Smith Chart

 In the previous video we saw how the standing wave is developed in transmission lines as a result of mismatch between the output load and the characteristic impedance of the line.

 We introduced the reflection coefficient, voltage and current standing wave ratios and impedance transformation.

 Today we will introduce an essential tool in microwave engineering that enable us to analyze impedance representation as well as standing wave characteristics on the transmission line. The smith chart.

Using this tool we can get the impedance transformation, the reflection coefficient, the vswr and so on.

 In transmission line theory we usually deal with normalized impedance as we saw in the previous video.

Normalized impedance means dividing the load impedance by the characteristic impedance of the line.

 So each time we say impedance we mean the normalized impedance so that the approach applies to any transmission line with any characteristic impedance.

 Impedance is a complex number consists of real and imaginary parts, resistance and reactance. 

Reflection coefficient is also a complex number with real and imaginary parts. It can also be represented in the polar form (magnitude and phase)

 Each load impedance corresponds to a reflection coefficient govern by this equation. If we have a load impedance represented by one point in the impedance plane, we have a corresponding reflection coefficient represented by a single point in the reflection coefficient plane.

Changing that normalized impedance to a different value, we got a different reflection coefficient. This is what the reflection coefficient equation is telling us. there's one to one relationship between impedance and reflection coefficient.

 In the impedance plane, each vertical line represents a constant resistance impedance. moving along the line we have impedance with the same resistance and variable reactance. That makes the reflection coefficient to be moving on a circle representing that resistance with certain center and radius.

Moving along different line forms a different circle on the reflection coefficient plane that represent that different resistance.

 For passive loads we don’t have negative resistance. We don’t have impedances with negative real parts. so we are limited by the right half of the impedance plane. Our limit is zero resistance line, the imaginary axis. This line represents all reactive loads and represented in the reflection coefficient plane by a circle whose radius is one and centered at the origin.

 That’s what expected all the reactive loads corresponds to a reflection coefficient of magnitude of one and phased determined by the reactance.

 The short circuit load for instance which is a point on the imaginary axis, lies one the unity circle and corresponds to a reflection coefficient of -1 meaning the magnitude is one and the phase is 180 degrees.

 For passive loads the reflection coefficient is less or equal to one or confined in that unity circle representing the imaginary axis of the impedance plane.

 So each resistance line in the complex impedance plane can be mapped to a circle in the reflection coefficient plane with radius and center determined by the resistance.

 Each horizontal line in the impedance plane represents a constant reactance impedance with variable resistance.

Moving along horizontal line forms a circle in the reflection coefficient plane whose radius and center is determined by the reactance value.

 The circles will be completed if we move from –inf to +inf on the impedance plane.

 All the reactance circles are centered on the vertical line u=1. Inductive loads correspond to the upper circles, while the capacitive loads correspond to the lower circles.

 The zero reactance line corresponds to a circle with an infinite radius whose center is also on the line u=1. But we are limited by the right half of the impedance plane for passive loads. And that correspond to reflection coefficient less or equal to 1.

 So we are limited by only the parts of the circles inside the unity circle. Each horizontal line or each reactance can be mapped to a circle in the reflection coefficient plane. The resistance and reactance circles are orthogonal.

 Resistance circles are aligned along the horizontal axis v=0 while the reactance circles are aligned along the vertical axis u=1. Impedance is found by finding the intersection point of the resistance and reactance lines. 

 In the reflection coefficient plane, impedance is found by finding the intersection point of the corresponding resistance and reactance circles. Finding the impedance in the reflection coefficient plane enable us to find the corresponding reflection coefficient on the same graph. So we managed to merge the 2 graphs.

 This is the smith chart. Each point on the smith chart determines the impedance as well as its corresponding reflection coefficient. So smith chart is a mapping of all the impedances in the impedance plane to the reflection coefficient plane.

 Now let’s see how these circles are built up mathematically. The derivation of smith chart in textbooks begin by the impedance - reflection coefficient formula. Substituting the impedance and reflection coefficient by their real and imaginary parts and doing some math rearranging, we got these 2 final equations.

 The real part of the impedance or the resistance is represented in the reflection coefficient plane by a circle having that radius and centered at this point. The imaginary part of the impedance is also represented by a circle have a certain radius and center.

These two equations build up the whole smith chart

For instance from the equation, the zero resistance circle from the equation is centered at the origin and have a radius of one. 

 Sweeping the resistance value we get different circles on the smith chart. As we increase the value of r, the center shifts toward the right and the radius becomes more and more smaller. when r approach infinity, the radius approaches zero and the center reaches the point (1,0).

So r=inf represented as a point at (1,0). This is the upper limit. So infinite resistance is represented as a point at (1,0).

 All the constant resistance, and reactance circles passes through the point (1,0). For the constant reactance circles, the zero reactance circle have an infinite radius and centered at (1, inf) at the same time it passes through the point (1,0) like all other circles.

 So it represented as a horizontal line passing through the point (1,0).

 As we increase x in the positive or inductive direction the center shifts down along the line u=1 and the radius gets smaller and smaller. that keeps happening until we get a dot centered at the point (1,0) representing infinite inductive reactance.

So infinite resistance and reactance are infinitely small circles lying on the same point. (u=1,v=0)

 Increasing x in the negative direction, the capacitive circles are doing the same thing in the opposite direction with a dot at (1,0) representing the infinite capacitive loads.

 Once again we’re interested only on the area inside the unity circle representing reflection coefficient less or equal to one. Other than that reflection coefficient is larger than one meaning negative resistance.

 The impedance is represented as the intersection point of a constant resistance circle and constant reactance circle. Or the solution of these 2 equations together at a specific reflection coefficient.

The short circuit load is the intersection of the zero resistance and zero reactance circles which is the very left point or the (-1,0) point.

 the open circuit load is at (1,0) representing infinite resistance and reactance.

 The upper half of the chart represents inductive loads. And the lower half represents capacitive loads.

The origin or the zero reflection coefficient represents the matched load.

In the next video we will see how to use smith chart in designing transmission lines