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 = R_g = r_i and r₂ = R_L = r_o. Substituting for R_L and r₁ in equation 3-13

Solving in terms of r₁₂r₂₁

Substituting for R_g and r_o in equation 3-21

Again solving in terms of r₁₂r₂₁

Equating equations 3-27 and 3-29

Cross multiplying and cancelling equal terms,

or

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,

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

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

In terms of the stability factor ; the output image resistance

For the typical point-contact transistor,

Forthe typical junction transistor

These values may be checked on the R_L vs r_i and R_g vs r_o characteristics plotted for these typical transistors in Figs. 3-11, 3-12, 3-14, and 3-15.