 Welcome to the session. Today's topic of the discussion is the transmission line parameters. So you have to study the different parameters and derive the different parameters. My name is Ajit Subash Suryanshi. I am from the WIT Walsh Institute of Technology, Sallapur. So what is the learning outcome of this today's session? At the end of the session, students will be able to derive the different parameters of the transmission line. Well, to proceed further, you should have a basic knowledge of solving the differential equation and you should have a knowledge of circuit analysis, that is, it is a basic level. And you should have that knowledge of a lossy and lossless transmission line equation that I have covered in the previous session. So reflection in the transmission line. So what is the reflection in the transmission line? As we have studied that there is two components in the transmission line equation. One is the transmitted component that is a normal component and other is the reflected components. So the wave travel from the left to right, it is denoted by the minus sign here, that is plus sign here. And wave travel from the opposite direction, it is denoted by the minus sign. So that is called as a reflected wave and this wave travel from the left to right is the transmitted waves. And these red lines are again showing the transmission line. And this is a generator, mostly in the case of the sinusoidal. And this is a generator resistance, which is a complex again. And this is a load, complex load is there. And this is a line of length of L. And please note that here the reference is taken at the side is equal to zero. So reference is starting from the load side, not from the generator side. So this is your generator side and this is your load side. So if we, this point is at the reference as the wave traveling from the left to right, that is in the z direction. So this will be the z, z is equal to zero, so it will be the zero point. And this is the end of the transmission line, it is at the z is equal to minus L. Because we are taking the z is equal to zero at this point. If you take the reference at the generator side, z is equal to zero. This will become z is equal to L. So it is always there is a reflection is studied from the load side. So that's why the load side is always taken as a reference. Please make note of that. And again, have a look at the equation of the transmission line. And this is the voltage equation of the transmission line for lossless transmission line. So there is no alpha parameter is there. Alpha is the attenuation and beta is the phase constant here. So here, as you can see this equation voltage as a function of the z. So this is a v plus e raise to minus j beta z. So which is traveling from the left to right. And this is the v minus e raise to plus j beta z traveling from the right to left. So this is the voltage equation. I have the proof of the equation already derived in the previous sessions. So this is the current equation. And in the current equation, this is z zero. And this is the characteristics impedance of the transmission line. This is the characteristics impedance. And the characteristics impedance is real for a lossless transmission line. Again, we are considering the lossless transmission line for the simplicity. So this characteristic impedance lines is complex in case of the lossy transmission line. So this is the voltage and current equation. Input impedance at z is equal to minus l, which can be given by the ratio of the voltage. So this is a ratio of the voltage at z is equal to minus l. And the ratio of the current divided by the current at z is equal to minus l. So as you can see, this is a ratio of the v at z is equal to minus l. And current i is equal to minus l. So deriving this at z is equal to minus l. Again these are the equations. After putting the z is equal to minus l in this equation, you will get these two equations here. Voltage and current. And after dividing these two equations, you will get the input impedance at the generator side. So this is an input impedance at the generator side. So this is a z in and this is a characteristic impedance. And this is a phase factor. As you can see, this is a plus j beta phase factor phase coefficient. And here minus j beta negative phase coefficient. And here, if you define a reflection coefficient. So what is meant by the reflection coefficient? It is a ratio of the reflected waves to the incident waves. So v minus is the reflected waves and v plus is the incident waves. So this ratio is always a complex number. And at the generator side that is v is equal to l, you will get this kind of the equation. So this is a capital gamma l. Gamma l is equal to v minus divided by v plus. So this is a reflection coefficient at the load. That is z is equal to 0. Also the reflection coefficient that is at the z is equal to 0 can be calculated by putting z is equal to 0 in these two equations. Once again, I will draw your attention towards these two equations. If you put z is equal to 0, you will get the voltage at this particular instant that is at the load side. So by putting z is equal to 0 in these two equations, which will be covered in the next slide. So by putting z is equal to 0 in this equation, you will get this reflection coefficient. So when the reflection coefficient is again, it is a complex number. It is giving the phase and the magnitude part. When it is a minus one, that means it is a complete negative reflection. That means the incident and the reflected waves are 180 degree phase shifted. And when there is a reflection coefficient is 0 in that case, there is a no reflection. That is perfectly matched condition. When reflection coefficient is 1 in that case, you will get the complete positive reflection. So again, putting z is equal to 0 in that voltage equation, you will get. So in this input impedance equation, you will get this too. So at z in at z is equal to 0 is equal to voltage at z is equal to 0 and the current at z is equal to 0. So you will get the z in at the z is equal to 0 is there nothing but the load impedance. So as you can see here, the impedance looking at left from this side is there nothing but the load impedance. When you put z is equal to 0, you will get such kind of the equation V plus V minus V minus I divide by z is equal to 0 and this is for the current. And when you divide this two equation, you will get the z L. Where z L is equal to z L is equal to this z 0 1 plus this reflection coefficient and divided by 1 minus reflection coefficient. After solving this equation for the reflection coefficient, you will get this kind of the equation that is z L minus z 0 is equal divided by the z L plus z 0. So putting this values in this input impedance equation, in this equation, you will get this set of equation. This two set of the equation as you can see here, this is z L minus z 0 z L plus z 0, this kind of the equation you will get. And after solving this equation by using Euler's identity, Euler's identity is e raised to i theta cos theta i plus i sin theta and my e raised to minus cos theta minus i theta cos theta plus minus i sin theta. So by using Euler's identity, this equation that is input impedance further simplified into this equation. So this is input impedance of the lossless transmission line. Once again, I will throw a light on this transmission line. Here this is the input impedance from the generator side and this is the impedance at the load side which is nothing but the z L. So different cases you have to study. In the first cases, when the length of the transmission line is the lambda by 2, that is half wavelength. In that case, you will get the input impedance is exactly equal to the load impedance. So very interesting fact you can have for the half wavelength and it is also called as half wavelength transformer. This kind of the transmission line is called as half wave transformer. When length of the transmission line is lambda by 4, that is also called as a quarter wave transformer. In that case, you will get the input impedance characteristic impedance square divided by z 0. As you can see this input impedance is inversely proportional to the load impedance. And what happens when the length of the transmission line is integer multiple of the lambda by 2. So take a paper piece of the paper and calculate input impedance in this case. When the length of the transmission line is n into lambda by. So again there is this reflection in the transmission line. So case when z L is equal to 0, this is a perfectly matched condition. There is no reflection from the load. And when load impedance is a purely imaginary that is purely reactive input is input impedance in that case also reactive. Voltage standing wave ratio. Next in the voltage standing wave ratio is the ratio which is defined as the voltage maximum magnitude of the voltage maximum divided by magnitude of the voltage minimum. When you solve for the voltage maximum and minimum, you will get something the voltage standing wave ratio is you will get this relation. 1 plus that is magnitude of the reflection coefficient and 1 divided by 1 minus magnitude of the reflection coefficient. So this is a reflection coefficient for the wave and this is at the generator side and this is at the reflector side. This is a load side. As you can see this voltage standing wave ratio is increases because voltage standing wave ratio is the ratio of the maximum divided by minimum. So there are 3 cases in which it has a different voltage standing wave ratio. So third one have a highest voltage standing wave ratio. So these are the reference.