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Impedance Transfer Ratio In RF and AF systems, transformers are employed to match impedances. Thus, you will sometimes hear or read about an impedance step-up transformer or an impedance step-down transformer. The impedance transfer ratio of a transformer varies according to the square of the turns ratio, and also according to the square of the voltage-transfer ratio. If the primary (source) and secondary (load) impedances are purely resistive and are denoted Zpri and Zsec, then the following relations hold:

Capacitive reactance can be plotted along a half line, or ray, just as can inductive reactance In fact, capacitive and inductive reactance, considered as one, form a whole line that is made of two half lines stuck together and pointing in opposite directions The point where they join is the zero-reactance point This was shown back in Fig 14-1 In a circuit containing resistance and capacitive reactance, the characteristics are two-dimensional, in a way that is analogous to the situation with the RL plane from the previous chapter The resistance ray and the capacitive-reactance ray can be placed end to end at right angles to make a quarter plane called the RC plane (Fig 14-5) Resistance is plotted horizontally, with increasing values toward the right Capacitive reactance is plotted downwards, with increasingly negative values as you go down The combinations of R and XC in this RC plane form impedances.

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Zpri/Zsec = (Tpri/Tsec )2 Zpri/Zsec = (Epri/Esec )2 The inverses of these formulas, in which the turns ratio or voltage-transfer ratio are expressed in terms of the impedance-transfer ratio, are: Tpri/Tsec = (Zpri/Zsec)1/2 Epri/Esec = (Zpri/Zsec)1/2

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

You ll learn about impedance in greater detail in the next chapter Each point on the RC plane corresponds to one and only one impedance Conversely, each specific impedance coincides with one and only one point on the plane Impedances that contain resistance and capacitance are written in the form R jXC Remember that XC is never positive, that is, it is always negative or zero Because of this, engineers will often write R jXC , dropping the minus sign from XC and replacing addition with subtraction in the complex rendition of impedance If the resistance is pure, say R 3 , then the complex impedance is 3 j0 and this corresponds to the point (3,0) on the RC plane You might at this point suspect that 3 j0 is the same as 3 j0 , and that you really need not even write the j0 part at all.

Problem 18-3 Consider a situation in which a transformer is needed to match an input impedance of 50.0 , purely resistive, to an output impedance of 300 , also purely resistive. What is the required turns ratio Tpri/Tsec The required transformer will have a step-up impedance ratio of Zpri/Zsec = 50.0/300 = 1/6.00. From the preceding formulas:

6

In theory, both of these notions are indeed correct But writing the j0 part indicates that you are open to the.

Tpri/Tsec = (Zpri/Zsec)1/2 = (1/6.00)1/2 = 0.166671/2 = 0.408 = 1/2.45

possibility that there might be reactance in the circuit, and that you re working in two dimensions It also underscores the fact that the impedance is a pure resistance 4 , then the complex impedIf you have a pure capacitive reactance, say XC ance is 0 j4, and this is at the point (0, 4) on the RC plane Again, it s important, for completeness, to write the 0 and not just the j4 The points for 3 j0 and 0 j4, and two others, are plotted on the RC plane in Fig 14-6 In practical circuits, all capacitors have some leakage resistance If the frequency goes to zero, that is, if the source is dc, a tiny current will flow because no insulator is perfect Some capacitors have almost no leakage resistance, and come close to being perfect But none are mathematically flawless.

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Problem 18-4 Suppose a transformer has a primary-to-secondary turns ratio of 4.00:1. The load, connected to the transformer output, is a pure resistance of 37.5 . What is the impedance at the primary The impedance-transfer ratio is equal to the square of the turns ratio. Therefore:

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