Smith chart :
The Smith chart is plotted on the complex reflection coefficient plane in two dimensions and is scaled in normalised impedance (the most common), normalised admittance or both, using different colours to distinguish between them. These are often known as the Z, Y and YZ Smith charts respectively.[7] Normalised scaling allows the Smith chart to be used for problems involving any characteristic impedance or system impedance, although by far the most commonly used is 50 ohms. With relatively simple graphical construction it is straighforward to convert between normalised impedance (or normalised admittance) and the corresponding complex voltage reflection coefficient.
The Smith chart has circumferential scaling in wavelengths and degrees. The wavelengths scale is used in distributed component problems and represents the distance measured along the transmission line connected between the generator or source and the load to the point under consideration. The degrees scale represents the angle of the voltage reflection coefficient at that point. The Smith chart may also be used for lumped element matching and analysis problems.
Use of the Smith chart and the interpretation of the results obtained using it requires a good understanding of AC circuit theory and transmission line theory, both of which are pre-requisites for RF engineers.
As impedances and admittances change with frequency, problems using the Smith chart can only be solved manually using one frequency at a time, the result being represented by a point. This is often adequate for narrow band applications (typically up to about 5% to 10% bandwidth) but for wider bandwidths it is usually necessary to apply Smith chart techniques at more than one frequency across the operating frequency band. Provided the frequencies are sufficiently close, the resulting Smith chart points may be joined by straight lines to create a locus.
A locus of points on a Smith chart covering a range of frequencies can be used to visually represent:
how capacitive or how inductive a load is across the frequency range
how difficult matching is likely to be at various frequencies
how well matched a particular component is.
The accuracy of the Smith chart is reduced for problems involving a large spread of impedances or admittances, although the scaling can be magnified for individual areas to accommodate these
Using transmission line theory, if a transmission line is terminated in an impedance (
) which differs from its characteristic impedance (
), a standing wave will be formed on the line comprising the resultant of both the forward (
) and the reflected (
) waves. Using complex exponential notation:
and
where
is the temporal part of the wave
is the spatial part of the wave and
where
is the angular frequency in radians per second (rad/s)
is the frequency in hertz (Hz)
is the time in seconds (s)
and 
are constants
is the distance measured along the transmission line from the generator in metres (m)
Also
is the propagation constant which has units 1/m
where
is the attenuation constant in nepers per metre (Np/m)
is the phase constant in radians per metre (rad/m)
The Smith chart is used with one frequency at a time so the temporal part of the phase (
) is fixed. All terms are actually multiplied by this to obtain the instantaneous phase, but it is conventional and understood to omit it. Therefore
and
The variation of complex reflection coefficient with position along the line
The complex voltage reflection coefficient 
is defined as the ratio of the reflected wave to the incident (or forward) wave. Therefore
where C is also a constant.
For a uniform transmission line (in which 
is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is lossy (![clip_image017[1]](http://lh6.ggpht.com/-PC6NPTXt2ks/TlknSsKqeVI/AAAAAAAAAs4/6akl3a1aBZc/clip_image0171_thumb.gif?imgmax=800 "clip_image017[1]")
is finite) this is represented on the Smith chart by a spiral path. In most Smith chart problems however, losses can be assumed negligible (
) and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes
The phase constant ![clip_image018[1]](http://lh4.ggpht.com/-5bySCIV1Jic/TlknaWRdujI/AAAAAAAAAtE/S5gztRoteWU/clip_image0181_thumb.gif?imgmax=800 "clip_image018[1]")
may also be written as
where 
is the wavelength within the transmission line at the test frequency.
Therefore
This equation shows that, for a standing wave, the complex reflection coefficient and impedance repeats every half wavelength along the transmission line. The complex reflection coefficient is generally simply referred to as reflection coefficient. The outer circumferential scale of the Smith chart represents the distance from the generator to the load scaled in wavelengths and is therefore scaled from zero to 0.50.
[edit] The variation of normalised impedance with position along the line
If 
and 
are the voltage across and the current entering the termination at the end of the transmission line respectively, then
and
.
By dividing these equations and substituting for both the voltage reflection coefficient
and the normalised impedance of the termination represented by the lower case z, subscript T
gives the result:
.
Alternatively, in terms of the reflection coefficient
These are the equations which are used to construct the Z Smith chart. Mathematically speaking ![clip_image022[1]](http://lh6.ggpht.com/-MLay2ESwxXY/TlkoJGVUkwI/AAAAAAAAAt0/cOtDqvOF_wU/clip_image0221_thumb.gif?imgmax=800 "clip_image022[1]")
and 
are related via a Möbius transformation.
Both ![clip_image022[2]](http://lh6.ggpht.com/-lht3lSdYoOE/TlkoLzGd9EI/AAAAAAAAAt8/UebaP7-hnMI/clip_image0222_thumb.gif?imgmax=800 "clip_image022[2]")
and ![clip_image038[1]](http://lh3.ggpht.com/-JsB16LEHCJQ/TlkoMg8XRMI/AAAAAAAAAuA/3hJCvycxUqY/clip_image0381_thumb.gif?imgmax=800 "clip_image038[1]")
are expressed in complex numbers without any units. They both change with frequency so for any particular measurement, the frequency at which it was performed must be stated together with the characteristic impedance.
![clip_image022[3]](http://lh6.ggpht.com/-RtCUP1wJibA/TlkoNoAuFjI/AAAAAAAAAuE/ypaIHL8s7f8/clip_image0223_thumb.gif?imgmax=800 "clip_image022[3]")
may be expressed in magnitude and angle on a polar diagram. Any actual reflection coefficient must have a magnitude of less than or equal to unity so, at the test frequency, this may be expressed by a point inside a circle of unity radius. The Smith chart is actually constructed on such a polar diagram. The Smith chart scaling is designed in such a way that reflection coefficient can be converted to normalised impedance or vice versa. Using the Smith chart, the normalised impedance may be obtained with appreciable accuracy by plotting the point representing the reflection coefficient treating the Smith chart as a polar diagram and then reading its value directly using the characteristic Smith chart scaling. This technique is a graphical alternative to substituting the values in the equations.
By substituting the expression for how reflection coefficient changes along an unmatched loss free transmission line
for the loss free case, into the equation for normalised impedance in terms of reflection coefficient
![clip_image036[1]](http://lh6.ggpht.com/-kE0aGVLR0Nk/TlkoRI10lxI/AAAAAAAAAuM/y-lFMMHOdfw/clip_image0361_thumb.gif?imgmax=800 "clip_image036[1]")
.
and using Euler's identity
yields the impedance version transmission line equation for the loss free case:[8]
where 
is the impedance 'seen' at the input of a loss free transmission line of length l, terminated with an impedance 
Versions of the transmission line equation may be similarly derived for the admittance loss free case and for the impedance and admittance lossy cases.
The Smith chart graphical equivalent of using the transmission line equation is to normalise ![clip_image043[1]](http://lh3.ggpht.com/-oxBw5jog7To/Tlkob6j2fhI/AAAAAAAAAug/1UUy-x2KR7Y/clip_image0431_thumb.gif?imgmax=800 "clip_image043[1]")
, to plot the resulting point on a Z Smith chart and to draw a circle through that point centred at the Smith chart centre. The path along the arc of the circle represents how the impedance changes whilst moving along the transmission line. In this case the circumferential (wavelength) scaling must be used, remembering that this is the wavelength within the transmission line and may differ from the free space wavelength.
[edit] Regions of the Z Smith chart
If a polar diagram is mapped on to a cartesian coordinate system it is conventional to measure angles relative to the positive x-axis using a counter-clockwise direction for positive angles. The magnitude of a complex number is the length of a straight line drawn from the origin to the point representing it. The Smith chart uses the same convention, noting that, in the normalised impedance plane, the positive x-axis extends from the center of the Smith chart at 
to the point 
. The region above the x-axis represents inductive impedances and the region below the x-axis represents capacitive impedances. Inductive impedances have positive imaginary parts and capacitive impedances have negative imaginary parts.
If the termination is perfectly matched, the reflection coefficient will be zero, represented effectively by a circle of zero radius or in fact a point at the centre of the Smith chart. If the termination was a perfect open circuit or short circuit the magnitude of the reflection coefficient would be unity, all power would be reflected and the point would lie at some point on the unity circumference circle.
Q-factor: The Q-factor is a measure of the frequency selectivity of a resonant or
Antiresonent circuit, and it is defined as
Maximum energy stored
Q= 2π ----------------------------------
Energy dissipated per cycle
ωW
= ------
P
Where W is the maximum energy stored and P is the average power loss.
At resonent frequency, the electric & magnetic energies are equal and intime quadrature.When the electric energy maximum, the magnetic energy is zero and vice-versa. The total energy stored in the resonator is obtained by integrating the energy density over the volume of the resonator.
W е =∫ ε/2 │E│²dv=wm=∫ μ/2 │H│²dv=W
v v
where , │E│ and │H│are the peak values of the field intensities.
The average power loss in the resonator can be obtained by integrating the power density over the inner surface of the resonator:
Hence, p=Rs/2∫ │Ht│²da
Where,Ht is the peak value of the tangential magnetic intensity and Rs is the surface of the resonator.
ωµ ∫│H│²dv
v
Now, Q= -------------------
Rs ∫│Ht│²da
s
Since the peak value of the magnetic intensity is related to its tangential and normal components by
│H│²=│Ht│²+│Hn│²
where, │Hn│ is the peak value of the normal magnetic intensity, the value of│Ht│²at the resonator walls is approximately twice the value of│H│²averaged over the volume.
So, the Q of a cavity resonator can be expressed approximately by
ωμ(volume)
Q= --------------------------
2Rs(surface areas)
An unloading resonant circuit can be represented by either a series or a parallel resonant ckt. The resonant frequency and the unloaded Qο of a cavity resonator are
fο=1/2π√2c
Qο=ωοL/R
Loaded Ql of the system is given by:
Ql=ω○L/(R+N²Zg) for [N²Ls]<<[R+N²Zg]
The coupling coefficient of the system is defined as:
K=N²Zg/R
And
The Ql=ωοL/R(1+K)=Qο/1+K
There are three types of coupling:
1. Critical coupling :if the resonator is matched to the generator, then K=1 then Ql=1⁄2Qο
2. Over coupling: if k>1, the cavity terminals are at a voltage maximum in the input line at the resonance.The normalized impedance at the voltage is the standing wave ratio.That is K=P
The loaded Ql=Qο/1+p
3 Under coupling : if k<1, the cavity terminals are at a voltage minimum and the input terminal impedance is equal to the reciprocal of the standing-wave ratio.
