 Welcome to the session. Today's topic of the discussion is the characteristics of the transmission line and its equation. My name is Ajit Subhash Suryanshi and I'm from the Valchian Institute of Technology. So today's topic of discussion that is the characteristics of the transmission line and its equation. So learning objective of this topic is to, at the end of the session, students will be able to derive the different constants of the transmission line. That is the characteristics of the transmission line. So before proceeding further, what is the prerequisite? That is, the knowledge before proceeding further is the student should have a knowledge of solving the differential equation and its basic level and electric circuits solutions. Again, it is a basic level. So transmission line equation. So the before session in the transmission line session, same as the transmission line equation. So in this transmission line equation, the two-wire parallel wire line can be modeled as a resistance, conductance, capacitance and inductance. So here, resistance inductors are distributed over the line and reserved for using the two parallel wire transmission line because it has a simplicity. So as you can see, the differential length can be modeled as inductance and along its fractional length of dz. So we are considering this line is transmitting from the, transmitting in the z direction from the left to right. You can consider in this slide. So Ldz is the differential amount of the inductance and Rdz is the differential amount of the resistance. As you know, the resistance can be modeled. Every conductor has its inbuilt resistance. So larger is the cross-sectional area, smaller is this element. So again, this is the capacitance. So in the two parallel wire lines, as two parallel lines are separated by the dielectric materials, in some time cases, air and other dielectric material. So there is a capacitance chances of the capacitance between these two wires. So and there is a, as there is no material is a perfect dielectric. So there is a leakage between these two wires and it is modeled as a conductance. So as you can see, this transmission line can be modeled as an inductor, registers capacitance and conductance. So conductance gives reason for using the conductance here is for the simplicity it can be added easily in the parallel circuits. So in the last session, we derived that rate of change of the current with respect to space with respect to change in the, so as the line is travelling in the z direction, rate of change of di with respect to dz is given by minus v r into minus r plus j omega l. And rate of change of the current with respect to time can be given by the voltage multiplied by the conductance plus j omega c. So change in the voltage is due to the change in the inductance and the resistance and change in the current is due to the change in the capacitance and the conductance. So these are the equations of the transmission line. And further derivative of this equation, this two equations of voltage and the current equation, it will use the double differential equation. So these are the two double differential equations, equation of the transmission line. And after solving this two equation that is d 2 v by dz square as a function of v. So again, this is a constant and in the current equation, this is a constant. So after solving this two equation differential, this two differential equation, especially the voltage equation, you will get this equation v as a function of z. And as you can see, there is a two components in the, in this waves. First is which is travelling in the left to right that is positive z direction and another component which is travelling in the opposite directions. So where this gamma is called as a propagation constant or it is a constant. So this gamma propagation constant can be given by the under root r plus j omega L multiplied by j plus j omega c. So this is a propagation constant and it is, again it is a characteristics of the transmission line. It depends upon line to line and it is changes to line to, if the parameter or physical parameter of the transmission line changes. One more point, interesting point you can note that this constant, this gamma is the complex quantity. It is not a real quantity. Yes, it is interesting to note that it is a complex quantity. And this gamma as it is a complex quantity, it can be resolved into the two components. That is one is a real component and that is one is the imaginary components. So real component is normally denoted as alpha and imaginary component is denoted by the beta. So gamma can be resolved or gamma can be given by alpha plus j beta. So this is about the propagation constant of the transmission line. So this propagation constant, again this is a propagation constant as you can, yes this is alpha. When you take a real part of this equation, you will get alpha and this alpha is normally called as a attenuation factor or attenuation constant. And this beta is called as a phase constant because as we have traveling from one point to another, it is get attenuated in the practical transmission line. So this is a attenuation factor and this is a phase factor. And you can resolve this to complex quantity into the two parts. That is one is the attenuation part and one more is the phase part. As you can see this alpha and beta, real and imaginary part of the gamma. So for that you have to solve this equation under root r plus j omega l, g plus j omega c. Resolute it with the two components, real component and the complex component, sorry imaginary components. So as you can see this is a diagram actually illustrating the phenomena how the waves is transmitted from one point to another. And here it is not intentionally, it is deliberately not illustrated the beta parts. It is illustrated only the alpha parts. So as wave travels from one point to another, it is let us take it here in the z direction. As you can see this is an input wave without attenuation. And this is the wave which is as we travel from the left to right of in the z direction. It is get attenuated, it is get decreased because due to alpha attenuation factor, the wave is get attenuated and its amplitude becomes less and less as we travel from the one point to from in the z directions. And for the beta, the phase shift of the wave form is changes because of the l and c components. And here the velocity of the wave can be given by the phase velocity and it is omega divided by beta where beta is equal to 2 pi by lambda. So very interesting question. In the practice if you want to example the lossless transmission line. So how you can example it? Think about it, write it on the paper. So how you can model or approximate model the lossless transmission line? So again lossless transmission line. So in the lossless transmission line as it can be modeled as R is equal to 0 and G is equal to 0, G is equal to 0 that means R is equal to infinity in these two cases. So R is 0 it is replaced by the short. So now this transmission line becomes only the inductors and the capacitors. In the lossless transmission line as you can see there is an inductor and the capacitors. And due to the inductor there is a voltage change and due to the capacitor there is a current change. And this is a single element in the transmission line. And writing the by a KVL we can write the for inductor. If you write the KVL for that inductor you will get this equation. And by KCL you will get the current equation. So by differentiating that voltage and current equation you will get the first differential equation of the transmission line. And again differentiating that equation you will get the second differential equation of that transmission line. So when you solve that differential equation you will get this equation. So this is the equation as you can see there is no alpha parts that is attenuation part. There is only the phase part in this equation. One is the traveling and another wave is traveling in the opposite direction. Positive is for the traveling from the z to traveling from the traveling from the left to right. And negative is for traveling from the right to left. And this beta can be given by omega under root I will say. So this is a propagation constant of the transmission line, lossless transmission line. And phase velocity again given by the omega divided by beta. So beta is equal to 2 pi by lambda. So this is the characteristics impedance. And the characteristics impedance of the transmission line can be calculated from the voltage and current equation. And for the lossless transmission line the characteristics impedance after solving this equation you will get the R up plus j omega under root R plus j omega L divided by g plus j omega c. So this is the characteristics impedance for the lossy transmission line. And similarly you can get the characteristics impedance for the lossless transmission line. It is at z0 is equal to under root L by c. So input impedance of the lossless transmission line. Suppose this is a line of a lossless transmission line. At the input side there is a sinusoidal and there is a generator impedance. And in this if you write the equation for the voltage and current you will get the input impedance. And after solving this equation you will get this input impedance. So my question to you is that at what length at what z is equal to L input impedance is equal to characteristics impedance. So in that case the output is the exact replica of the input. Think about it write that. So these are the comparisons of the lossy and the lossless transmission line. And these are some of the few references here.