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CASCADE CONNECTION OF TWO-PORTS
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By de nition, two-port networks are said to be connected in cascade if the output of the rst is the input to the second, the output of the second is the input to the third, and so on. As an example, a cascade connection of three two-ports is shown in Fig. 288, in which we re using, for each stage, the standard notation of Fig. 277. Note that V1 and I1 are the input voltage and current to the cascade, and we re assuming the cascade is terminated in a load impedance of ZL ohms.
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Fig. 288. Cascade connection of three two-ports.
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In terms of the standard notation of Fig. 277, note that the negative of the output current of each network equals the input current to the next network; that is, from inspection of Fig. 288, I2 I3 I4 I5 I6 I7
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CHAPTER 11 Matrix Algebra. Networks
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and also, from inspection of the gure, V2 V3 ; V4 V5 ; V6 V7
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and so on, for any number of cascaded networks. When two-ports are connected in cascade, it s convenient to use the a parameters, in which the output current of each network is written with the minus sign, that is, as a negative current, in the form of eq. (514) in section 11.6. Thus, letting V1 and I1 denote input voltage and current and V2 and I2 denote output voltage and current, the matrix equation for each individual network in a cascade of two-port networks will be, using a parameters from eq. (514), ! ! V1 V2 a I1 I2 where a11 a a21 a12 a22 !
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Referring to Fig. 288 we therefore have, starting at the left (the input end of the cascade), that the matrix equation for the rst network is ! ! ! V2 V3 V1 a1 a1 538 I1 I2 I3 and then, since V3 I3 eq. (538) becomes V1 I1 and then, since V5 I5 eq. (539) becomes V1 I1 ! a3 ! a1 a2 a3 ! a2 ! a1 a2 ! a3 V4 I4 ! a2 ! 539 ! V5 I5 !
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V7 I7 !
V7 I7
540
which is the nal matrix equation for the 3-network two-port cascade of Fig. 288. It s clear that the foregoing procedure can be continued for any number of two-ports in cascade. Thus, if n two-ports are connected in cascade, and if V1 and I1 are the input voltage and current to the cascade and Vo and Io are the nal output voltage and current, then eq. (540) extends to the general form ! ! Vo V1 a1 a2 a3 an 541 I1 Io that is V1 I1 ! A Vo Io ! 542
Matrix Algebra. Networks
where A is the overall network matrix for n two-ports in cascade. We have thus deduced the important fact that The overall network matrix for two-port networks in cascade is equal to the PRODUCT of the matrices of the individual two-ports if a parameter values are used. Let us note that the nal signs of the currents and voltages in Fig. 288 will depend, in any given case, upon the networks inside the boxes (see third item discussion in nal paragraph of section 11.7). Problem 252 Prove that the two two-ports in Fig. 286 can be replaced by a single equivalent twoport whose h-parameters are equal to the sums of the corresponding h-parameters of the individual two-ports. Problem 253 If three individual two-port networks are connected in parallel, express the parameters of the single equivalent two-port in terms of the z-parameters of the individual two-ports. Problem 254 Two identical two-ports are connected in cascade. Write the matrix expression for the single equivalent two-port in terms of h-parameters. Problem 255 In problem 254, show that the value of the input current I1 , in terms of the hparameters, is equal to I1 1 h11 h22 1 dh h22 VL V1 1 dh h11 dh 2 h11 h22 ZL
Problem 256 Find the expression for the inverse matrix, h 1 , if the answer to problem 254 is written in the form ! ! V1 Vo h Io I1
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