26 vb zb ib 727 equations 724 726 and 727 can be

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Unformatted text preview: a - V1 b I1 + - a I2 V1b - I2 + + V2a - V2 b I2 b b Z11 Z12 b b Z21 Z22 + V2b - - Figure 7.5: Two 2-ports interconnected such that the input and output ports are in series. The series connection results in the relationships: a b a b I1 = I1 = I1 , I2 = I2 = I2 V1 = V1a + V1b , V2 = V2a + V2b . In vector notation the voltage and current vectors must satisfy I = Ia = Ib (7.24) V = Va + Vb . (7.25) The voltage and current vectors for the individual 2-ports must satisfy: Va = Za Ia (7.26) Vb = Zb Ib (7.27) Equations 7.24, 7.26, and 7.27 can be used in equation 7.26 to show: V = {Za + Zb }I. (7.28) Thus, the Z-parameter matrix for the series combination of 2-ports is the sum of the constituent Z-parameter matrices. 7.3.3 Cascaded 2-ports I1 Y1a Y1a 1 2 a a Y2 1 Y2 2 + V1a a I1 + I1 V1 +- + b I1 V1a - + V1b b I1 V1 Y1a Y1a 1 2 Y2a Y2a 1 2 Y1b1 Y1b2 Y2b1 Y2b2 I2 + V2a a I2 - + I2 + b I2 V2a -+ b I2 V2 +- V2b - V2 + + Y1b1 suc1b2 that the input and output ports are in parallel. Yh Figure 7.4: Two 2-ports interconnected V1b V2b Y2b1 Y2b2 - The parallel connection results in the following relationships: Figure 7.4: Two 2-ports interconnec= d su=hVtb ,atVth= iV put anbd output ports are in paral.l17. e na = V V te V a c h (7 el ) 1 1 1 2 2 2 a b a I1 = I1 + I1 , I2 = I2 + I b . The parallel connection results in the following relation2 ships: In vector notation, e.g. V1 = V1a = V1b , V2 = V2a = V2b V1 I Y Y12 V= , a = b 1 , Ya= b 11 I V Y I12 = I1 + I1 ,I2I2 = I2 + I2 .21 Y22 (7.18) (7.17) , (7.18) In vectationstat1onandg7.18 can be written as: equ or no 7. i 7 , e. . V1 V2 I1 a = b Y11 Y12 , I = V = V, Y V = , I2 Y21 Y22 I = Ia + Ib . equations 7.17 and 7.18 can be written as: The voltage and current vectors asso ciated with the individual 2-ports must satisfy V =aVa =aVb I = Y Va I = Ia + b b . b I Ib = Y V . The voltage and current vectors asso ciated with the individual 2-ports must satisfy Equations 7.19, 7.21, and 7.22 can be used in equation 7.20 to show that V= Ia = Ya Va b I = {Y a + Y }V . (7.19) (7.20) (7.19) (7.21) (7.20) (7.22) (7.21) (7.23) Ib = Yb Vb . (7.22) Thus, the Y-parameter matrix for the parallel combination of 2-ports is the sum of the con o i s ent , - . ara a et r 22 c i n b Equatistntu7.19Y7p21, mnde7.matraces.e used in equation 7.20 to show that I = {Y a + Y b }V . (7.23) The voltage and current vectors for the individual 2-ports must satisfy: Va = Za Ia (7.26) a Ib I2 Vb = Zb Ib 1 a I1 b I2 Equations 7.24, 7.26, anda7.27 a an be used in equation 7.26 to show: c + + + AB Ab B b a a{Za + Zb }I. b V V2 = V1 a a V b b CD - - CD 1 - (7.27) + V2b - (7.28) Thus, the Z-parameter matrix for the series combination of 2-ports is the sum of the constituent Z-parameter matrices. 7.3.3 Figure 7.6: Two 2-ports in cascade. Cascaded 2-ports Figure 7.6 shows two 2-ports connected in cascade, i.e. the output of the first 2-port drives 218 llowingf equations aCHAPTER 7. y Ithe iODUCTION-ports:PORT PARAMETERS Thetho input o the second 2rposa. fe - e rt tisfied b NTRndividual 2 TO 2The cascade connection results in the following relationships: a I1 a INa = ABCDa Ob Ta U b a a V Ib I22 = V1 , I2 1= −I1 b b b b I2 (7.29) (7.31) (7.32) V IN = ABCD OUT V2 + + ector IN = a 1 , + utput vector OUT = + a b b Define the input v , and chain parameter AB1 o AB −I2 b Using equation 7V1a2 in equatioa I7.30, V2and Vhen using the resultVin equation 7.31 we find .3 n a tb a b 1 2 Db B - A C D Then t- relationshipsC iven in equation 7.29 can be written matrix ABCD = .a he gb C D IN = ABCDa ABCD OUTb . (7.33) as: OUTa = INb (7.30) Fiector7.6:f Twe o-portll i2-pasrt de.e related by the matrix pro duct gure s o th o 2vera s n c o ca ar Thus, the input and output v of the constituent ABCD matrices. The following equations are satisfied by the individual 2-ports: 7.4 Power Gain DefiNaitiABCDaOUTa I n = ons b b (7.31) b (7.3 l ) I e = ABCD OUT One very useful application for thN 2-port parameters defined earlier is the calcu2ation of the Using t quadion 7putinmquatances 0,nand he power gainresultain2eportionhen1 tweminated with inpu e an t out.32 i e pedion 7.3 a d t then using the s of -qua t w 7.3 er find a i arbitrary source and load termina tions, ZS a nd ZL .b Consbder a system consisting of a 2-port a (7.nce IN = ABCD ermina e UT . driven by a source with impedance ZS and tABCD tOd with a load having impeda33) ZL as showhuin tFe iure t7an.d output vectors of the overall 2-port are related by the matrix product T n s, h ig npu .7 of the constituent ABCD matrices. 7.4 ZS Pin Power Gain Definitions Pout DABCD = AD − BC 7.7.1 242 ￿ ￿ Y12 Y22 Z11 Z21 ￿ ￿ Z22 DZ − Z12 DZ 1 h11 − h12 h11 h11 h11 D B − DABCD B ￿ Z12 Z22 DZ DZ Y22 DY Y − D12 Y Y − D21 Y = B B Y11 DY = Dh h22 h12 h22 − h21 h22 1 h22 A C = 1 C DABCD C D C Converting to h-parameters h11 h21 7.7.4 ￿ h12 h22 ￿ = DZ Z22 Z12 Z22 21 − Z22 Z 1 Z22 = 1 Y11 12 − Y11 Y Y21 Y11 DY Y11 = B D DABCD D 1 −D C D Converting to ABCD-parameters A C B D ￿ = = Dh Z11 A h21 CHAPTER 7. 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