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# Input and Output Impedance Matching

Author: Leonard Krugman

Equations 3-13 and 3-21 indicate that the input resistance is affected by the load resistance and, conversely, the output resistance depends on the generator internal resistance.

Thus, starting with a given load resistance, if the generator resistance is changed to match the input resistance, the output resistance of the transistor changes, thus requiring a change in load resistance, and so on. In the following analysis, the proper values of generator and load resistance which satisfy both the input and output matching conditions at the same time are determined. Let r1 equal the proper value of input resistance and generator resistance. Let r2 equal the image matched value for the transistor output resistance and the load resistance. Then: r1 = Rg = ri and r2 = RL = ro. Substituting for RL and r1 in equation 3-13

 Eq. (3-26)

Solving in terms of r12r21

 (r1 - r11) (r2 + r22) = - r12r21 Eq. (3-27)

Substituting for Rg and ro in equation 3-21

 Eq. (3-28)

Again solving in terms of r12r21

 (r2 - r22) (r1 + r11) = - r12r21 Eq. (3-29)

Equating equations 3-27 and 3-29

 (r1 - r11) (r2 + r22) = (r2 - r22) (r1 + r11) Eq. (3-30)

Cross multiplying and cancelling equal terms,

 r1r2 - r2r11 + r1r22 -r11r22 = r1r2 - r1r22 + r2r11 - r11r22 2r1r22 = 2r2r11 Eq. (3-31)

or

 Eq. (3-32)

This latter equation indicates that matching the input and output resistances for maximum power gain requires their values to be in the same ratio as the open-circuit characteristics of the transistor.

The absolute value of the generator internal resistance and its matched input resistance in terms of transistor open-circuit parameters can now be determined. Substituting the equality into equation 3-26,

 Eq. (3-33) Eq. (3-34) Eq. (3-35) Eq. (3-36)

In terms of the stability factor, , which will be defined later in the chapter, the input image resistance

 Eq. (3-37)

For the typical point-contact transistor previously considered, when r11 = 250 ohms, r12 = 100 ohms, r21 = 24,000 ohms, and r22 = 12,000 ohms, the numerical value of r1 is

For the typical junction transistor, when r11 = 550 ohms, r12 = 500 ohms, r21 = 1,900,000 ohms, and r22 = 2,000,000 ohms,

The output image resistance of a transistor can be determined in a similar fashion from the ratio

Substituting this equality into equation 3-28

 Eq. (3-38)
 Eq. (3-39A) Eq. (3-39B) Eq. (3-40)

In terms of the stability factor ; the output image resistance

 Eq. (3-41)

For the typical point-contact transistor,

Forthe typical junction transistor

These values may be checked on the RL vs ri and Rg vs ro characteristics plotted for these typical transistors in Figs. 3-11, 3-12, 3-14, and 3-15.

Last Update: 2010-11-17