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is the admittance matrix for the equivalent network. Another useful relationship can be found as follows. From eq. (497) we have z I V and hence (by section 11.3) I z 1 V Comparison of the last equation with eq. (503) shows that y z 1 504
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that is, the admittance matrix is the inverse of the impedance matrix (and vice versa, the impedance matrix is the inverse of the admittance matrix). Equations for the experimental determination of the values of the y coe cients can be found in the same general way as for the z coe cients. For example, if we apply a known voltage V1 to terminals 1; 1 in Fig. 277 and measure I1 with terminals 2; 2 shortcircuited, for which V2 0, solution of eq. (500) gives y11 I1 V1 V2 0 505
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which thus de nes an experimental procedure for nding the value of y11 .
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CHAPTER 11 Matrix Algebra. Networks
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Another very useful description of the contents of the box is in terms of the hybrid or h parameters. In this system I1 and V2 are taken as the independent variables, and the simultaneous equations for the equivalent network are written in the form V1 h11 I1 h12 V2 I2 h21 I1 h22 V2 or, in matrix form, V1 I2 506 507
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h12 h22
I1 V2
! 508
Note that eq. (506) says that V1 volts volts volts, and thus, by Ohm s law, we see that h11 has the dimension of ohms, while h12 is simply a dimensionless ratio. Likewise, eq. (507) says that I2 amperes amperes amperes, and thus, by Ohm s law, we see that h21 is a dimensionless ratio while h22 has the dimension of reciprocal ohms ; that is, h22 has the dimension of 1=ohms mhos: In regard to the above, it should be mentioned that the following notation is also often used with the h parameters: h11 hi input impedance in ohms; h12 hr reverse voltage feedback factor; h21 hf forward current transfer ratio, h22 ho output admittance in mhos. It should also be noted here that the h parameters (and also the y parameters) nd especially wide use in practical work involving transistors. One reason for this is that it is relatively easy to nd the values of these parameters by direct experiment in the laboratory. From eq. (508) note that the h-parameter matrix for the equivalent circuit is ! h11 h12 h h21 h22 So far we ve listed three pairs of simultaneous equations that can be used to describe the network inside the box in Fig. 277. Thus, eqs. (494) and (495) constitute the impedance or z form of the equations, eqs. (500) and (501) the admittance or y form, and eqs. (506) and (507) the hybrid or h form. It s also possible to write three more pairs of such equations. Of these three, the two pairs of principal interest are written in terms of what are generally called the g and a parameters, as follows, beginning with the g parameters. I1 g11 V1 g12 I2 V2 g21 V1 g22 I2 and hence, in matrix form, I1 V2 509 510
g11 g21
g12 g22
V1 I2
! 511
Matrix Algebra. Networks
Lastly, the a parameters are de ned by the equations V1 a11 V2 a12 I2 I1 a21 V2 a22 I2 or, in matrix form, V1 I1 ! ! ! 514 * 512 * 513 *
a11 a21
a12 a22
V2 I2
The equations show that the g and a coe cients are hybrid-type parameters; some are measured in units of impedance z, some in units of admittance y, and some are ratios of like quantities and thus dimensionless. Problem 240 Keeping in mind that only like quantities can be added together or set equal to each other, and using the basic relationships i v=z yv and v iz i=y, nd the dimensions of each of the g and a coe cients. Problem 241 Write a relationship that can be used to experimentally nd the value of the y22 coe cient. Repeat for the g21 coe cient. Problem 242 Solve eq. (514) for V2 I2
Problem 243 If A is a square matrix of order n, and if c is a constant coe cient, show that 1 cA 1 A 1 c (In the above, you may nd it convenient to refer to eqs. (482) and (483).) Problem 244 Express the value of each of the four impedance parameters in terms of the admittance parameters. Problem 245 For a certain two-port operating at 30 megahertz (30 MHz), it is found that the values of the admittance parameters are, in mhos, equal to y11 5 4 j3 10 3 y21 10 4 j9 10 3 y12 2 j 10 3 y22 j 6 10 3
Making use of the results of problem 244, nd the value of each of the four z parameters at the same frequency.
* It is convenient, with a coe cients, to use the minus signs as shown. The reason for doing so is explained later on in the discussion of the cascade connection of two-ports. Note that the minus sign appears with I2 in the matrix equation.
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