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[[File:quarter wave impedance transformer.svg|thumb|350px|Using a transmission line as an impedance transformer.]] | |||
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 [[electrical impedance|impedance]]. The device presents at its input the [[dual impedance|dual]] of the impedance with which it is terminated. | |||
It is a similar concept to a [[stub (electronics)|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, ''Z''<sub>0</sub>, input, ''Z''<sub>in</sub> and load, ''Z''<sub>L</sub>, impedances is: <math>\frac{Z_\mathrm{in}}{Z_0}=\frac{Z_0}{Z_L}</math> | |||
==Applications== | |||
[[File:LC to J-inverter filter.svg|thumb|350px|The [[lumped element model|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 model|distributed element]] design.]] | |||
At [[radio frequency|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. It is especially useful for making an [[inductor]] out of a [[capacitor]], since designers have a preference for the latter.<ref name=Matthaei>Matthaei et al, pp.144-149.</ref> | |||
Another application is when [[direct current|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.<ref name=Bhat>Bhat & Koul, p.686.</ref> Likewise, an open circuit can be transformed into a short circuit.<ref>Bhat & Koul, pp.601-602.</ref> | |||
The device can be used as a component in a [[distributed element filter|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 [[power 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<ref>A 2-element-kind network is one consisting of only two kinds of elements, that is, ''[[LC circuit|LC]]'', ''[[RC circuit|RC]]'' or ''[[RL circuit|RL]]'' circuits.</ref> ''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<ref>The ''K'' and ''J'' notation originates in archaic symbols for impedance and admittance respectively. The ''K'' is the same ''K'' that makes an appearance in the well known [[constant-k filter]] and the ''K'' parameter is defined for inverters of [[lumped element model|lumped element]] design in exactly the same way as it is defined for the constant-k filter. For a λ/4 transformer the difference is moot, the same device will serve as a ''K''-inverter with an inverter parameter of ''K''=''Z''<sub>0</sub> or equally as a ''J''-inverter with an admittance inverter parameter of ''J''=''Y''<sub>0</sub>, the characteristic admittance (=1/''Z''<sub>0</sub>).</ref> depending on whether they are inverting a series impedance or a shunt [[admittance]].<ref name=Matthaei/> 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.<ref>Matthaei et al, pp.434-435.</ref> | |||
==Theory of operation== | |||
[[File:SmithChartLineLength.svg|thumb|right|400px|Quarter-wave transformers are illustrated in an [[Smith chart|impedance Smith chart]]. Looking towards a load through a length {{math|''l''}} of lossless transmission line, the normalized impedance changes as {{math|''l''}} increases, following the blue circle. At {{math|''l''{{=}}λ/4}}, the normalized impedance is reflected about the center of the chart.]] | |||
[[File:Transmission line animation open short.gif|thumb|right|300px|Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Colors represent voltages, and black dots represent electrons. 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. This reflects the fact that open circuit ({{math|''Z''{{=}}∞}}) is dual to short circuit ({{math|''Z''{{=}}0}}).]] | |||
A transmission line that is terminated in some impedance, ''Z''<sub>L</sub>, that is different from the [[characteristic impedance]], ''Z''<sub>0</sub>, 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;<ref name=Connor>Connor, pp.13-16.</ref> | |||
:<math>Z_\mathrm{in}=Z_0 \frac{Z_L + Z_0\tanh(\gamma l)}{Z_0 + Z_L\tanh(\gamma l)}</math> | |||
:where ''γ'' is the line [[propagation constant]]. | |||
A very short transmission line, such as those being considered here, in many situations will have no appreciable [[insertion loss|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,<ref name=Connor/> | |||
:<math>Z_\mathrm{in}=Z_0 \frac{Z_L + iZ_0\tan(\beta l)}{Z_0 + iZ_L\tan(\beta l)}</math> | |||
Since ''β'' is the same as the angular [[wavenumber]], | |||
:<math>\beta=\frac{2\pi}{\lambda}\ ,</math> | |||
for a quarter-wavelength line, | |||
:<math>l=\frac{\lambda}{4}\ ,</math> <math>\beta l = {\pi \over 2}\ ,</math> | |||
and the impedance becomes, | |||
:<math>Z_\mathrm{in}=Z_0 \frac{Z_L + iZ_0\tan({\pi \over 2})}{Z_0 + iZ_L\tan({\pi \over 2})}=Z_0 \frac{iZ_0\tan({\pi \over 2})}{iZ_L\tan({\pi \over 2})}=\frac{{Z_0}^2}{Z_L}</math> | |||
which is the same as the condition for dual impedances; | |||
:<math>\frac{Z_\mathrm{in}}{Z_0}=\frac{Z_0}{Z_L}</math> | |||
==Notes== | |||
{{reflist|2}} | |||
==References== | |||
*Bharathi Bhat, Shiban K. Koul, ''Stripline-like transmission lines for microwave integrated circuits'', New Age International, 1989 ISBN 81-224-0052-3. | |||
*F.R. Connor, ''Wave Transmission'', Edward Arnold Ltd., 1972 ISBN 0-7131-3278-7 | |||
*George L. Matthaei, Leo Young and E. M. T. Jones, ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'' McGraw-Hill 1964. | |||
[[Category:Linear filters]] | |||
[[Category:Filter theory]] | |||
[[Category:Analog circuits]] | |||
[[Category:Electronic design]] | |||
[[Category:Distributed element circuits]] | |||
Revision as of 02:35, 30 January 2014
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:
Applications
At 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. It is especially useful for making an inductor out of a capacitor, since designers have a preference for the latter.[1]
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.[2] Likewise, an open circuit can be transformed into a short circuit.[3]
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 power 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[4] 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[5] depending on whether they are inverting a series impedance or a shunt admittance.[1] 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.[6]
Theory of operation
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There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
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There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
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Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.).
A transmission line that is terminated in some impedance, ZL, that is different from the characteristic impedance, 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;[7]
- where γ is the line propagation constant.
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,[7]
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;
Notes
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References
- Bharathi Bhat, Shiban K. Koul, Stripline-like transmission lines for microwave integrated circuits, New Age International, 1989 ISBN 81-224-0052-3.
- F.R. Connor, Wave Transmission, Edward Arnold Ltd., 1972 ISBN 0-7131-3278-7
- George L. Matthaei, Leo Young and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964.
- ↑ 1.0 1.1 Matthaei et al, pp.144-149.
- ↑ Bhat & Koul, p.686.
- ↑ Bhat & Koul, pp.601-602.
- ↑ A 2-element-kind network is one consisting of only two kinds of elements, that is, LC, RC or RL circuits.
- ↑ The K and J notation originates in archaic symbols for impedance and admittance respectively. The K is the same K that makes an appearance in the well known constant-k filter and the K parameter is defined for inverters of lumped element design in exactly the same way as it is defined for the constant-k filter. For a λ/4 transformer the difference is moot, the same device will serve as a K-inverter with an inverter parameter of K=Z0 or equally as a J-inverter with an admittance inverter parameter of J=Y0, the characteristic admittance (=1/Z0).
- ↑ Matthaei et al, pp.434-435.
- ↑ 7.0 7.1 Connor, pp.13-16.