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The T and Pi Equivalent Networks
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It is often helpful, in the analysis of complicated networks, to replace an actual network with a simpler network that is the EQUIVALENT of the actual network. A second network is said to be equivalent to a rst network if, when the rst is replaced by the second, there is NO CHANGE in the values of the voltages and currents appearing AT THE INPUT AND OUTPUT TERMINALS) (1, 1) and (2, 2). Consider now, Figs. 172 and 173. Let the box in Fig. 172 contain the actual network, which can be any linear, bilateral network we might be interested in. (For linear and bilateral, see footnote in section 4.7). " " Let Vg be the generated voltage of the generator, Zg be the internal impedance of the " " " " "L , be the external load impedance. Also let V1 , I1 , V2 , and I2 be the generator, and Z voltages and currents at the input and output terminals of the actual network, as shown in Fig. 172. Since we re dealing with sinusoidal steady-state analysis, these voltages and
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* The Greek letter m (mu) is here read as micro, and indicates multiplication by 10 6 . Thus, 28.5 mH 28:5 10 6 H (henrys).
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CHAPTER 9 Impedance Transformation
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Fig. 172
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Fig. 173*
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currents will be expressed, as usual, in the form of COMPLEX NUMBERS, each such number requiring the speci cation of TWO SEPARATE VALUES, the magnitude, and the phase angle. Now, in Fig. 172, let us remove the actual network and replace it with the proposed EQUIVALENT NETWORK inside the box in Fig. 173. If the network inside the box in Fig. 173 is a true equivalent network, there will be, at the operating frequency, NO " CHANGE in either magnitude or phase angle in any of the four complex numbers V1 , " " " I1 , V2 , or I2 . Let us now consider what the circuit of an equivalent network would have to be like. We begin with the reasonable assumption that the equivalent network should be as simple as possible. The simplest form of an equivalent network can then be arrived at as follows. First, we have the fact that a complex number is composed of two independent parts, the magnitude and the phase angle. Since we are dealing here with four complex numbers " " " " (V1 , I1 , V2 , I2 ), it follows that there are eight separate quantities (four magnitudes and four phase angles) involved in setting up an equivalent network. Note, however, from Fig. 172, that by Ohm s law " " " V2 I2 ZL " and thus, if the external load impedance ZL is taken into account, the four complex quantities become " " " " V1 ; I1 ; I2 ZL ; and " I2
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Thus only six of the quantities (three magnitudes and three phase angles) need be determined for the network itself, the other two (one magnitude and one phase angle) " being supplied by the known value of the load impedance ZL . Hence the simplest equivalent network need have only three independent, adjustable impedances. Some thought will show that there are only two ways that three such impedances can be arranged, one arrangement being called the T network, the other being called the pi (), as in Figs. 174 and 175. Notes: In all our discussions of T and pi networks we ll use the same standard notation " shown in Figs. 174 and 175. Also, in these gures, each Z can represent any simple series, parallel, or series-parallel connection of impedances. Now let us take up the problem of how to nd the values of the impedances in the equivalent networks above, beginning with the T network.
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* Fig. 173 represents a four-terminal or two-port network. Terminals (1, 1) constitute the input port and (2, 2) the output port.
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