TABLE OF CONTENTS Over limited bandwidth there are important lumped (L-C) and distributed (transmission-line) equivalences. For example, a shunt inductor may be replaced with a shorted transmission line stub. The equivalent inductive reactance of a shorted stub less than 90 long is
| (2.27) |  |
where
| (2.28) |  |
| (2.29) |  |
Similarly, an open stub less than 90 long may replace a capacitor. The equivalent capacitive reactance is
| (2.30) |  |
The reactance of inductors and capacitors vary linearly with frequency over the entire frequency range for which component parasitics are not a problem. From the above expressions we see the reactance of transmission line stubs are trigonometric functions of frequency which are linear and therefore simulate lumped reactance when the electrical length is short. The error is about 1% at 10 and 10% at 30 . The reactance is predicted accurately by the above equations for any length less than 90 . It is not absolutely necessary that the reactance varies linearly with frequency unless the oscillator is to be tuned over a wide frequency range. Electrical lengths of 45 or even 60 are sometimes used. However, as the length approaches 90 , the reactance approaches infinity. Unlike lumped elements, transmission line elements do not have unique solutions for Z o and ? e. For example, 50 ohms of inductive reactance is simulated with a 50 ohm shorted stub 45 long or a 100 ohm shorted stub 26.56 long.
The equations above describe the equivalence between a single lumped and a distributed element. A distributed...
