Explanation
Reactance is the imaginary part of the complex electrical impedance. The specification that the network must be passive and lossless implies that there are no resistors (lossless), or amplifiers or energy sources (passive) in the network. The network consequently must consist entirely of inductors and capacitors and the impedance will be purely an imaginary number with zero real part. Other than that, the theorem is quite general, in particular, it applies to distributed element circuits although Foster formulated it in terms of discrete inductors and capacitors. Foster's theorem applies equally to the admittance of a network, that is the susceptance (imaginary part of admittance) of a passive, lossless one-port monotonically increases with frequency. This result may seem counterintuitive since admittance is the reciprocal of impedance, but is easily proved. If an impedance,
- where,
- is impedance
- is reactance
- is the imaginary unit
then the admittance is given by
- where,
- is admittance
- is susceptance
If X is monotonically increasing with frequency then 1/X must be monotonically decreasing. −1/X must consequently be monotonically increasing and hence it is proved that B is increasing also. It is often the case in network theory that a principle or procedure apply equally to impedance or admittance as they do here. It is convenient in these circumstances to use the concept of immittance which can mean either impedance or admittance. The mathematics are carried out without stating which it is or specifying units until it is desired to calculate a specific example. Foster's theorem can thus be stated in a more general form as,
-
- Foster's theorem (immittance form)
- The imaginary immittance of a passive, lossless one-port monotonically increases with frequency.
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