 Hello everyone, myself Piyusha Shedkar. In today's session, we will discuss the characteristics of a wave. These are the learning outcome. At the end of this session, students will be able to define and derive the equations of transmission line parameters. These are the contents. So before going to start the characteristics of the transmission line which are the primary and secondary constant used in transmission line, it should be known to you. So you can pause video here and recall that which are the primary and secondary constants used in transmission line. Yes, the elements used in equivalent model of transmission line are called as primary constants. These primary constants are R, L, C and G that is resistance, inductance, capacitance and conductance. And the constants obtained in analysis process using these primary constants are called as secondary constants which can be denoted with gamma and z0 where gamma is nothing but the propagation constant whereas z0 is the characteristics impedance. By taking the reference of these primary constant that is R, L, C and G, we can define the equations for propagation constant as well as for the characteristics impedance that is gamma and z0. Now what is transmission line? Transmission line is a device for transmitting or guiding electromagnetic energy from one place to another place. And this equation in terms of the R, L, C and G can be defined for the transmission line. The transmission line equation can be defined in terms of voltage as well as in terms of the current by taking the circuit theory and Ohm's law. Now you can refer this figure which is nothing but the equivalent model used for the transmission line whereas R and L are connected in series and capacitor and conductance which are connected in parallel with the input and output. So R and L this is denoted as the primary constant whereas C and G are also the primary constant used in transmission line whereas the R plus j omega L is known as the series impedance and C plus j omega C is known as the shunt admittance. Series impedance can be defined with the letter z whereas C plus j omega C that is shunt admittance can be defined with the y letter. Input can be applied to this one and output can be taken across this C plus j omega C that is across the y admittance. Now let us consider the types of transmission line. There are the different types of the transmission line through which the energy can be transmitted from one place to the other place or from one end to the other end of all these transmission lines. So this is the coaxial cable that is one of the type of a transmission line, microstrip line and also the two wire transmission line is also considered to transmit the energy from one place to other place. Now let us discuss about the different characteristics of the transmission line. So let us consider the first characteristics is nothing but the wavelength of a wave. So what is wavelength? Wavelength is defined as the distance that a wave travels along the line. In order to that the phase shift you are getting equal to 2 pi radians and it is denoted with the letter lambda. So you can refer this figure, the figure which having the sinusoidal waveform which is drawn with respect to the amplitude and the distance. So if you observe this wave, so here you are getting the total amplitude, max value of the amplitude at the positive side and max value of the amplitude at the negative side, generally it is denoted with the plus Vm and the minus Vm sign. So the distance covered from this first max value at the positive side to the second max value at the negative side. This distance is covered by the wave which is denoted with the lambda letter. So lambda is nothing but the wavelength. So here this is the distance along the line given for which the wave travel along this line for the total phase shift equal to 2 pi. So here it is the total phase shift is covered from this zero point to this zero point is total 2 pi radian. This is known as the wavelength of the wave. So the equation which is given by the lambda can be defined with beta times of the lambda equal to 2 pi. So phase shift is denoted with the beta. The lambda times of the wavelength equal to it covers the total distance equal to 2 pi therefore lambda can be defined as 2 pi by beta where beta is nothing but the phase shift constant and the lambda is the wavelength. Second characteristics of the transmission line is nothing but the velocity of propagation. So what is velocity of propagation of the wave? It is defined as the velocity with which a signal of single frequency propagates along the line at a particular frequency. That particular frequency can be considered as just F. F is nothing but the any frequency. So again you can refer this electromagnetic wave. What is electromagnetic wave? The wave which having the highest frequency range. Generally it is in terms of the gigahertz. It starts with the 3 gigahertz to the thousands of gigahertz. So you can refer this sinusoidal wave so which having the one wave cycle for this first max value to the second max value which covers the 2 pi radian as a phase shift. This is known as the one wave cycle. So which travels with the velocity that is for a single frequency you can calculate it. So the frequency can be calculated by knowing the time is it is considered as the 1 by T. Now the velocity of propagation continuing with that one it is denoted with the letter Vp phase velocity therefore it is denoted as a letter P. The change of 2 pi in phase angle represents one cycle in time t and which is occur in a distance of one wavelength lambda. Therefore the previous figure which having the distance which covers the one total cycle is covered the phase shift of 2 pi radian. So it is denoted in terms of the velocity of propagation like this one. So lambda is given by Vp by f. So you can define the equation for the velocity of propagation equal to lambda times of the f. So wavelength is multiplied by any frequency f is nothing but the Vp phase velocity. So if you are putting the value of the lambda, lambda equation you know that in terms of the 2 pi by beta it is nothing but the 2 pi by beta. So if you are putting the value of lambda here as a 2 pi by beta Vp equal to 2 pi by beta times of f whereas this equation can be written as equal to omega by beta. So 2 pi f is nothing but the angular frequency which can be denoted with the omega. So you are getting the equation for Vp equal to omega by beta. Next characteristics is the group velocity. So the group velocity generally we are taking the grouping of the variations in the sinusoidal waveform. So if you refer this figure this is nothing but the sinusoidal waves which travels in this direction and it forms one envelope. So envelope of this wave is formed by taking the grouping of the waveforms. This is called as the group velocity. So the group velocity is called defined as the velocity of the envelope of the complex signal. So if you refer this sinusoidal signal this is nothing but called as a phase velocity which changes its phase while travelling in the line. Whereas the envelope which is formed in grouping of the waves which is known as the group velocity. Generally the group velocity is defined with the letter Vg. Now let us consider the two angular frequencies omega 1 and omega 2 and with respect to that the phase constants are given by beta 1 and beta 2. Now the group velocity is nothing but group velocity is denoted with the letter Vg equal to angular frequency for the second wave minus first. So it can be written as omega 2 minus omega 1 upon beta 2 minus beta 1. So the change in omega can be written with d omega and change in beta value can be written with d beta. So Vp is nothing but omega by beta whereas the Vg is nothing but change in omega by change in beta value. So change in angular frequency with respect to the change in phase shift that is d omega by d beta is nothing but the Vg. So difference let us discuss the difference between this phase and group velocity. So if you observe this waveform there will be the phase changes will occur in the sinusoidal signal and if you observe it neatly so you can get that the envelope is travelling in the right hand side direction whereas the phase velocity will travel in the left direction. This is actually the figure shows animated format for the phase velocity as well as the group velocity. So envelope shows the group velocity whereas the simple sinusoidal waveform can be defined for the phase velocity. So the phase velocity is the rate at which the phase of the wave propagates in the space whereas the group velocity it is nothing but the rate at which the envelope of the wave packet propagates in the given line. So how do you can define the difference between these two? So the phase velocity is defined for both single wave and superimposed wave whereas the group velocity is defined only for the superimposed wave. Another difference the group velocity is the velocity of the wave with lower frequency whereas the phase velocity is the velocity of the wave with higher frequency. One equation is there which gives the relation between this vp and vg how it relates to each other. The phase velocity and group velocity relationship is given by this equation vp into vg is equal to c square where c is nothing but the velocity of light vg is the group velocity and vp is the phase velocity. So this equation gives you the relation between phase velocity and group velocity. So these are the characteristics of the transmission line equations. These are the references used for today's session. Thank you.