^ The K and J notation originates in archaic symbols for impedance and admittance respectively.^ A 2-element-kind network is one consisting of only two kinds of elements, that is, LC, RC or RL circuits.The quarter wave transformer is a subset of series line (section) matching methods. The quarter wave transformer is an alternative to a stub but, whereas a stub is terminated in a short (or open) circuit and the length is chosen so as to produce the required impedance transformation, the λ/4 transformer is in series with the load and its length and characteristic impedance are designed to produce the required impedance transformation. The result is that the input impedance of the line (ratio of voltage to current) differs from the characteristic impedance and for a line of length l is given by Z i n = Z 0 Z L + Z 0 tanh ( γ l ) Z 0 + Z L tanh ( γ l ) At the input to the line the reflected voltage adds to the incident voltage and the reflected current subtracts (because the wave is travelling in the opposite direction) from the incident current. This reflects the fact that open circuit ( Z=∞) is dual to short circuit ( Z=0).Ī transmission line that is terminated in some impedance, Z L, that is different from the characteristic impedance, Z 0, will result in a wave being reflected from the termination back to the source. A quarter-wavelength away from the open-circuit, the current and voltage oscillations are exactly the same as at a short-circuit, and vice versa. Black dots represent electrons, and arrows show the electric field. Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). The further the signal is from this frequency the less accurately the impedance transformer will be reproducing the impedance inverter function and the less accurately it will be representing the element values of the original lumped-element filter design. This is because the impedance transformer line only has the correct electrical length of λ/4 at one specific frequency. Filters incorporating λ/4 inverters are only suitable for narrow band applications. Inverters are classified as K-inverters or J-inverters depending on whether they are inverting a series impedance or a shunt admittance. Equally, for bandpass filters, a two-resonator-kind (resonators and anti-resonators) filter can be converted to a one-resonator-kind. The most common use for inverters is to convert a 2-element-kind LC filter design such as a ladder network into a one-element-kind filter. There are many such circuits and the term does not necessarily imply a λ/4 transformer. Inverter is a general term for the class of circuits that have the function of inverting an impedance. Impedance inverters are not to be confused with the more common meaning of power inverter for a device that has the inverse function of a rectifier. The device can be used as a component in a filter, and in this application it is sometimes known as an inverter because it produces the mathematical inverse of an impedance. Likewise, an open circuit can be transformed into a short circuit. Feeding in the DC via a λ/4 transformer will transform the short circuit into an open circuit which has no effect on the signals on the line. An ideal DC voltage source has zero impedance, that is, it presents a short circuit and it is not useful to connect a short circuit directly across the line. Īnother application is when DC power needs to be fed into a transmission line, which may be necessary to power an active device connected to the line, such as a switching transistor or a varactor diode for instance. It is especially useful for making an inductor out of a capacitor, since designers have a preference for the latter. The lumped-element low-pass filter (top) can be converted to a design that eliminates the inductors and contains capacitors only by the use of J-inverters, resulting in a mixed lumped-element and distributed-element design.Īt radio frequencies of upper VHF or higher up to microwave frequencies one quarter wavelength is conveniently short enough to incorporate the component within many products, but not so small that it cannot be manufactured using normal engineering tolerances, and it is at these frequencies where the device is most often encountered.
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