In order to represent negative resistances we need to compress the conventional Smith Chart to be a subset of a larger chart, which typically has a radius of |Γ| = 3.16, this value being chosen to represent 10 dB return gain. Pozar).Negative resistance values plotted on a Smith Chart lie outside the |Γ| = 1 boundary of the conventional Smith Chart. (Microwave Engineering, 4th edition, David M. The slotted line of the manufacturer Festo. And knowing the reflection coefficient we can find the load impedance. The voltage minimum repeats every half wavelength. The voltage minimum should occur when e ( θ – 2 β l ) = – 1, s o θ = π – 2 β l m i n. So, the reflection coefficient of the transmission line is | Г | = S W R – 1 S W R + 1 , where the SWR is the standing wave ratio. Let’s assume that we measured the SWR and l m i n, the distance to the first voltage minimum across the terminated line. This allows us to determine the load impedance, and as the load impedance is the complex number, then the slotted line should measure two quantities.įigure 2 depicts the usual slotted line.
#USE THE SMITH CHART TO DETERMINE THE INPUT IMPEDANCE HOW TO#
Below are examples of how to use the Smith Chart with real components.Ī slotted line, used in the RF measurements, consists of a probe (waveguide or coaxial line), allowing the sampling of the electric field amplitude of the standing wave of the terminated line. The Smith Chart usually has both impedance and admittance scales. The Smith Chart can also be used for resolving admittance of the transmission line, as admittance and impedance are related to each other. The rotation angles of the plot are related to the wavelength of the transmission line. This result is similar to the normalised impedance, the difference is only in the rotating angle of the plot. The Smith Chart can be very useful to resolve graphically the transmission line equation, so the input impedance of the transmission line is: Z i n = Z 0 1 + Г e – 2 j β l 1 – Г e ( – 2 j β l, where Г is the reflection coefficient of the transmission line and l is the length of the transmission line. We can see that all the reactance circles lie on the vertical line Г r = 1 , and all the resistance circles lie on the horizontal line Г i = 0. These equations can be rearranged to represent the resistance and reactance circles on the Smith Chart: Г r – r L ( 1 + r L ) 2 + Г i 2 = 1 ( 1 + r L ) 2, ( Г r – 1 ) 2 + ( Г i – 1 x L ) 2 = ( 1 x L ) 2. Resolving the real and imaginary parts of impedance are: r L = 1 – Г r 2 – Г i 2 ( 1 – Г r ) 2 + Г i 2, x L = 2 Г i ( 1 – Г r ) 2 + Г i 2. r L + j x L = 1 + Г r + j Г i 1 – Г r – j Г i. So let’s represent the reflection coefficient and load resistance as complex values: Г = Г r + j Г i, z L = r L + j x L. So the reflection coefficient Г = z L – 1 z L + 1 = | Г | e j θ, here z L = Z L Z 0, or z L = 1 + | Г | e j θ 1 + | Г | e j θ.Īt the beginning of the RF and Microwave course we showed that all the quantities can be represented with their real and imaginary parts. Regarding the impedance, it is better to use normalised quantities z = Z Z 0, where is an impedance, Z 0 is a lossless transmission line impedance. To do so, impedance or admittance circles should be depicted in the chart.
A useful aim of the Smith Chart is that it can convert reflection coefficients to the normal impedances and reverse. And θ is the angle between the reflection coefficient vector and the chart horizontal diameter, -180<θ<180. The reflection coefficient is in the polar form Г = | Г | e j θ, where | Г | is the reflection coefficient magnitude radius from the plot centre.
The Smith Chart is based on the polar system of the voltage reflection coefficient Г. Smith, and despite of being developed so long ago, it is still in use and is the most famous and helpful tool among engineers. It is a very useful tool for solving the problems of RF engineers with regards to transmission lines.
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