hello and welcome to this presentation understanding the Smith chart in the short presentation will give you an overview of what the Smith chart is and how complex impedance azar represented on a Smith chart if you don't already have a background in network analysis you may want to watch the presentations understanding s-parameters and understanding viz warren return loss before beginning this presentation the smith chart is named after its inventor philip Hager Smith who originally described it in an article in electronics magazine in January 1939 there are many applications of the Smith chart Smith himself wrote a 200 page book describing different ways of using his creation the most common applications of the Smith chart involved in peat in Smashing or the design of matching networks before modern computational resources this is a non-trivial math intensive task so the greatest advantage of the Smith chart was that it allowed promise to be solved graphically that is with a compass a ruler and a pencil as you might imagine this is somewhat less important now than it used to be so why study or use the Smith chart in modern times as we'll see the Smith chart is still very useful in terms of visualizing complex impedances especially as a function of frequency and the Smith chart is widely used when tuning or verifying the performance of matching networks the Smith chart is used when making one-port measurements that is when measuring the reflection coefficients another way of saying this is that the Smith chart shows us the load impedance Z sub L relative to the source impedance C sub 0 a Smith chart makes it easy to visualize these complex impedance values either as a single point or as a line that represents impedance as a function of frequency why not just show these values on our old friend the Cartesian coordinate plane as you should already know a complex impedance consists of a purely resistive part R as well as a reactive part X this reactance can be either inductive or capacitive with the inductance being a positive reactance and capacitance being a negative reactance there are a couple of issues when trying to plot these values using the standard Cartesian ordinate plane first since resistance is always positive will only ever use the right-hand plane more importantly both the impedance and resistance can range from 0 to infinity the Smith chart essentially bends the right half of our Cartesian coordinate plane such that the positive and negative reactance axes are curved around to meet the resistance axis the top half of the Smith chart represents the inductive region and the bottom half represents the capacitive region with a purely resistive axis separating them a complex impedance appears as a point on the Smith chart in the remainder this presentation we're going to go step by step through the Smith chart and explain what all these curved lines and points actually mean we'll start our tour of the Smith chart by looking at the point directly in the middle of the chart let's zoom in a little bit this point is often called the prime center and corresponds to our source impedance e sub 0 in most RF systems our source impedance is a purely resistive 50 ohm load when using the Smith chart we normalize the source impedance to 1 in this case by dividing by 50 so the center of our Smith chart 1.0 corresponds to our purely resistive 50 ohm load if our point moves along the resistive access to 2.0 this would correspond to apo resistance of 2 times 50 or 100 ohms and moving the point to zero point 4 would correspond to a pure resistance of 50 times 0.4 or 20 ohms keep in mind that all values on the Smith chart both resistive and reactive are normalized by the same value this allows the same Smith chart to be used regardless of system impedance for example in systems that have a standard system impedance of 75 ohms remember that in most cases we want the load impedance Z sub L to be matched as closely as possible to the source impedance Z sub 0 since this minimizes the level of reflected power and maximizes power transfer from source to load measured values of Z sub L are plotted on the Smith chart where our normalized source impedance Z sub 0 is always in the center so the closer are measured and plotted Z sub L values are to the center the better our impedance match if our measured load impedance falls in the center of the Smith chart z sub l equals Z Sub Zero and we have a perfect match the farther away our measured value is from the center the higher the degree of mismatch a common golden impedance matching therefore is to find a way of moving Z sub L as close to the center as possible if we look at a trace of Z sub L as a function of frequency then the load is resonant at the frequency where the trace moves through or near the center of the Smith chart let's take a closer look at the resistance axis the only straight line on the Smith chart remember that our normalized purely resistive source impedance is represented by the one in the center and corresponds to a visible R of one moving to the left resistance decreases until it reaches the edge of the circle where resistance equals zero in other words a short-circuit moving to the right the resistance increases until it reaches infinity or an open circuit therefore vis wores infinite on either end of the resistance axis meaning 100% reflected power this is also true at any point along the outer edge of the Smith chart one last note points on the resistance axis are pure resistance with no reactive part most loads however have both a resistive and a reactive part so a complex Z sub L will not lie in the resistive axis how then do we represent the resistive part of a complex impedance the resistive part of a complex load will instead lie somewhere along a so-called resistance circle recall that our prime Center has a normalized resistance of 1 a circle tangent to the right side of the chart which passes through the prime center represents a constant normalized resistance circle of 1 in other words every point along the circle has a normalized real or resistive part with magnitude 1 a similar circle which passes through the resistance axis at zero point 2 represents a normalized resistance of zero point 2 at every point on that circle the same is true if the circle passes through the point 4 on the resistance axis every impedance that lies on this circle will a real part are equal to 4 for any point on the Smith chart we can determine its resistive part by simply following the corresponding resistance circle until it intercepts the horizontal resistance axis and then reading off the value now that we've covered resistance let's explain how reactance is represented on the Smith chart as we just saw the resistance axis is the horizontal line we converted from Cartesian coordinates to the Smith chart by bending or curving the vertical reactance axis into a circle this means that our reactance axis lies along the circumference of the Smith chart if we zoom in a bit we can see the values of normalized reactants indicated along the circumference of the chart for example 0.7 1.0 1.6 3.0 10 etc notice that the values are increasing rapidly as we move towards the right-hand side of the chart similar to the resistant circles the Smith chart also contains reactance curves that show normalized reactance every point along a reactance curve has the same reactive or imaginary part for example every point along this curve will have a normalized reactive part equal to one point zero points lying on this curve will all have a normalized reactance of 3.0 we're looking at the upper half of the Smith chart here so all values of reactants are positive if we're in the lower half of the Smith chart then all values of reactants are negative now that we know what resistance circles and reactance curves are it's easy for us to either plot or interpret complex impedances on the Smith chart let's use the complex impedance 100 plus 75 J as our example first we need to normalize this impedance by dividing both real and imaginary parts by our source impedance C sub 0 we'll assume the standard 50 ohms so our normalized impedance is 2 plus 1.5 J we now find and plot the resistance circle for this normalized resistance this is circle that passes through the point 2.0 on the resistance axis then we find and plot the reactance curve for our normalized reactance this is the curve that touches the circular impedance axis at 1.5 we now can find our impedance at the intersection of these two lines we can reverse this procedure to read a complex and peanuts from a Smith chart we just determined which resistance circular point lies on then determine which reactance curve are on once we have these values we simply multiply by our source impedance C sub zero to obtain the actual value of our complex impedance let's summarize what we've learned first the Smith chart is a way of displaying complex impedances these impedances may be either individual points or lines showing values as a function of frequency the initial motivation for the creation of the Smith chart was that it enables impedance matching and many other things to be done graphically instead of algebraically this was particularly important prior to modern computational methods a Smith chart consists of resistance and reactance axes and resistance circles and reactance curves and lastly remember that values on the Smith chart are always normalized to the system impedance this concludes our presentation understanding the Smith chart thanks for watching