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Fig. 300
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Fig. 301
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As Figs. 301 and 302 show, Fig. 300 can be considered to consist of two two-ports in series; in Fig. 302 note that N2 denotes the single shunt impedance Z and N1 denotes the transistor.
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* See note 30 in Appendix.
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Matrix Algebra. Networks
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From our work in section 11.9 concerning series-connected two-ports, we know that the general matrix representation for Fig. 302 will be equal to the SUM of the matrix representations of N1 and N2 . Thus, since we ll be dealing with the sum of two matrices, and since the matrix representation for N2 is in terms of impedance (eq. (543)), it follows that the matrix representation for N1 (the transistor) must also be expressed in terms of impedance. Therefore, upon making use of eq. (543) and the impedance matrix representation of a transistor (note 30 in Appendix), we have that the matrix equation for Fig. 300 is ! ! ! V1 Z11 Z Z12 Z I1 545 Z21 Z Z22 Z I2 V2 To continue with another example, let us nd the matrix representation for a network consisting of a transistor, in the CE mode, using a collector-to-base feedback impedance of Z ohms (or, if we wish, Y 1=Z mhos), as shown in Fig. 303.
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Fig. 303
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Now, after some thought, we see that the feedback admittance Y can actually be considered to be a two-port network in parallel with the two-port representation of the transistor. This can be seen by redrawing Fig. 303 in the form of Fig. 304, in which we ve added a ctitious ground lead G, to show more clearly that Y can be considered to be a two-port network in its own right. Thus, in Fig. 304, the upper two-port network N1 is the feedback admittance Y, while the lower two-port N2 is the transistor itself; as the gure shows, N1 is in parallel with N2 . Since the two two-ports are connected in parallel, the admittance matrix of the overall equivalent two-port is the sum of the admittance matrices of the individual two-ports. Hence, if Y11 , Y12 , Y21 , and Y22 are the admittance parameters of the transistor, the matrix equation for Fig. 303 is (making use of eq. (544)) ! ! ! Y11 Y Y12 Y V1 I1 546 Y21 Y Y22 Y V2 I2 Lastly, as another example, consider two impedances Z and Z 0 , connected in the L con guration of Fig. 305. As Fig. 306 shows, the L-network can be considered to be a cascade connection of two two-ports in the manner of Fig. 288. Hence the transmission characteristics for the Lnetwork can be expressed in terms of the product of the a parameter matrices of the individual networks 1 and 2; thus ! ! Vo V1 a1 a2 547 I1 Io by eq. (541) in section 11.9.
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CHAPTER 11 Matrix Algebra. Networks
Fig. 305
Fig. 306
Now let us make use of the conversion chart in section 11.8 to write the above aparameter matrices in terms of Z and Z 0 (or Y and Y 0 if we wish). To do this, note that, for network 1, which is the single shunt form of Fig. 298, we ve already found that (eq. (543)) ! ! Z11 Z12 Z Z Z Z Z Z21 Z22 and thus, for the particular case of network 1, it is true that Z11 Z12 Z21 Z22 Z; hence dZ Z 2 Z 2 0 and upon substituting these values into the fth row of the conversion chart we have, for network 1, that ! 1 0 a1 1=Z 1 In a like manner for network 2 (which is of the single series form of Fig. (299)), we ve found that (eq. (544)) ! ! Y11 Y12 Y 0 Y 0 Y 0 Y21 Y22 Y 0 Y0 and thus, for the particular case of network 2, it is true that Y11 Y 0 ; Y12 Y 0 ; Y21 Y 0 ; Y22 Y 0 ; hence dY 0
and upon substituting these values into the fth row of the conversion chart we have, for network 2, that ! ! 1 1=Y 0 1 Z0 a2 0 1 0 1 Thus, substituting into eq. (547), we have that the transmission characteristics for the L-network of Fig. 305 can be expressed in the matrix form ! ! ! ! V1 1 0 1 Z 0 Vo I1 Io 1=Z 1 0 1 that is, V1 I1 ! 1 1=Z Z0 1 Z 0 =Z ! Vo Io ! 548
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