# Impedance Matching

• C. W. Davidson
Chapter

## Abstract

The condition for maximum power transfer between generator and load for a simple system with fixed generator resistance is well known (figure 5.1a). The power transferred to the load is
$$W = {{{V^2}} \over 2}{{{R_1}} \over {{{({R_1} + {R_{\rm{g}}})}^2}}}$$
(5.1)
Differentiating with respect to R1 to find the value that maximises the load power, we have
$${{{\rm{d}}W} \over {{\rm{d}}{R_{\rm{1}}}}} = {{{V^2}} \over 2}\left[ {{1 \over {{{({R_1} + {R_{\rm{g}}})}^2}}} - {{2{R_1}} \over {{{({R_1} + {R_{\rm{g}}})}^3}}}} \right] = 0$$
(5.2)
for a maximum, so that
$${{{R_1}} \over {({R_1} + {R_{\rm{g}}})}} + {1 \over 2},\;\;{\rm{or}}\;\;\;{R_{\rm{1}}} = {R_{\rm{g}}}$$
(5.3)

## References

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J. R. Whinnery, et al., ‘Coaxial-line Discontinuities’, Proc. I.R.E., 32 (1944) p. 695.
2. 2.
B. Easter, The Equivalent Circuit of Some Microstrip Discontinuities’, I.E.E.E. Trans. Microwave Theory and Techniques, 23 (1975) p. 655.
3. 3.
P. I. Somlo, ‘A Logarithmic Transmission-line Calculator’, I.E.E.E. Trans. Microwave Theory and Techniques, 8 (1960) p. 463.
4. 4.
P. I. Day, ‘Transmission-line Transformation between Arbitrary Impedances Using the Smith Chart’, I.E.E.E. Trans. Microwave Theory and Techniques, 23 (1975) p. 772.
5. 5.
T. A. Milligan, ‘Transmission-line Transformation between Arbitrary Impedances’, I.E.E.E. Trans. Microwave Theory and Techniques, 24 (1976) p. 159.
6. 6.
P. Bramham, ‘A Convenient Transformer for Matching Coaxial Lines’, Electronic Engineering (January, 1961) p. 42.Google Scholar
7. 7.
B. J. Minnis, ‘A Printed Circuit Stub Tuner for Micro Integrated Circuits’, I.E.E.E. Trans. Microwave Theory and Techniques, 35 (1987) p. 346.