Bài giảng Electric circuit theory - Chapter XV: Two-port Networks - Nguyễn Công Phương

Nguy ễn Công Ph ươ ng Electric Circuit Theory Two-port Networks Contents I. Basic Elements Of Electrical Circuits II. Basic Laws III. Electrical Circuit Analysis IV. Circuit Theorems V. Active Circuits VI. Capacitor And Inductor VII. First Order Circuits VIII.Second Order Circuits IX. Sinusoidal Steady State Analysis X. AC Power Analysis XI. Three-phase Circuits XII. Magnetically Coupled Circuits XIII.Frequency Response XIV.The Laplace Transform XV. Two-port Networks Two-port Networks - sites.google.com/site/ncpdhbkhn 2 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 3 Introduction I1 I2 + + Linear V1 V2 – – Network I1 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 4 Two-port Network 1. Introduction 2. Parameters a) Impedance z b) Admittance y c) Hybrid h d) Inverse Hybrid g e) Transmission T f) Inverse Transmission t 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 5 Impedance Parameters (1) I1 I2 + + V= zI + zI Linear 1 111 122 V1 V2 –  – Network = + V2 zI 211 zI 222 I1 I2 V1 zzI 11 12  1  I 1 =  = []z  V2 zzI 21222   I 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 6 Impedance Parameters (2) I1 I2 + + V= zI + zI Linear 1 111 122 V1 V2 –  – Network = + V2 zI 211 zI 222 = I I2 I2 0 1  V = = 1 I1 I2 0 z11 + V= z I  I1 I =0 → 1 11 1 → 2 + Linear   – V= z I V1 V2  Network – 2 211  = V2 z21 I1 =  I2 0 I Two-port Networks - sites.google.com/site/ncpdhbkhn1 7 Impedance Parameters (3) I1 I2 + + V= zI + zI Linear 1 111 122 V1 V2 –  – Network = + V2 zI 211 zI 222 = I I2 I1 0 1  V = = 1 I1 0 I2 z12 + V= z I  I2 I =0 → 1 122 → 1 Linear +   – V= z I V1 V2  2 22 2  V – Network z = 2  22 I  2 I =0 1 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 8 Impedance Parameters (4) I1 I2 + + V= zI + zI Linear 1 111 122 V1 V2 –  – Network = + V2 zI 211 zI 222 I1 I2   =VV1 = 1 z11 z 12  I1= I 2 = I20 I 1 0    VV z=2 z = 2  21 22  I1= I 2 = I20 I 1 0  Two-port Networks - sites.google.com/site/ncpdhbkhn 9 Ex. Impedance Parameters (5) Find [z] ? 10 Ω 30 Ω = Ω I1 I2 0 20 + + 10 Ω 30 Ω – I I V1 V2 1 2 20 Ω – + + [z] V V1 V2 – z = 1 – 11 I 1 I =0 2 I I = + = 1 2 V1(10 20) I 1 30 I 1 V= zI + zI 30 I 1 11 1 12 2 → =1 = Ω  z11 30 = + V2 zI 211 zI 222 I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 10 Ex. Impedance Parameters (6) Find [z] ? 10 Ω 30 Ω = Ω I1 I2 0 20 + + 10 Ω 30 Ω – I I V1 V2 1 2 20 Ω – + + [z] V V1 V2 – z = 2 – 21 I 1 I =0 2 I I = = 1 2 V2 VR 220 I 1 V= zI + zI 20 I 1 11 1 12 2 → =1 = Ω  z21 20 = + V2 zI 211 zI 222 I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 11 Ex. Impedance Parameters (7) Find [z] ? 10 Ω 30 Ω = Ω I1 0 I2 20 + 10 Ω 30 Ω + – I I V1 V2 1 2 – 20 Ω + + [z] V V1 V2 – z = 1 – 12 I 2 I =0 1 I I = = 1 2 V1 VR 220 I 2 V= zI + zI 20 I 1 11 1 12 2 → =2 = Ω  z12 20 = + V2 zI 211 zI 222 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 12 Ex. Impedance Parameters (8) Find [z] ? 10 Ω 30 Ω = Ω I1 0 I2 20 + 10 Ω 30 Ω + – I I V1 V2 1 2 – 20 Ω + + [z] V V1 V2 – z = 2 – 22 I 2 I =0 1 I I = + = 1 2 V2(20 30) I 2 50 I 2 V= zI + zI 50 I 1 11 1 12 2 → =2 = Ω  z22 50 = + V2 zI 211 zI 222 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 13 Ex. Impedance Parameters (9) Find [z] ? 10 Ω 30 Ω 20 Ω I1 I2 30 20  + + [z] z = V1 V2 –   – 20 50  I1 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 14 Ex. Impedance Parameters (10) Find [z] ? 10 Ω 30 Ω 20 Ω I1 I2 I1 I2 + + + + [z] [z] V1 V2 V1 V2 – – – – I I1 I2 I1 2 = 30 20  z ? z =   20 50  Two-port Networks - sites.google.com/site/ncpdhbkhn 15 Ex. Impedance Parameters (11) I Find [z] ? I1 2 + + Method 2 10 Ω 30 Ω V1 Ω V2 – – 20 = + = + + = + + VVV1 10 20 10I1 20( I 1 I 2 ) (10 20)I1 20 I 2 = + = + + = + + VVV2 30 20 30I2 20( I 1 I 2 ) 20I1 (20 30) I 2 = + + V1(10 20) I 1 20 I 2 z =10 + 20 = 30 Ω →  11 = + +  V220 I 1 (20 30) I 2 z =20 = 20 Ω →  12 = + = = Ω V zI zI z21 20 20  1 111 122 = +  = + = Ω V2 zI 211 zI 222 z22 20 30 50 Two-port Networks - sites.google.com/site/ncpdhbkhn 16 Two-port Network 1. Introduction 2. Parameters a) Impedance z b) Admittance y c) Hybrid h d) Inverse Hybrid g e) Transmission T f) Inverse Transmission t 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 17 Admittance Parameters (1) I1 I2 + + I= y V + y V Linear 1 11 1 12 2 V1 V2 –  – Network = + I2 y 211 V y 222 V I1 I2 I1 y 1112 y  V 1  V 1 =  = []y  I2 y 2122 y  V 2  V 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 18 Admittance Parameters (2) I1 I2 + + I= y V + y V Linear 1 11 1 12 2 V1 V2 –  – Network = + I2 y 211 V y 222 V = I I2 V2 0 1  I = 1 I1 I2 y11 I= y V  V1 V =0 → 1 11 1 → 2 + Linear   – = = V1 V2 0 I2 y 211 V  I Network y = 2  21 V  1 V =0 2 I I2 Two-port Networks - sites.google.com/site/ncpdhbkhn1 19 Admittance Parameters (3) I1 I2 + + I= y V + y V Linear 1 11 1 12 2 V1 V2 –  – Network = + I2 y 211 V y 222 V = I I2 V1 0 1  I = 1 I1 I2 y12 = V2 = I y V  V1 0 →  1 12 2 →  Linear + = – = V1 0 V2 I2 y 22 V 2  I Network y = 2  22 V  2 V =0 1 I I2 Two-port Networks - sites.google.com/site/ncpdhbkhn1 20 Admittance Parameters (4) I1 I2 + + I= y V + y V Linear 1 11 1 12 2 V1 V2 –  – Network = + I2 y 211 V y 222 V I1 I2   =I1 = I 1 y11 y 12  VV1= 2 = V20 V 1 0    I I y=2 y = 2  21 22  VV1= 2 = V20 V 1 0  Two-port Networks - sites.google.com/site/ncpdhbkhn 21 Ex. Admittance Parameters (5) Ω Find [y] ? Ω 2 Ω I2 1 3 + + 2Ω = I1 V V2 0 I I – 1 1 2 1Ω 3Ω – + + I Linear y = 1 V1 V2 – 11 – Network V1 = V2 0 1× 2 I I V=(1// 2) I = I = 0.67 I 1 2 1 11+ 2 1 1  = + I I1 y 111 V y 122 V → =1 =  y11 1.5S = + 0.67 I1 I2 y 211 V y 222 V Two-port Networks - sites.google.com/site/ncpdhbkhn 22 Ex. Admittance Parameters (6) Ω Find [y] ? Ω 2 Ω I2 1 3 + + 2Ω = I1 V V2 0 I I – 1 1 2 1Ω 3Ω – + + I Linear y = 2 V1 V2 – 21 – Network V1 = V2 0 I = = =− I1 2 VV11Ω V 2 Ω 2 I 2  = + I I1 y 111 V y 122 V →y =2 =− 0.5S  21 − = + 2I2 I2 y 211 V y 222 V Two-port Networks - sites.google.com/site/ncpdhbkhn 23 Ex. Admittance Parameters (7) Ω Find [y] ? Ω 2 Ω I1 1 3 + + = 2Ω V1 0 V2 I2 I I – 1 2 – 1Ω 3Ω + + I Linear y = 1 V1 V2 – 12 – Network V2 = V1 0 I = = =− I1 2 VV23Ω V 2 Ω 2 I 1  = + I I1 y 111 V y 122 V →y =1 =− 0.5S  12 − = + 2I1 I2 y 211 V y 222 V Two-port Networks - sites.google.com/site/ncpdhbkhn 24 Ex. Admittance Parameters (8) Ω Find [y] ? Ω 2 Ω I1 1 3 + + = 2Ω V1 0 V2 I2 I I – 1 2 – 1Ω 3Ω + + I Linear y = 2 V1 V2 – 22 – Network V2 = V1 0 2× 3 I I VIVV=(2 // 3) = = 1.2 1 2 2 22+ 3 22  = + I I1 y 111 V y 122 V → =2 =  y22 0.83S = + 1.2 I2 I2 y 211 V y 222 V Two-port Networks - sites.google.com/site/ncpdhbkhn 25 Ex. Admittance Parameters (9) 2Ω Find [y] ? 1Ω 3Ω I1 I2 1.5− 0.5  + + = Linear y   V1 V2 – −0.5 0.83  – Network I1 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 26 Two-port Network 1. Introduction 2. Parameters a) Impedance z b) Admittance y c) Hybrid h d) Inverse Hybrid g e) Transmission T f) Inverse Transmission t 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 27 Hybrid Parameters V= h I + h V I I  1 11 1 12 2 1 2 +  I= h I + h V + 2 211 222 Linear V1 V2 – – Network VhhI1 1112  1  I 1 =  = []h  I h h V V 2 21 22  2  2 I1 I2   =VV1 = 1 h11 h 12  IV1= 2 = V20 I 1 0    I I h=2 h = 2  21 22  IV1= 2 = V20 I 1 0  Two-port Networks - sites.google.com/site/ncpdhbkhn 28 Inverse Hybrid Parameters  I= gV + gI I I  1 11 1 12 2 1 2 + V= gV + gI + 2 21 1 22 2 Linear V1 V2 – – Network I1 h 1112 h  V 1  V 1 =  = []h  V hhI I 2 21222   2 I1 I2   =VV1 = 1 h11 h 12  IV1= 2 = V20 I 1 0    I I h=2 h = 2  21 22  IV1= 2 = V20 I 1 0  Two-port Networks - sites.google.com/site/ncpdhbkhn 29 Transmission Parameters V= AV − BI I I  1 2 2 1 2 +  I= CV − DI + 1 2 2 Linear V1 V2 – – Network VVV1AB   2  2 =   = []T  ICD − I − I 1   2  2 I1 I2 VV  AB=1 = 1  V2= I 2 = I20 V 2 0    I I CD=1 = 1    V2= I 2 = I20 V 2 0  Two-port Networks - sites.google.com/site/ncpdhbkhn 30 Inverse Transmission Parameters V= aV − bI I I  2 1 1 1 2 +  I= cV − dI + 2 1 1 Linear V1 V2 – – Network VVV2a b   1  1 =   = []t  Ic d − I − I 2   1  1 I1 I2 VV  a=2 b = 2  V1= I 1 = I10 V 1 0    I I c=2 d = 2    V1= I 1 = I10 V 1 0  Two-port Networks - sites.google.com/site/ncpdhbkhn 31 Two-port Networks V= zI + zI V= h I + h V V= AV − BI  1 111 122  1 111 122  1 2 2 = + = + = − V2 zI 211 zI 222  I2 h 211 I h 222 V  I1 CV 2 DI 2 I= y V + y V  I= gV + gI V= aV − bI  1 111 122  1 111 122  2 1 1 = + = + = − I2 y 21 V 1 y 22 V 2 V2 gV 211 gI 222  I2 cV 1 dI 1 Two-port Networks - sites.google.com/site/ncpdhbkhn 32 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 33 Relationships between Parameters (1) V1 zzI 1112  1  I 1 =  = []z  V2 zzI 21222   I 2 IV1−1  1 → = []z  IV  2 2 − →[y] = [ z ] 1 IV1  1 = []y  IV2  2 Two-port Networks - sites.google.com/site/ncpdhbkhn 34 Relationships between Parameters (2) − [y] = [ z ] 1 − [g] = [ h] 1 − [t] = [ T] 1 Two-port Networks - sites.google.com/site/ncpdhbkhn 35 Relationships between Parameters (3) = + V1 h 11 I 1 h 12 V 2  h 1 = + → =−12 +  I2 h 21 I 1 h 22 V 2 V2 I 1 I 2 h22 h 22 h h  h →=−12 21 + 12 Vh1 11  I 1 I 2 h22  h 22    h h h  = −h12 h 21 + h 12 = −12 21 = 12 Vh1 11  I 1 I 2 z11 h 11 z 12   h  h h22 h 22 →  22 22 →   h 1   h12 1 = −12 = V= − I + I z21 z 22  2 1 2 h h   h22 h 22 22 22  Two-port Networks - sites.google.com/site/ncpdhbkhn 36 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 37 Ex. 1 Two-port Network Analysis (1) = o = Ω I1 I2 E 220 0 V;ZL j 50 ; + + 10j 20  + = z   ;find currents? – j20 40  E V1 [z] V2 ZL – – I I1 2 V=10 I + j 20 I  1 1 2 V=j20 I + 40 I 0  2 1 2 220 0= 10I + j 20 I I =14.09 + j 4.94 A →  1 2 →  1 = = o − = + I = −2.47 − j 3.96 A V1 E 220 0 V  j50I2 j 20 I 1 40 I 2  2 =− =− V2 ZIL 2j50 I 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 38 Ex. 2 Two-port Network Analysis (2) I I Write equations? 1 2 + + Z + V1 [T] V2 ZL – – E – I I1 2 V= AV − BI  1 2 2 = −  I1 CV 2 DI 2 + = ZI1 V 1 E = V2 ZL I 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 39 Ex. 3 Two-port Network Analysis (3) Write equations? Method 1 I I IL S I1 F ZF I2 = + + + I1 y 11 V 1 y 12 V 2 Z S V [y] V – – 1 2 + Z I= y V + y V L 2 211 222 E – − − = IS I1 I F 0 − − = IF I2 I L 0 IS I IF Z I IL + = 1 F 2 ZIVES S 1 V1 V2 + + ZS – ZIVV− + = 0 – F F 1 2 + ZL − = E – V2 ZL I L 0 Two-port Networks - sites.google.com/site/ncpdhbkhn 40 Ex. 3 Two-port Network Analysis (4) Write equations? Method 2 I I IL S I1 F ZF I2 + + = + =− Z I1 yV 111 yV 122 IA I B S V [y] V – – 1 2 + ZL = + =− – I2 yVyV 211 222 IB I C E + = ZIVES A 1 I I I − = S I F ZF I L V2 ZL I C 0 1 2 V1 V2 + + ZS – − + = I – ZIVVF B 1 2 0 A IC + IB ZL E – Two-port Networks - sites.google.com/site/ncpdhbkhn 41 Ex. 3 Two-port Network Analysis (5) Write equations? Method 3 I I IL S I1 F ZF I2 + + − − = Z a :IS I1 I F 0 S V [y] V – – 1 2 + Z − − = L b :IF I2 I L 0 E – = + = + I1 yV 111 yV 122 yV 11a yV 12 b = + = + I2 yV 211 yV 222 yV 21a yV 22 b I I IL S I1 F ZF I2 I− I = yV + yV V1 V2 + → SF11 a 12 b a + b  ZS – I− I = yV + yV –  FL21 a 22 b + ZL E – Two-port Networks - sites.google.com/site/ncpdhbkhn 42 Ex. 3 Two-port Network Analysis (6) Write equations? Method 3 IS I IF Z IL − = + 1 F I2 ISF I yV11 a yV 12 b − = + + + IFL I yV21 a yV 22 b Z S V [y] V – – 1 2 − + Z = E V a L IS E – ZS = Vb IL ZL V− V = a b IS I IF Z I IL IF 1 F 2 ZF V1 V2 + a + b EV− V − V ZS – a− a b =yV + y V –  11a 12 b + Z  Z Z L →  S F E – V− V V  a b− b =yV + y V  21a 22 b  ZF Z L Two-port Networks - sites.google.com/site/ncpdhbkhn 43 Ex. 3 Two-port Network Analysis (7) Write equations? Method 3 I I IL S I1 F ZF I2 + + Z EV− V − V S V [y] V – – 1 2 a− a b = + + ZL  yV11a yV 12 b  Z Z E –  S F V− V V  a b− b =yV + y V  21a 22 b  ZF Z L (ZZy++ Z ZV )( + ZZy −= ZV ) ZE →  SF11 SFa SF12 SbF − + ++ =  (ZZyLF21 ZV La )( ZZy LF 22 Z LFb ZV )0 Two-port Networks - sites.google.com/site/ncpdhbkhn 44 Ex. 4 Two-port Network Analysis (8) I I Z Write equations? 1 2 A + + Z ZM + V1 [T] V2 ZB – – E – I I1 2 V= AV − BI  1 2 2 = −  I1 CV 2 DI 2 + = ZI1 V 1 E = + − V2( ZZA B 2 ZI M ) 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 45 Ex. 5 Two-port Network Analysis (9) i1 i2 e = 10 + 20cos5 t V. Find v1? + + 2µ F 20 Ω 30 20  z = + v   v 1 20 50  2 – – e – 10 Ω Two-port Networks - sites.google.com/site/ncpdhbkhn 46 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks a) Series b) Parallel c) Cascade d) Hybrid 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 47 Series (1) I1 I1a I2a I2 + + + Z1 Z2 + Network V V1a a 2a – I = I1 = I2 – V = V1 + V2 I1a I2a V V I= I = I 1 2 1 1a 1 b I1b I2b +  = + + VVV1 1a 1 b  Network = = V1b b V2b – I I I – –  2 2a 2 b –  = + VVV2 2a 2 b I1 I1b I2b I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 48 Series (2) Network a: = + V1a zI 11 aa 1 zI 12 aa 2  = + V2a zI 211 aa zI 222 aa = = = + I I I V1a zI 111 a zI 122 a 1 1a 1 b   V= zI + zI = + Network b:  2a 211 a 222 a VVV1 1a 1 b  V= zI + zI = =  1b 111 bb 122 bb I2 I 2a I 2 b = + V= zI + zI V2b zI 211 bb zI 222 bb 1b 111 b 122 b  = +  = + VVV2 2a 2 b V2b zI 21 b 1 zI 22 b 2 = = I1 I 1a I 1 b = = I2 I 2a I 2 b Two-port Networks - sites.google.com/site/ncpdhbkhn 49 Series (3) V= zI + zI  1a 11 a 1 12 a 2 Network a: = + V2a zI 211 a zI 222 a  = + = = V1b zI 11 b 1 zI 12 b 2 I1 I 1a I 1 b Network b:  V= zI + zI  = +  2b 21 b 1 22 b 2 VVV1 1a 1 b  VVV= + = = 1 1a 1 b I2 I 2a I 2 b = +  = + VVV2 2a 2 b VVV2 2a 2 b VVV=+=( z + zIz )( + + zI ) → 1 1ab 1 11 ab 111 12 ab 122  =+= + + + VVV2 2ab 2( z 21 ab zIz 211 )( 22 ab zI 222 ) Two-port Networks - sites.google.com/site/ncpdhbkhn 50 Series (4) =+= + + + VVV1 1ab 1( z 11 ab zIz 111 )( 12 ab zI 122 )  =+= + + + VVV2 2ab 2( z 21 ab zIz 211 )( 22 ab zI 222 ) Vz+ z z + z   I I ↔1 =11a 11 b 12 a 12 b 1 = []z 1 + +   V2 z21a z 21 b z 22 a z 22 b   I2 I 2 zz   zz  [][]=11aa 12 = 11 bb 12 za ; z b   zz21aa 22   zz 21 bb 22  [ ] =[ ] + [ ] z za z b Two-port Networks - sites.google.com/site/ncpdhbkhn 51 Series (5) I1 I1a I2a I2 + + + + Network V V1a a 2a – – I1 I2 + + I I 1a 2a [z] V1 V2 V1 V2 – I1b I2b – + + I I Network 1 2 V1b b V2b – – – – [ ] =[ ] + [ ] z za z b I1 I1b I2b I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 52 Parallel I1a I2a + + [y ] a V2a V1a I – 1 – I2 I1 I2 + + + + I1a I2a V [y] 1 V2 V1 V2 – – – I1b I2b – + + I1 I2 [yb] V V1b 2b – – [ ] =[ ] + [ ] y ya y b I1b I2b Two-port Networks - sites.google.com/site/ncpdhbkhn 53 Cascade I1 I1a I2a I1b I2b I2 + + + + + + [Ta] [Tb] V1 V1a V2a V1b V2b V2 – – – – – – I1 I1a I2a I1b I2b I2 I1 I2 + + [T] [TTT] = [ ][ ] V1 V2 a b – – I1 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 54 Hybrid 1 I1 I1a I2a + + + [ha] V1a V2a – – I 2 I1 I2 + + + I I 1a 2a V [h] V V V2 1 2 – 1 – I1b I2b – + + I1 I2 [hb] I2 V1b V2b – – – [ ] =[ ] + [ ] h ha h b I1 I1b I2b Two-port Networks - sites.google.com/site/ncpdhbkhn 55 Hybrid 2 I1a I2a I2 + + + [ga] V1a V2a I – 1 – I1 I2 + + + I I 1a 2a [g] V V2 V1 V2 – 1 – – I1b I2b + + I1 I2 I1 [gb] V1b V2b – – – [ ] =[ ] + [ ] g ga g b I1b I2b I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 56 Ex. 1 Interconnection of Networks (1) o + E = 100 0V;Z = 5; Ω 30 20  + 1 z = = Ω =− Ω   I1 I2 Z2j10; Z 3 j 20; 20 50  V [z] V 1– 2 Find currents? Z1 – Method 1 Z2 + – 1 Z3 2 E ++ + = 1:ZIVZIIE11 1 31 ( 2 ) I 2:ZI++ V ZI ( + I )0 =  1 22 2 31 2 → I2 V=30 I + 20 I  1 1 2 I= I + I  = + Z 3 1 2 V220 I 1 50 I 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 57 Ex. 1 Interconnection of Networks (2) o + E = 100 0V;Z = 5; Ω 30 20  + 1 z = = Ω =− Ω   I1 I2 Z2j10; Z 3 j 20; 20 50  V [z] V 1– 2 Find currents? Z1 – + + Method 2 Z2 I1b I2b + = + – V1b V V1bz 111 bb I z 122 bb I Z [zb] 2b  E 3 – – = + Z3 V2b zI 21 bb 1 zI 22 bb 2 == += + + + V1bb V 2 ZI 31( bb I 2 ) ZI 31 b ZI 32 b I I 1 [z ] = 2 Z Z  −j20 − j 20  n → =3 3 = Z1 zb     Z Z −j20 − j 20  V' [z + zb] V' 3 3  1 2 Z2 − − + 30j 20 20 j 20  – →[] = zn   – 20−j 20 50 − j 20  E – Two-port Networks - sites.google.com/site/ncpdhbkhn 58 Ex. 1 Interconnection of Networks (3) o + = = Ω + E 100 0V;Z1 5; 30 20  z = I = Ω =− Ω   1 I2 Z2j10; Z 3 j 20; 20 50  [zn] = Find currents? Z1 V' [z + zb] V' Method 2 1 2 Z2 + ZI+ V = E – – 1 1 1 – E + = ZIV2 2 2 0 30−j 20 20 − j 20  [] = V=−(30j 20) I +− (20 j 20) I zn    1 1 2 20−j 20 50 − j 20  =− +− V2(20j 20) I 1 (50 j 20) I 2 I →  1 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 59 Ex. 2 Interconnection of Networks (4) = o = Ω E 200 0V;Z S 5; 1.50− 0.50  y =   = Ω =− Ω − IS I IL ZFj10; Z L j 20; 0.50 0.83  I1 F ZF I2 Find I ? + s Z + Z IVV'''=−( ) /j 10 =− j 0.10 V ' + j 0.10 V ' S L 112 1 2 V [y] V –  – 1 2 '''=− = ' − ' + IVV221( ) /j 10 j 0.10 V 1 j 0.10 V 2 − j0.10 j 0.10  – →y ' =   E j0.10− j 0.10  1.50−j 0.10 − 0.50 + j 0.10  Z → =+ = IF F [y' ] yT y y '   −0.50 +j 0.10 0.83 − j 0.10  I I S L I I S I1 I2 L + + ZS + + ZS ZL + V1 [yT] V2 ZL V [y] V – – 1 2 + – – E – – E Two-port Networks - sites.google.com/site/ncpdhbkhn 60 Ex. 2 Interconnection of Networks (5) = o = Ω E 200 0V;Z S 5; 1.50− 0.50  y =   = Ω =− Ω − IS I IL ZFj10; Z L j 20; 0.50 0.83  I1 F ZF I2 Find I ? + s Z + Z − − + S L 1.50j 0.10 0.50 j 0.10  V [y] V – = – 1 2 yT   + −0.50 +j 0.10 0.83 − j 0.10   – E + = o 5IS V 1 200 0 = V2 j20 I L I I = − +−+ S L  IS (1.50j 0.10) V1 ( 0.50 j 0.10) V 2  + + −=−+ + −  IL ( 0.50j 0.10) V1 (0.83 j 0.10) V 2 ZS + V1 [yT] V2 ZL – – E – → = + IS 0.0049j 0.86A Two-port Networks - sites.google.com/site/ncpdhbkhn 61 Ex. 1 Interconnection of Networks (4) o + E = 100 0V;Z = 5; Ω 30 20  + 1 z = = Ω =− Ω   I1 I2 Z2j10; Z 3 j 20; 20 50  V [z] V 1– 2 Find currents? Z1 – Method 3 Z2 + – Z Z Z3 a b ? E Zc I a 1 Za Zb I2 Z E  1 I1 Z Z+ Z  c Z = 1 a → 2 V I + a 1 1 1 2 + +  – ZZ+ Z + Z ZZ + I3 Z 1a b 2 c 3 E 3 Two-port Networks - sites.google.com/site/ncpdhbkhn 62 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 63 T & П Networks (1) I1 I2 + + 1. Find the [ x’] of the T (or П) [x] V1 V2 network – – 2. Let [ x] = [x’] I1 I2 3. Find impedances of the T (or П) network Two-port Networks - sites.google.com/site/ncpdhbkhn 64 Ex. 1 T & П Networks (2) 30 20  I I z =   ; find the equivalent T network? 1 2 + 20 50  + ZZZ+  [z] = a c c V1 V2 – z' – +  ZZZc b c  I1 I2 z= z ' I [z’] I + = = Ω 1 2 ZZa c 30 Za 10 +   + → = → = Ω Z Z Zc 20 Zc 20 a b V– V1– Z 2  + =  = Ω c ZZb c 50 Zb 30 Two-port Networks - sites.google.com/site/ncpdhbkhn 65 Ex. 2 T & П Networks (3) 1.5− 0.5  I I y = ; 1 2 −  find the equivalent П network? + 0.5 0.83  + +   Z+ Z V [y] V Za Z c − 1 a c = 1 2 –  1.5 –   Z Z Za Z c Z c  a c y ' =   +   1 − 1 Zb Z c → − =− I    0.5 I1 2 Zc Z b Z c   Zc  Z+ Z b c = I [y’] y= y '  0.83 1 I2  Zb Z c + + Z  = Ω c Za 1 Z Z V– V1– a b 2 → = Ω Zc 2  = Ω Zb 3 Two-port Networks - sites.google.com/site/ncpdhbkhn 66 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 67 Equivalent Two-port Networks of Magnetically Coupled Circuits (1) I1 M I2 + + V=jω L I + j ω M I → 1 11 2 V L1 L2 V  1 2 V=jMω I + jL ω I – 2 1 22 – LLM= −  a 1 → = − LLMb 2 I  1 La Lb I2 LM=  =ω + ω +  c + + V1jLa I 1 jL c ( II 12 )   =jLω( + L ) I + jL ω I V L V →  a c1 c 2 2 c 2 =ω + + ω – – V2jLc( II 12 ) jL b I 2  =ω + ω +  jLcI1 jL( c L b ) I 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 68 Equivalent Two-port Networks of Magnetically Coupled Circuits (2) I M I 1 2 La Lb + + = − LLMa 1 L L LLM= − V1 1 2 V2 Lc b 2 – – = LMc LLM− 2 L = 1 2 Lc a − LM2 LLM− 2 L = 1 2 L L b − a b LM1 LLM− 2 L = 1 2 c M Two-port Networks - sites.google.com/site/ncpdhbkhn 69 Equivalent Two-port Networks of Magnetically Coupled Circuits (3) Ex. Find currents? M R2 L1 a R3 C M L1 L2 L + a C 2 – + + – R3 E3 E1 – + E1 R2 – E b 3 b Two-port Networks - sites.google.com/site/ncpdhbkhn 70 Equivalent Two-port Networks of Magnetically Coupled Circuits (4) Ex. Find currents? M R2 L L C a b R2 L1 L2 L a C c + + R3 – R – E 3 E1 + 1 + – E – 3 E3 b I M I 1 2 La Lb + + = − LLMa 1 = − L1 L2 LLM V1 V2 Lc b 2 – – = LMc Two-port Networks - sites.google.com/site/ncpdhbkhn 71 Equivalent Two-port Networks of Magnetically Coupled Circuits (5) Ex. Find currents? L1 a R3 L L M a b R2 + L Lc C 2 – C + + R3 E3 – – E1 + E1 R2 – E b 3 = − LLMa 1 = − LLMb 2 = LMc Two-port Networks - sites.google.com/site/ncpdhbkhn 72 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 73 Input Impedance (1) I1 I2 + + Z Zeq + V1 V2 Z2 + Z2 – – – E – Eeq I I1 2 Z= Z * Maximum power for Z2 ? 2 eq I1 + = Z Zeq Z in + V1 – E – I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 74 Input Impedance (2) I1 + Z Z Z a b Z Z= Z eq in Zc + V1 – E – (Z+ Z ) Z Z=a c + Z in+ + b Z Za Z c I1 + + 1 Z = + V V Zin 1 open− circuit – – I2 E – I I 1 2 V − Z = open circuit + in + Ishort− circuit Z + – Z V1 + V1 Ishort− circuit – – 1V E – I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 75 Input Impedance (3) I1 I2 + + Z Zeq + V1 V2 Z2 + Z2 – – – E – Eeq I I1 2 Z= Z * Maximum power for Z2 ? 2 eq = ZIVE+ = I1 I2 0 1 1 1 + + I = 0 Z 2 = → V + V V E 2 1 2 eq V= zI + zI – 1 111 122 – E –  = + V2 zI 211 zI 222 I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 76 Ex. 1 Input Impedance (4) 30 20  E = 220 V I1 I2 z =   ; 20 50  Z =15 +j 25 Ω + + Z What Z2 will absorb maximum power from the circuit? + V1 V2 Z2 – – = * E – ZZ2 eq 1 = I Method 1 Zeq I1 2 I2 + + = I I (15j 25)I1 V 1 0 1 2 = + V2 1 Z + V=30 I + 20 I – Zeq 1 1 2 V1  + Z V=20 I + 50 I – 1V 2  2 1 2 – Eeq → = − I2 0.023j 0.002 A I1 →= + Ω →= − Ω Zeq 43.15j 3.75 Z2 43.15j 3.75 Two-port Networks - sites.google.com/site/ncpdhbkhn 77 I1 + Ex. 1 Input Impedance (5) + 30 20  E = 220 V Z = + z   ; V1 Voc = + Ω – – 20 50  Z 15j 25 E – What Z2 will absorb maximum power from the circuit? = * I ZZ2 eq 1 V + = oc Method 2 Zeq Isc Z + + = + + = + V1 Isc (15j 25)I1 V 1 220 (15j 25)I1 V 1 220 – E – = V = 0 I2 0 2 V=30 I + 20 I V=30 I + 20 I  1 1 2  1 1 2 = + = + V220 I 1 50 I 2 V220 I 1 50 I 2 →= − = →=− + =− V2 74.72j 41.51V V oc I2 1.63j 1.10A I sc 74.72− j 41.51 →=Z =+Ω43.31j 3.77 →=Z 43.31 −j 3.77 Ω eq 1.63− j 1.10 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 78 Ex. 1 Input Impedance (5) 30 20  E = 220 V I1 I2 z =   ; 20 50  Z =15 +j 25 Ω + + Z What Z2 will absorb maximum power from the circuit? + V1 V2 Z2 – – = * E – ZZ2 eq (Z+ Z ) Z Method 3 Z=a c + Z I I eq+ + b 1 2 Z Za Z c = Ω Za 10 Z Z Z =20 Ω Z a b c Z = Ω c Zb 30 (15+j 25 + 10)20 →=Z +=+Ω30 43.21j 3.77 →=Z 43.21 −j 3.77 Ω eq 15+j 25 + 10 + 20 2 Two-port Networks - sites.google.com/site/ncpdhbkhn 79 Input Impedance (6) = − V1 AV 2 BI 2  I1 I2 = −  I1 CV 2 DI 2 + + − =V1 = AV 2 BI 2 Linear Z1in V1 V2 Z2 – − – I1 CV 2 DI 2 Network = V2 Z 2 I 2 AZ− B I I2 →Z = 2 1 1in − CZ2 D I1 I2 V DV− BI Z =2 = 1 1 + + 2in − − + I2 CV 1 AI 1 Linear = − Z2 V V V1 Z 1 I 1 1 2 – – Network −DZ − B →Z = 1 2in + CZ1 A I1 I2 Two-port Networks - sites.google.com/site/ncpdhbkhn 80 Input Impedance (7) AZ− B Z = 2 1in − → = B CZ2 D Z1sc = D Z2 0 (short-circuit) AZ− B Z = 2 1in − → = A CZ2 D Z1oc → ∞ C Z2 (open-circuit) −DZ − B Z = 1 − 2in + → = B CZ1 A Z2sc = A Z1 0 (short-circuit) −DZ − B Z = 1 − 2in + → = D CZ1 A Z2oc → ∞ C Z1 (open-circuit) Two-port Networks - sites.google.com/site/ncpdhbkhn 81 Input Impedance (8) B Z = 1sc D  Z Z A = 1sc 1 oc  − A  Z2sc( Z 1 oc Z 1 sc ) Z = 1oc B= − AZ C  2sc →  A − C = = B Z Z2sc  1oc A  B D = − Z −D  1sc Z = 2oc C Two-port Networks - sites.google.com/site/ncpdhbkhn 82 Input Impedance (9) Z1 cZ3 d Z7 e Ex. 2 Find T? a Z4 V1 Z A = Z2 6 Z8 V 2 I =0 2 Z5 b b Z Z A = 1sc 1 oc − Z1sc = ? Z2sc( Z 1 oc Z 1 sc ) = − B AZ 2sc A C = Z1sc = Zab = {[( Z7// Z6// Z5)+ Z3]// Z4// Z2}+Z1 Z1oc B D = − Z1sc Two-port Networks - sites.google.com/site/ncpdhbkhn 83 Input Impedance (10) Z1 cZ3 d Z7 e Ex. 2 Find T? a Z4 Z2 Z6 Z8 Z5 b b Z Z A = 1sc 1 oc − Z1oc = ? Z2sc( Z 1 oc Z 1 sc ) = − B AZ 2sc A C = Z1oc = Zab = [{[( Z7+Z8)// Z6// Z5]+ Z3}// Z4// Z2]+Z1 Z1oc B D = − Z1sc Two-port Networks - sites.google.com/site/ncpdhbkhn 84 Input Impedance (11) Z1 cZ3 d Z7 e Ex. 2 Find T? a Z4 Z2 Z6 Z8 Z5 b b Z Z A = 1sc 1 oc − Z2sc = ? Z2sc( Z 1 oc Z 1 sc ) = − B AZ 2sc A C = Z2sc = Zeb = [{[( Z1// Z2// Z4)+ Z3]// Z5// Z6}+ Z7]// Z8 Z1oc B D = − Z1sc Two-port Networks - sites.google.com/site/ncpdhbkhn 85 Input Impedance (12) Z1 cZ3 d Z7 e Ex. 2 Find T? a Z4 Z2 Z6 Z8  Z Z Z A = 1sc 1 oc 5 Z1sc  − b b  Z2sc( Z 1 oc Z 1 sc )  = − B AZ 2sc →  Z1oc  A C =  Z1oc  B Z2sc D = −  Z1sc Two-port Networks - sites.google.com/site/ncpdhbkhn 86 Two-port Network 1. Introduction 2. Parameters 3. Relationships between Parameters 4. Two-port Network Analysis 5. Interconnection of Networks 6. T & П Networks 7. Equivalent Two-port Networks of Magnetically Coupled Circuits 8. Input Impedance 9. Transfer Function Two-port Networks - sites.google.com/site/ncpdhbkhn 87 Transfer Function (1) = V2 • Voltage transfer function: Kv V1 = I2 • Current transfer function: Ki I1 = V2 • Voltage – current transfer function: Kvi I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 88 Ex. 1 Transfer Function (2) 30 20  = I1 I2 = E 220V z   ; + 20 50  Z =15 +j 25 Ω L + [z] Find Kv, Ki, Kvi ? – E V2 ZL V= zI + zI –  1 111 122 = + I1 I2 V2 zI 211 zI 222 = VE1 V= − Z I  z+ Z 2L 2 IE= 22 L  1 − + E= zI + zI  z1122 z z 12 z 21 z 11 ZL → 111 12 2 →   − −ZI = zI + zI z21  L 2 211 222 IE=  2 − +  z1122 z z 12 z 21 z 11 ZL Two-port Networks - sites.google.com/site/ncpdhbkhn 89 Ex. 1 Transfer Function (3) 30 20  = I1 I2 = E 220V z   ; + 20 50  Z =15 +j 25 Ω L + [z] Find Kv, Ki, Kvi ? – E V2 ZL z+ Z – IE= 22 L 1 − + I z11 z 22 z 12 z 21 z 11 ZL I1 2 −z IE= 21 2 − + z Z z11 z 22 z 12 z 21 z 11 ZL →VE = 21 L 2 − + = − z11 z 22 z 12 z 21 z 11 ZL V2 ZL I 2 V z Z →==K 2 21 L =+0.28j 0.19 v − + V1 z 1122 z z 1221 z z 11 ZL Two-port Networks - sites.google.com/site/ncpdhbkhn 90 Ex. 1 Transfer Function (4) 30 20  = I1 I2 = E 220V z   ; + 20 50  Z =15 +j 25 Ω L + [z] Find Kv, Ki, Kvi ? – E V2 ZL z+ Z – IE= 22 L 1 − + I z11 z 22 z 12 z 21 z 11 ZL I1 2 − z21 − = z21 IE2 →=K =−0.27 + j 0.10 z z− z z + z Z i + 11 22 12 21 11 L ZZ22 L = I2 Ki I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 91 Ex. 1 Transfer Function (5) 30 20  = I1 I2 = E 220V z   ; + 20 50  Z =15 +j 25 Ω L + [z] Find Kv, Ki, Kvi ? – E V2 ZL z+ Z – IE= 22 L 1 − + I z11 z 22 z 12 z 21 z 11 ZL I1 2 z21 ZL = z21 ZL VE2 →K = z z− z z + z Z vi + 11 22 12 21 11 L z22 ZL = V2 =6.60 +j 5.15 Ω K vi I1 Two-port Networks - sites.google.com/site/ncpdhbkhn 92 Ex. 2 Transfer Function (6) = = + Ω I I EZ380V;L 15j 25 ; 1 2 + K =0.28 + j 0.19; Find V2? v + [z] – E V2 ZL – I1 I2 V K = 2 v V →= = + × 1 VKE2 v (0.28j 0.19) 380 = VE1 =107.7 + j 70.5 V → = V2 128.7 V Two-port Networks - sites.google.com/site/ncpdhbkhn 93