The Smith Chart - Part I
Why do we need it?

By David J. Jefferies
D.Jefferies email

mpedance measured at a point along a transmission line depends not only on what is connected to the line, but also on the properties of the line, and where the measurement is made physically, along the transmission line, with respect to the load (possibly an antenna).

The SMITH chart is a graphical calculator that allows the relatively complicated mathematical calculations, which use complex algebra and numbers, to be replaced with geometrical constructs, and it allows us to see at a glance what the effects of altering the transmission line (feed) geometry will be. If used regularly, it gives the practitioner a really good feel for the behaviour of transmission lines and the wide range of impedance that a transmitter may see for situations of moderately high mismatch (VSWR).

Waves and reflection coefficients
At any point along a transmission line, waves travel past an observer in both directions (say from left to right and from right to left). If we add up all the right-travelling waves to get a total resultant forward wave complex amplitude, and then do the same for all the left-travelling waves to get a total resultant backward wave complex amplitude, we can divide the total backward wave amplitude by the total forward wave amplitude to derive a complex dimensionless number gamma which is related to how much is reflected at the end of the line (to the right) and how far away the reflecting point is.

We use complex arithmetic and algebra, as is normal in ac theory, to express both amplitude and phase angle information with a single symbol. A complex number is really just an ordered pair of numbers, having certain algebraic properties. Because there are two quantities we plot a complex number as a point on the x-y plane, sometimes called the Argand diagram, with the x-distance as one of the two quantities (the real part) and the y-distance as the other (the imaginary part). Any point on the Argand diagram with co-ordinates (x,y) has distance (technically called the modulus) from the origin, where the modulus r = sqrt(x.x+y.y), by Pythagoras' theorem, and also an angle (technically called the argument ) where we can say arg(x,y)= phi = arctan (y/x) around from the x-axis. All these quantities may be found with a calculator. Since an ac waveform has magnitude (equivalent to r) and phase angle (equivalent to phi) we see immediately that complex arithmetic and algebra is uniquely equipped to add, subtract, multiply, and divide ac quantities. The multiplication and division is most readily carried out in the polar form of the complex number (r,phi) whereas the addition and subtraction is more easily carried out using the rectangular form (x,y). Most scientific calculators have a key for converting easily between polar and rectangular forms. Impedance also has a polar form as well as a rectangular form; but the rectangular form of (resistance, reactance) is more common.

The SMITH chart lets us relate the complex dimensionless number gamma at any point P along the line, to the normalised load impedance zL = ZL/Zo which causes the reflection, and also to the distance we are from the load in terms of the wavelength of waves on the line.

• Let us digress to discuss "normalised impedance". We find that transmission lines have a property called "characteristic impedance" Zo, which is the square root of the inductance/metre divided by the square root of the capacitance per metre of the cable. The SMITH chart is presented in terms of dimensionless normalised impedance, where the actual impedance of interest is divided by the Zo of the particular line being used. In this way, we can get away with a single SMITH chart calibration for all possible line characteristic impedances. As the reflection coefficient gamma is also a dimensionless number this has the added advantage of simplifying the mathematics and making the understanding easier. Transmission lines having differing values of Zo all behave the same, as far as their normalised impedance properties are concerned.

We can then read off the normalised impedance z at the point P along the line, where the actual impedance zZo is the local ratio of total voltage to total current taking into account the phase angles as well as the sizes. This impedance is what a generator would "see" if we cut the line at this point P and connected the remaining transmission line and its load to the generator terminals.

Why should the impedance we see vary along the transmission line? Well, the impedance we measure is really the total voltage on the line (formed from the sum of forward and backward wave voltages) divided by the total current on the line (formed by the sum of forward and backward wave currents). But taking the direction of positive current to be from left to right (on the centre conductor of a coax cable, eg) then in the backward wave, the current is oppositely directed to the way the wave is flowing and so is a negative current. The voltages however, being measured from centre conductor to braid in each wave, are in the same sense in forward and backward waves. On a lossless line the current in the forward wave is in phase with the voltage, whereas the current in the backward wave is oppositely directed according to our sign convention. When we combine this "sum and difference" of phasors with the progressive phase shift between forward and backward waves (because of the transit time it takes to travel to the load and back compared to a cycle of ac) the ratio of total voltage to total current necessarily depends on where we are along the line. Of course, if we happen to be at the load impedance ZL then the ratio of total voltage to total current is just equal to ZL as it has to be.

Looking at the Figure above, the forward wave voltage and forward wave current phasors are fixed (their ratio is the characteristic impedance of the line, perhaps 50 ohms or 75 ohms for a coax cable) but the backward wave phasors swing around, always with current and voltage oppositely directed, in a circular arc depending on where we are along the line. Thus the total voltage and the total current on the line are both functions of position, and their ratio (the impedance) varies along the line.

The value of gamma (backward wave voltage divided by forward wave voltage) at the load can be taken to be the intrinsic reflection coefficient and we derive gamma at a distance d/(lambda) number of wavelengths away from the load by allowing for the round trip loss, and multiplying by the phase factor exp -{j 2*360 *d/(lambda)} where the angle is in degrees.

Thus if we know the loss per wavelength on the line, and also the reflection coefficient at the end of the line, we can find out the ratio of backward wave amplitude to forward wave amplitude at all points along the line, together with information about the relative phase of the backward wave with respect to the forward wave.

Now, below is a small view of the Smith Chart (for a more legible close-up view, click on the link below this example).

Click Here for a Large View of the SMITH Chart (151 Kb)

In this figure we show a load impedance of 0.3 + j 0.5 (normalised) transformed a distance of 0.12 wavelengths towards the generator, where it measures as 1.6 + j 1.7. Thus, on a coaxial cable of 75 ohms Zo at a frequency of 146 MHz, if we assume the velocity factor of the cable is 0.67 then the velocity of waves on the cable is 20 cms per nanosecond and the wavelength on the cable is 20 * 1000/146 cms or 137 cms or 1.37 metres. Therefore, a load impedance of 22.5 + j 37.5 ohms produces an input impedance on the cable, at a distance of 137 * 0.12 cms or 16.4 cms from the load, of 120 + j 127.5 ohms. We see the dramatic effect of even short lengths (in terms of a wavelength) of cable.

Note 22.5 = 75 * 0.3 etc

What is the SMITH chart?
It is a polar plot of the complex reflection coefficient (called gamma herein), or also known as the 1-port scattering parameter s or s11, for reflections from a normalised complex load impedance z = r + jx; the normalised impedance is a complex dimensionless quantity obtained by dividing the actual load impedance ZL in ohms by the characteristic impedance Zo (also in ohms, and a real quantity for a lossless line) of the transmission line.

The contours of z = r + jx (dimensionless) are plotted on top of this polar reflection coefficient (complex gamma) and form two orthogonal sets of intersecting circles. The centre of the SMITH chart is at gamma = 0 which is where the transmission line is "matched", and where the normalised load impedance z=1+j0; that is, the resistive part of the load impedance equals the transmission line impedance, and the reactive part of the load impedance is zero.

The complex variable z = r + jx is related to the complex variable gamma by the formula

                             1 + gamma 
               z = r+jx =   -----------
                             1 - gamma


       and of course, the inverse of this relationship is

                          z - 1        (r-1) + jx
	       gamma  =  --------   =  ------------
                          z + 1        (r+1) + jx

From this chart we can read off the value of gamma for a given z, or the value of z for a given gamma. The modulus of gamma, which is written |gamma|, is the distance out from the centre of the chart, and the phase angle of gamma, written arg(gamma), is the angle around the chart from the positive x axis. There is an angle scale at the perimeter of the chart.

On a lossless transmission line the waves propagate along the line without change of amplitude. Thus the size of gamma, or the modulus of gamma, |gamma|, doesn't depend on the position along the line. Thus the impedance "transforms" as we move along the line by starting from the load impedance z = ZL/Zo and plotting a circle of constant radius |gamma| travelling towards the generator. The scale on the perimeter of the SMITH chart has major divisions of 1/100 of a wavelength; by this means we can find the input impedance of the loaded transmission line if we know its length in terms of the wavelength of waves travelling along it.

• The plot you usually see is the inside of the region bounded by the circle |gamma| = 1. Outside this region there is reflection gain; in this outside region, the reflected signal is larger than the incident signal and this can only happen for r less than 0 (negative values of the real part of the load impedance). Thus the perimeter of the SMITH chart as usually plotted is the r=0 circle, which is coincident with the |gamma| = 1 circle.
• The r=1 circle passes through the centre of the SMITH chart. The point gamma = 1 angle 0 is a singular point at which r and x are multi-valued.
• The SMITH chart represents both impedance and admittance plots. To use it as an admittance plot, turn it through 180 degrees about the centre point. The directions "towards the generator" and "towards the load" remain in the same sense. The contours of constant resistance and constant reactance are now to be interpreted as constant (normalised) conductance g, and (normalised) susceptance s respectively.
• To see this property of the SMITH chart we note first that the admittance y is the reciprocal of the impedance z (both being normalised). Thus inverting the equation above we see that

  
                               1 - gamma 
                  y = g+js =  ------------
                               1 + gamma

  and this is the same formula that we had above if we make the substitution gamma --> (-gamma). Of course, inverting the SMITH chart is the same as rotating it though 180 degrees or pi radians, since (-gamma) = (gamma)(exp{j pi}).

• Admittance plots are useful for shunt connected elements; that is, for elements in parallel with the line and the load.

Why is one circuit of the SMITH chart only half a wavelength?
We remember that the SMITH chart is a polar plot of the complex reflection coefficient, which represents the ratio of the complex amplitudes of the backward and forward waves.

Imagine the forward wave going past you to a load or reflector, then travelling back again to you as a reflected wave. The total phase shift in going there and coming back is twice the phase shift in just going there. Therefore, there is a full 360 degrees or 2 pi radians of phase shift for reflections from a load HALF a wavelength away. If you now move the reference plane a further HALF wavelength away from the load, there is an additional 360 degrees or 2 pi radians of phase shift, representing a further complete circuit of the complex reflection (SMITH) chart. Thus for a load a whole wavelength away there is a phase shift of 720 degrees or 4 pi radians, as the round trip is 2 whole wavelengths. Thus in moving back ONE whole wavelength from the load, the round trip distance is actually increasing by TWO whole wavelengths, so the SMITH chart is circumnavigated twice.

A note on the precision of the SMITH chart
It might be thought that the SMITH chart is only a rough and ready calculator since points can only be determined and plotted on it to within a certain tolerance depending on the size of the print copy of the chart. However, the angular scale at the edge has divisions of 1/500 of a wavelength (0.72 degrees) and the reflection coefficient scale can be read to a precision of 0.02. A little thought shows that this is quite sufficient for most purposes. For example, if the wavelength in coaxial cable at 1 GHz is 20 cm, the SMITH chart locates the position along the cable to 20/500 cm or 0.4 mm and it is clear to anyone who has handled cable at 1GHz that it cannot be cut to this precision.

Should more precision be required, an enlarged section of the chart can easily be made with most photocopy machines. A corollary of these remarks about precision is that many students over-specify the accuracy of their answers to transmission line problems. Normally 3 significant figures in the reflection coefficient is more than ample; angles can be quoted to the nearest degree and normalised impedances and admittances to about 1%. For, it is going to be very difficult to construct a real circuit which is accurately described by more precision than this.

Since many people now rely on computer modelling of transmission lines, they have lost sight of the precision limits of the descriptions of their physical circuit implementations. If your matching circuit requires parameters to be chosen more closely than about a percent in order to work, you probably won't be able to make it physically.

What are the main advantages of the SMITH chart?
Several other graphical transmission line calculators have been proposed. The SMITH chart is particularly elegant for the following reasons.

• It is a direct graphical representation, in the complex plane, of the complex reflection coefficient.
• It is a Reimann surface, in that it is cyclical in numbers of half-wavelengths along the line. As the standing wave pattern repeats every half wavelength, this is entirely appropriate. The number of half wavelengths may be represented by the winding number.
• It may be used either as an impedance or admittance calculator, merely by turning it through 180 degrees.
• The inside of the unity gamma circular region represents the passive reflection case, which is most often the region of interest.
• Transformation along the line (if lossless) results in a change of the angle, and not the modulus or radius of gamma . Thus, plots may be made quickly and simply.
• Many of the more advanced properties of microwave circuits, such as noise figure and stability regions, map onto the SMITH chart as circles.
• The "point at infinity" represents the limit of very large reflection gain, and so therefore need never be considered for practical circuits.
• The real axis maps to the Standing Wave Ratio (SWR) variable. A simple transfer of the plot locus to the real axis at constant radius gives a direct reading of the SWR.

The list above is by no means exhaustive. There was a very good tutorial introduction to the SMITH chart, published in the UK magazine "Wireless World" in January, February and March 1960, and written by R.A Hickson of the Belling and Lee Company Ltd. In the reprinted form it is 16 pages long and contains all the mathematics and some detailed applications. Recommended.

Below are some useful links to other tools and devices for employing the SMITH chart:

• Postscript SMITH charts and tools.
• GIF and Postscript SMITH charts DJJ site..
• http://www.educatorscorner.com/experiments/spectral/SpecAn9.html
• http://www.ee.surrey.ac.uk/Personal/D.Jefferies/transmission.html

This article is an extended version of David Jefferies' web page at http://www.ee.surrey.ac.uk/Personal/D.Jefferies/smith.html with additional original graphics (edited by antenneX) by the Author

Dr. David J. Jefferies
School of Electronic Engineering, Information Technology and Mathematics
University of Surrey
Guildford GU2 7XH
Surrey
England
Click Here for the Authors' Biography

Send mail to [email protected] with questions or comments.
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Last modified: December 31, 2010