𝑅 𝐿 𝐶) 𝑉 𝑆
𝑉 𝐿 (jw)(출력) = H(jw) X 𝑉 𝑆 (jw)
Thevenin

5. H(jw) H(jw) → Transfer function or Frequency response (전달함수 또는 주파수응답)
(전달함수 또는 주파수응답)
H(jw) = 𝐻 ∠ ΔH = VL(jw) Vs(jw)
Vs(jw)
VL(jw) = H(jw)Vs(jw)
H(jw) 입력 출력

6. Fourier analysis 1 𝑇 = f : fundamental angular frequency
Fourier series : periodic function
Fourier transform : aperiodic
x(t) = x(t + nT), n = 1, 2, 3, …
T : period
1 𝑇 = f : fundamental angular frequency
(기본주파수)
ω = 2πf : fundamental angular frequency
nf : Harmonics (고조파)
x(t) = a0 + ∑ ancos(n ω t) + ∑ bnsin(n ω t)
*그림 6.11 참고

7. x(t) = 𝑐 0 + ∑ 𝑐 𝑛 cos (nωt - ψn)
𝑐 𝑛 = 𝑥= 𝑎 𝑛 𝑏 𝑛 2
Ψn = tan −1 𝑏𝑛 𝑎𝑛
𝑎 0 = 1 𝑇 0 𝑇 𝑥 𝑡 𝑑𝑡
𝑎 𝑛 = 2 𝑇 0 𝑇 𝑥 𝑡 cos 𝑛wt 𝑑𝑡
= 2 𝑇 − 𝑇 2 𝑇 2 𝑥 𝑡 cos 𝑛𝑤𝑡 𝑑𝑡
𝑏 𝑛 = 2 𝑇 0 𝑇 𝑥 𝑡 sin⁡(𝑛𝑤𝑡)𝑑𝑡
= 2 𝑇 − 𝑇 2 𝑇 2 𝑥(𝑡)sin⁡(𝑛𝑤𝑡)𝑑𝑡

8. y(t) x(t) R,C,L 선형회로 x(t) = 𝑛=1 ∞ 𝑐 𝑛 𝑐𝑜𝑠(𝑛𝑤𝑡− 𝜑 𝑛 )
𝑐 0 는 입력 → 𝑦 0 ;DC회로해석
𝑐 1 ∠ 𝜑 1 인 정현파(𝑤) → 𝑦 1
𝑐 2 ∠ 𝜑 2 인 정현파 2𝑤 → 𝑦 2
︙ ︙
∴ y(t) = 𝑦 𝑛=1 ∞ 𝑑 𝑛 cos (nωt - θn)
𝑑 𝑛 ∠ θ 𝑛 (nw)
phasor 해석

9. 𝑉 𝐿 jw = 𝐻 (jw) 𝑉 𝑆 jw
𝑑 1 ∠ θ 1 = 𝐻 (𝑗𝑤) ∠ ΔH(jw) 𝑐 1 ∠ φ 1
= 𝑐 1 𝐻 (𝑗𝑤) ∠ φ 1 + ∠ 𝐻 (𝑗𝑤)
𝑑 2 ∠ θ 2 = 𝑐 2 𝐻 (𝑗2𝑤) ∠ φ 2 + ∠ 𝐻 (𝑗2𝑤) ︙
𝑑 𝑛 ∠ θ 𝑛 = 𝑐 𝑛 𝐻 (𝑗𝑛𝑤) ∠ φ 𝑛 + ∠ 𝐻 (𝑗𝑛𝑤)

10. 𝑉 𝑖 (𝑡) = 𝑉 1 sin 𝜔 1 𝑡 + 𝑉 2 sin 𝜔 2 𝑡 w1
w1 + w2 w2

11. Low pass Filter
H(jw) = 𝑉 𝑜 (𝑗𝑤) 𝑉 𝑖 (𝑗𝑤) = 1 𝑗𝑤𝑐 𝑅+ 1 𝑗𝑤𝑐 = 1 1+𝑗𝑤𝑅𝐶

12. 𝑯 (𝒋𝒘) = 𝟏 𝟏+ 𝒘 𝟐 𝑹 𝟐 𝑪 𝟐
∠ 𝑯 (𝒋𝒘) = - 𝐭𝐚𝐧 −𝟏 𝒘𝑹𝑪

13. w1 RC w2
w1 + w2 w1
Low pass Filter

14. High pass Filter
H(jw) = 𝑉 𝑜 (𝑗𝑤) 𝑉 𝑖 (𝑗𝑤) = 𝑅 𝑗𝑤𝑐 𝑅+ 1 𝑗𝑤𝑐 = 𝑗𝑤𝑅𝐶 1+𝑗𝑤𝑅𝐶

15. 𝑯 (𝒋𝒘) = 𝒘𝑹𝑪 𝟏+ 𝒘 𝟐 𝑹 𝟐 𝑪 𝟐
∠ 𝑯 (𝒋𝒘) = 𝝅 𝟐 - 𝐭𝐚𝐧 −𝟏 𝒘𝑹𝑪

16. w1 RC w2
w1 + w2
High pass Filter w2

17. Band pass Filter (Notch Filter)
𝑯(𝒋𝒘)
w w2
Band pass Filter (Notch Filter)
𝑯(𝒋𝒘) w1

18. <Low pass Filter> <High pass Filter>
𝑤 𝑐
① ② ③
𝑤 𝑐
③ ② ①
𝑤 𝑐 = cutoff frequency(차단주파수)
① : Pass Band
② : Transition Band
③ : Stop Band
Bandwidth(대역폭) = 𝒘 𝒄
Bandwidth(대역폭) = 이론적으로 ∞

19. 𝑤 𝐿 𝑤 𝐻 𝑤 𝐿 𝑤 𝐻 <Band pass Filter> <Band stop Filter>
𝑤 𝐿 𝑤 𝐻
𝑤 𝐿 𝑤 𝐻
① ② ③ ③ ② ①
③ ② ① ② ③
Bandwidth(대역폭) = 𝒘 𝑯 - 𝒘 𝑳
① : Pass Band
② : Transiton Band
③ : Stop Band