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SERIES CONNECTION OF TWO-PORTS
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Two two-port networks, a and b, are said to be connected in series if the two input circuits are in series and the two output circuits are in series. The basic series connection of two-ports is thus as shown in Fig. 284. In Fig. 284 note that V1 and V2 are the input and output voltages for the overall composite network. It follows that V1 will divide between the two series-connected inputs and V2 will divide between the two series-connected outputs, as shown. Let us now apply the z-form of eqs. (494) and (495), to the individual networks a and b in the gure. Let subscript a apply to network a and subscript b apply to network b. In the gure, note that the same current I1 ows through both inputs and the same
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CHAPTER 11 Matrix Algebra. Networks
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Fig. 284.
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Series connection of two-ports.
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current I2 ows through both outputs; thus, from eqs. (494) and (495) we have eqs. (526) and (527): 8 V1a z11a I1 z12a I2 > > 526 > > > < V2a z21a I1 z22a I2 527 for network a #" # ! " > I1 z11a z12a > V1a > > that is; > 528 : V2a z21a z22a I2 8 > > > > < for network b > > > > that is; : V1b z11b I1 z12b I2 V2b z21b I1 z22b I2 ! ! ! V1b z I z 11b 12b 1 V2b z21b z22b I2 529 530 531
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Now write down the sum of eqs. (526) and (529) and also the sum of eqs. (527) and (530). Doing this, and noting, from Fig. 284, that V1a V1b V1 , and also that V2a V2b V2 , we nd, for the composite network consisting of two series-connected two-ports, that V1 z11a z11b I1 z12a z12b I2 V2 z21a z21b I1 z22a z22b I2 or, in matrix form, V1 V2 ! z11a z11b z12a z12b z21a z21b z22a z22b ! ! 534 532 533
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Equations (532), (533), and (534) show that series-connected two-ports can be replaced with a single equivalent two-port whose z parameters are the sum of the corresponding zparameters of the individual two-ports. This fact can be summarized by writing, for seriesconnected two-ports: z za zb 535 where za and zb are the impedance matrices of the individual series two-ports in eqs. (528) and (531), and where z is the impedance matrix of the single equivalent two-port, which appears in eq. (534). Note that (in accordance with the rule for addition of matrices laid down in section 11.1) the sum of the two impedance matrices in eqs. (528) and (531) does produce the impedance matrix of eq. (534), which is what eq. (535) says. It should
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Matrix Algebra. Networks
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also be apparent from the preceding work that any number n of series two-ports can be replaced by a single equivalent two-port whose impedance matrix is equal to the sum of the impedance matrices of the n individual two-ports. If the values of the z-parameters are not known in a given case but other parameter values are known, the values of the z-parameters can then be found by use of the conversion chart in section 11.8. For example, suppose the h-parameter values are known but the z-parameter values are unknown, inspection of the conversion chart shows that the required z-parameter values can be found, given the h-values, by means of the formulas z11 dh=h22 ; z12 h12 =h22 ; z21 h21 =h22 ; z22 1=h22
PARALLEL CONNECTION OF TWO-PORTS
Two two-port networks, a and b, are connected in parallel if the input lines are connected in parallel and the output lines are connected in parallel. The basic parallel connection of two-ports is therefore as shown in Fig. 285.
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