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'''Mismatch loss''' in [[transmission line]] theory is the amount of power expressed in [[decibels]] that will not be available on the output due to impedance mismatches and [[Reflections of signals on conducting lines|signal reflections]].  A transmission line that is properly terminated, that is, terminated with the same impedance as that of the [[characteristic impedance]] of the transmission line, will have no reflections and therefore no mismatch loss.  Mismatch loss represents the amount of power wasted in the system.  It can also be thought of as the amount of power gained if the system was perfectly matched.  [[Impedance matching]] is an important part of RF system design; however, in practice there will likely be some degree of mismatch loss.<ref>Daniels, David J. (2004). Ground penetrating radar (2nd Edition).  Institution of Engineering and Technology. ISBN 978-0-86341-360-5</ref> In real systems, relatively little loss is due to mismatch loss and is often on the order of 1dB.
 
== Calculation ==
Mismatch loss (ML) is the ratio of incident power to the difference between incident and reflected power:
 
[[File:T line img.jpg|thumb|right|Figure 1. Simple circuit showing characteristic impedance Z<sub>o</sub> and the load impedance Z<sub>L</sub>. In a perfectly matched system Z<sub>L</sub>=Z<sub>o</sub>, and there is no mismatch loss.]]
 
:<math>
ML_\mathrm{dB} = 10 \log_{10} \bigg(\frac{P_i}{P_i-P_r}\bigg) \,
</math>
 
:<math>
P_r=P_i - P_d\,
</math>
 
where
 
<math>P_i</math> = incident power<br />
<math>P_r</math> = reflected power<br />
<math>P_d</math> = delivered power (also called the ''accepted power'')
 
The fraction of incident power delivered to the load is
 
[[File:T line img2.jpg|thumb|right|Figure 2. Simple circuit showing incident power, P<sub>i</sub>, on a load. The reflected power will be the difference between P<sub>i</sub> and the power delivered, P<sub>d</sub>.]]
:<math>
\frac{P_d}{P_i}=1-\rho^2
</math>
<br />
where
<math>\rho</math> is the magnitude of the [[reflection coefficient]].  Note that as the reflection coefficient approaches zero, power to the load is maximized.
 
If the reflection coefficient is known, mismatch can be calculated by<br />
<br />
:<math>
ML_\mathrm{dB} = - 10 \log_{10} \bigg(1-\rho^2\bigg) \,
</math>
In terms of the voltage standing wave ratio ([[VSWR]]):
 
:<math>
ML_\mathrm{dB} = - 10 \log_{10} \bigg(1-\bigg(\frac{VSWR-1}{VSWR+1}\bigg)^2\bigg) \,
</math>
<ref>Lo YT and Lee SW 1988 Antenna handbook: Theory, Applications, and Design</ref>
 
==Sources of mismatch loss==
 
Any component of the transmission line that has an input and output will contribute to the overall mismatch loss of the system.  For example, in mixers mismatch loss occurs when there is an impedance mismatch between the RF port and IF port of the mixer.<ref>Carr, Joseph J. (2002). RF components and circuits.  Oxford: Newnes. ISBN 0-7506-4844-9</ref> This is one of the principal reasons for losses in mixers.  Likewise, a large amount of the loss in [[amplifiers]] comes from the mismatch between the input and output.  Consequently, not all of the available power generated by the amplifier gets transferred to the load.<ref>Skolnik, Merrill I. (2001). Introduction to radar systems (3rd Edition).  New York: McGraw-Hill. ISBN 0-07-288138-0</ref> This is most important in antenna systems where mismatch loss in the transmitting and receiving antenna directly contributes to the losses the system—including the system [[noise figure]].  Other common RF system components such as [[Filter (signal processing)|filters]], [[attenuator]]s, [[splitters]], and [[combiner]]s will generate some amount of mismatch loss.  While completely eliminating mismatch loss in these components is near impossible, mismatch loss contributions by each component can be minimized by selecting quality components for use in a well designed system.
 
==Mismatch error==
<ref>White, Joseph F. (2004). High frequency techniques: An introduction to RF and microwave engineering.  Hoboken: Wiley. ISBN 0-471-45591-1</ref> If there are two or more components in cascade as is often the case, the resultant mismatch loss is not only due to the mismatches from the individual components, but also from how the reflections from each component combine with each other.  The overall mismatch loss cannot be calculated by just adding up the individual loss contributions from each component.  The difference between the sum of the mismatch loss in each component and total mismatch loss due to the interactions of the reflections is known as mismatch error.  Depending on how the multiple reflections combine, the overall system loss may be lower or higher than the sum of the mismatch loss from each component.  Mismatch error occurs in pairs as the signal reflects off of each mismatched component.  So for the example in Figure 3, there are mismatch errors generated by each pair of components.<ref>Briggs, John N. (2004). Target detection by marine radar.  Institution of Engineering and Technology. ISBN 978-0-86341-359-9</ref> The mismatch uncertainty increases as the frequency increases, and in wide-band applications.  The phasing of the reflections makes it particularly harder to model.
 
The general case for calculating mismatch error (ME) is:
 
[[File:T line img3.jpg|thumb|Figure 3. Simple circuit showing multiple reflections due to more than 1 mismatched device.]]
 
:<math>
ME_\mathrm{dB} = 20 \log_{10} \bigg(1-\rho_1\rho_2\,e^{-j2\theta}\bigg) \,
</math>
 
where <math>\theta</math> is the complex phase change due to the second reflection
 
==See also==
*[[VSWR]]
*[[Insertion loss]]
*[[Return loss]]
*[[Reflection coefficient]]
 
==References==
{{Reflist}}
 
[[Category:Telecommunications engineering]]

Revision as of 00:33, 1 May 2013

Mismatch loss in transmission line theory is the amount of power expressed in decibels that will not be available on the output due to impedance mismatches and signal reflections. A transmission line that is properly terminated, that is, terminated with the same impedance as that of the characteristic impedance of the transmission line, will have no reflections and therefore no mismatch loss. Mismatch loss represents the amount of power wasted in the system. It can also be thought of as the amount of power gained if the system was perfectly matched. Impedance matching is an important part of RF system design; however, in practice there will likely be some degree of mismatch loss.[1] In real systems, relatively little loss is due to mismatch loss and is often on the order of 1dB.

Calculation

Mismatch loss (ML) is the ratio of incident power to the difference between incident and reflected power:

Figure 1. Simple circuit showing characteristic impedance Zo and the load impedance ZL. In a perfectly matched system ZL=Zo, and there is no mismatch loss.
MLdB=10log10(PiPi−Pr)
Pr=Pi−Pd

where

Pi = incident power
Pr = reflected power
Pd = delivered power (also called the accepted power)

The fraction of incident power delivered to the load is

Figure 2. Simple circuit showing incident power, Pi, on a load. The reflected power will be the difference between Pi and the power delivered, Pd.
PdPi=1−ρ2


where ρ is the magnitude of the reflection coefficient. Note that as the reflection coefficient approaches zero, power to the load is maximized.

If the reflection coefficient is known, mismatch can be calculated by

MLdB=−10log10(1−ρ2)

In terms of the voltage standing wave ratio (VSWR):

MLdB=−10log10(1−(VSWR−1VSWR+1)2)

[2]

Sources of mismatch loss

Any component of the transmission line that has an input and output will contribute to the overall mismatch loss of the system. For example, in mixers mismatch loss occurs when there is an impedance mismatch between the RF port and IF port of the mixer.[3] This is one of the principal reasons for losses in mixers. Likewise, a large amount of the loss in amplifiers comes from the mismatch between the input and output. Consequently, not all of the available power generated by the amplifier gets transferred to the load.[4] This is most important in antenna systems where mismatch loss in the transmitting and receiving antenna directly contributes to the losses the system—including the system noise figure. Other common RF system components such as filters, attenuators, splitters, and combiners will generate some amount of mismatch loss. While completely eliminating mismatch loss in these components is near impossible, mismatch loss contributions by each component can be minimized by selecting quality components for use in a well designed system.

Mismatch error

[5] If there are two or more components in cascade as is often the case, the resultant mismatch loss is not only due to the mismatches from the individual components, but also from how the reflections from each component combine with each other. The overall mismatch loss cannot be calculated by just adding up the individual loss contributions from each component. The difference between the sum of the mismatch loss in each component and total mismatch loss due to the interactions of the reflections is known as mismatch error. Depending on how the multiple reflections combine, the overall system loss may be lower or higher than the sum of the mismatch loss from each component. Mismatch error occurs in pairs as the signal reflects off of each mismatched component. So for the example in Figure 3, there are mismatch errors generated by each pair of components.[6] The mismatch uncertainty increases as the frequency increases, and in wide-band applications. The phasing of the reflections makes it particularly harder to model.

The general case for calculating mismatch error (ME) is:

Figure 3. Simple circuit showing multiple reflections due to more than 1 mismatched device.
MEdB=20log10(1−ρ1ρ2e−j2θ)

where θ is the complex phase change due to the second reflection

See also

References

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  1. ↑ Daniels, David J. (2004). Ground penetrating radar (2nd Edition). Institution of Engineering and Technology. ISBN 978-0-86341-360-5
  2. ↑ Lo YT and Lee SW 1988 Antenna handbook: Theory, Applications, and Design
  3. ↑ Carr, Joseph J. (2002). RF components and circuits. Oxford: Newnes. ISBN 0-7506-4844-9
  4. ↑ Skolnik, Merrill I. (2001). Introduction to radar systems (3rd Edition). New York: McGraw-Hill. ISBN 0-07-288138-0
  5. ↑ White, Joseph F. (2004). High frequency techniques: An introduction to RF and microwave engineering. Hoboken: Wiley. ISBN 0-471-45591-1
  6. ↑ Briggs, John N. (2004). Target detection by marine radar. Institution of Engineering and Technology. ISBN 978-0-86341-359-9
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