Quarter wave impedance transformer
Encyclopedia
A quarter-wave impedance transformer, often written as λ/4 impedance transformer, is a component used in electrical engineering consisting of a length of transmission line
or waveguide
exactly one-quarter of a wavelength
(λ) long and terminated in some known impedance
. The device presents at its input the dual of the impedance with which it is terminated. It is a similar concept to a stub; but whereas a stub is terminated in a short (or open) circuit and the length designed to produce the required impedance, the λ/4 transformer is the other way around; it is a pre-determined length and the termination is designed to produce the required impedance. The relationship between the characteristic, Z0, input, Zin and load, ZL, impedances is;
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. It is especially useful for making an inductor
out of a capacitor
, since designers have a preference for the latter. Another 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. 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. 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. Likewise, an open circuit can be transformed into a short circuit.
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. Impedance inverters are not to be confused with the more common meaning of inverter
for a device that has the inverse function of a rectifier
. Inverter is a general term for the class of circuits that have the function of inverting an impedance. There are many such circuits and the term does not necessarily imply a λ/4 transformer. 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. Equally, for bandpass filters, a two-resonator-kind (resonators and anti-resonators) filter can be converted to a one-resonator-kind. Inverters are classified as K-inverters or J-inverters depending on whether they are inverting a series impedance or a shunt admittance
. Filters incorporating λ/4 inverters are only suitable for narrow band applications. This is because the impedance transformer line only has the correct electrical length of λ/4 at one specific frequency. 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.
, Z0, will result in a wave being reflected from the termination back to the source. 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. 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;
A very short transmission line, such as those being considered here, in many situations will have no appreciable loss
along the length of the line and the propagation constant can be considered to be purely imaginary phase constant, iβ and the impedance expression reduces to,
Since β is the same as the angular wavenumber
,
for a quarter-wavelength line,
and the impedance becomes,
which is the same as the condition for dual impedances;
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
or waveguide
Waveguide
A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...
exactly one-quarter of a wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
(λ) long and terminated in some known impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
. The device presents at its input the dual of the impedance with which it is terminated. It is a similar concept to a stub; but whereas a stub is terminated in a short (or open) circuit and the length designed to produce the required impedance, the λ/4 transformer is the other way around; it is a pre-determined length and the termination is designed to produce the required impedance. The relationship between the characteristic, Z0, input, Zin and load, ZL, impedances is;
Applications
At radio frequenciesRadio frequency
Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...
of upper VHF or higher up to microwave
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
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. It is especially useful for making an inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
out of a capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
, since designers have a preference for the latter. Another application is when DC
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...
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
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...
or a varactor diode for instance. An ideal DC voltage source has zero impedance, that is, it presents a short circuit
Short circuit
A short circuit in an electrical circuit that allows a current to travel along an unintended path, often where essentially no electrical impedance is encountered....
and it is not useful to connect a short circuit directly across the line. 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. Likewise, an open circuit can be transformed into a short circuit.
The device can be used as a component in a filter
Distributed element filter
A distributed element filter is an electronic filter in which capacitance, inductance and resistance are not localised in discrete capacitors, inductors and resistors as they are in conventional filters. Its purpose is to allow a range of signal frequencies to pass, but to block others...
and in this application it is sometimes known as an inverter because it produces the mathematical inverse of an impedance. Impedance inverters are not to be confused with the more common meaning of inverter
Inverter
Inverter may refer to* Inverter , a device that converts direct current to alternating current** Inverter , an air conditioner that can continuously regulate its output by altering the compressor speed in response to cooling demand** Uninterruptible power supply, which often are based on an...
for a device that has the inverse function of a rectifier
Rectifier
A rectifier is an electrical device that converts alternating current , which periodically reverses direction, to direct current , which flows in only one direction. The process is known as rectification...
. Inverter is a general term for the class of circuits that have the function of inverting an impedance. There are many such circuits and the term does not necessarily imply a λ/4 transformer. 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. Equally, for bandpass filters, a two-resonator-kind (resonators and anti-resonators) filter can be converted to a one-resonator-kind. Inverters are classified as K-inverters or J-inverters depending on whether they are inverting a series impedance or a shunt admittance
Admittance
In electrical engineering, the admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance . The SI unit of admittance is the siemens...
. Filters incorporating λ/4 inverters are only suitable for narrow band applications. This is because the impedance transformer line only has the correct electrical length of λ/4 at one specific frequency. 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.
Theory of operation
A transmission line that is terminated in some impedance, ZL, that is different from the characteristic impedanceCharacteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
, Z0, will result in a wave being reflected from the termination back to the source. 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. 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;
- where γ is the line propagation constantPropagation constantThe propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...
.
A very short transmission line, such as those being considered here, in many situations will have no appreciable loss
Insertion loss
In telecommunications, insertion loss is the loss of signal power resulting from the insertion of a device in a transmission line or optical fiber and is usually expressed in decibels ....
along the length of the line and the propagation constant can be considered to be purely imaginary phase constant, iβ and the impedance expression reduces to,
Since β is the same as the angular wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
,
for a quarter-wavelength line,
and the impedance becomes,
which is the same as the condition for dual impedances;