 Hello everyone, in this video lecture we are going to discuss with the wave equation for transmission line, myself Dr. Rupali Shalke working as an associate professor in Waltzens Institute of Technology. Learning outcomes, at the end of this session students are able to derive the wave equation for transmission line and also presents its equivalent circuit diagram. A transmission line is a device for transmitting or guiding a energy from one place to another. There are various types of transmission line, few of them are parallel plate transmission line, two wire transmission line or co-excel transmission line. The knowledge of values of the electric circuit parameters associated with the transmission line is necessary for analyzing and designing. So, on a transmission line when voltage and current is present which results in the electric field and magnetic field in a space in around the conductor. The circuit quantities like voltage and current and field quantities like electric field and magnetic field both are present on the transmission line. This gives a two ways to analyze the transmission line. One way is by we can express the wave parameter by voltage and current using the Ohm's law and by applying the Maxwell's equation for the field quantities for expressing the wave parameters. Now, let us see how they are being used for the analysis purpose. Now, starting with the circuit theory parameters like voltage and current, let us consider a transmission line where voltage V is applied and I is the current flowing through it. Now, we will consider a small length of the transmission line as a dx and assuming that the signal is transmitting in a x direction. The according to the Ohm's law by the relation between the V and R, V is equal to IR and I is given by V by R. The voltage for a small distance dx is given by dV by dx which is equal to IR as it is considered as a transmission line we are considering the resistance in terms of impedance. So, if the equation will become dV by dx is equal to I z and the current equation V is equal to IR will become di by dx is equal to V where y is a admittance. Now, consider or differentiate the equation one with respect to x and using the UV rule we get d square V by dx square will be equal to I dz by dx plus z di by dx. As this impedance z is a constant therefore, the value of the dz by dx will be equal to 0, the derivation of the constant is equal to 0 that is why we get this term is then get 0 and only we remain with the z di by dx therefore, this equation will reduce in this format. Now, substituting the value of di by dx from the equation which we have seen in the previous slide which is equal to V y here after substituting it will get you d square V by dx square will be equal to z y into V and as we see that on the both side the voltage term is present that is why we called it as a wave equation for transmission line in the voltage term. Similarly, let us consider the equation to differentiate the equation to with respect to x then it will be d square I by dx square will be equal to V dy by dx plus y dV by dx here also as V is the constant term therefore, its derivations will be equal to 0. dV by dx will be equal to 0 when we substitute in this above equation then the equation will reduce in the following format. Now, from the equation 1 we will substitute the value by of dV by dx in this particular equation which is equal to dV by dx is equal to I z then it will be d square I by dx square will be equal to y sorry z y into I where if you see on both the side LHS and RHS this equation consist of the current terms therefore, we called it as a wave equation for transmission line in terms of current. This is by deriving the wave equation by considering the voltage and current that is a circuit theory. Now, the value of R and y R given by z is equal to R plus j omega L which is called as a series impedance and the y is given by G plus j omega C which is called as a shunt admittance. Now, if we draw the equivalent circuit for this the equivalent circuit of the transmission line will be a series impedance that is z plus the L and R are connected into the series while G and C are connected in a parallel form therefore, it is called as a series impedance and this is called the shunt admittance. If we see here the distribution of these terms z and y is continuously throughout the transmission line that is why this is called as a distributed transmission line. This is a equivalent circuit representing the transmission line. Now, considering the field theory by considering the field theory let us consider a transmission line with the upper and lower surface equivalent as a script. Here we are conducting strip here we are considering the two transmission line whose length is W is the width of the transmission line and L is the length of the transmission line. The wave is propagating in a x direction as shown in the figure for the field theory we will replace the following terms. We will replace epsilon is equal to L in the which is nothing but a induction per unit length which unit is Henry per meter and mu as a C is equal to capacitance per unit length whose value is f by m that is Faraday's per meter and sigma is equal to G which is nothing but a conduction per unit length which whose unit is Mahos per meter. Using in place of the epsilon mu and sigma we will substitute the L, C and G respectively. Now, according to the Maxwell's equation we are derived the wave equation for the conducting media in the electric field which is nothing but a dow square e by dow x square is equal to j omega epsilon sigma plus j omega mu and e bar L that is we will integrate over the length then this will be integrating e dx dl and but integral of e dl is equal to V according to the Faraday's law and substituting the value which we are considering in the previous slide. This equation will become here we will substitute as a V dow square V by dow x square will be equal to j omega L G plus j omega C into integral e dl will be equal to V which is nothing but dow square V x square plus R plus j omega L plus G plus j omega C. This equation will be considered for when R is equal to 0 here we are substituting the value for the if there is a value for the L then this will be R plus j omega L plus G plus j omega C into V which is nothing but a R plus j omega L will be the series impedance and G plus j omega will be the shunt admittance Z and Y. This is nothing but a which will be equal to the gamma square V where gamma is nothing but a propagation constant. Therefore, the gamma value will be under root of Z Y this is a wave equation for transmission line. Now using the wave equation derived from the MPS law which is in a magnetic field terms. This is a wave equation for conducting media in a magnetic field that is h bar. We will integrate this equation over the length then this equation will be integral del square h del x square del dl will be equal to j omega in bracket sigma plus j omega mu integral of h dL we are integrating h dL but integral of h dL is equal to i which is given by the MPS law then we will substitute this values del square i by del x square plus j plus in place of omega we will substitute L in place of sigma we will substitute G in place of mu we will substitute C and integral of h dL will be equal to i. After substitution we will expand by adding the R term to the first term R plus j omega L plus G plus j omega C into i. If we see that on the and R plus j omega L as a Z and G plus j omega C as a Y. If you observe this equation that on the both side there is a current term therefore, this equation we can say as a transmission line equation in terms of current which is also written with the propagation constant. Now pause this video and consider what will be the equivalent circuit diagram if R and G is equal to 0. Since R and G are the lossy component if R and G are equal to 0 then the transmission line is called as a lossless transmission line and its equivalent circuit will be here there will be no value for the the only there will be a conduction and the capacitance factor. These are the references.