 Welcome to the 11th lecture in the course Engineering Electromagnetics. The main topic of discussion today is the transmission line parameters which are also known as the primary constants for the transmission lines. Before we take up this main topic we consider a small extension of the impedance matching on the transmission lines utilizing stub transmission lines. This impedance matching was discussed in the last lecture and we considered two possible schemes, one using a single stub and the other using a double stub and the single stub configuration can be considered to be like this and somewhere here we connect a length of a transmission line usually short circuited called the stub. The concept we consider today is the situation when the characteristic admittance of the main transmission line and of the stub transmission line are different. Now we have seen that we calculate the normalized load impedance the corresponding normalized load admittance and then we transform it up to a point where the input impedance seen looking into that plane is equal to input admittance seen is equal to 1 plus J B because the stub can only alter the imaginary portion of the admittance. Now when we are dealing with the situation when the characteristic admittances are different then let us say that the value of the susceptence that we have obtained here is 1 plus J B1 B1. The subscript 1 emphasizing the fact that this B1 is the value normalized with respect to the characteristic admittance Y1 which means that B1 is equal to the actual susceptence that is required here that is B divided by Y1 so that the actual susceptence required at this plane is B equal to B1 Y1 and therefore the normalized susceptence required on the stub transmission line is going to be B2 which is B upon Y2 the characteristic admittance of the stub transmission line alright. So when we have common characteristic impedances or common characteristic admittances then this problem does not arise if they are different then we have to recalculate the normalized susceptence corresponding to the stub transmission line characteristic value and then use this in the Smith chart alright. So this should clarify if any confusion was created earlier. Now we go on to the main topic of discussion today and that is the transmission line parameters. You would recall that we have been modeling the transmission line in the general case in the form of a distributed parameter circuit which looks like this. Each small length of the transmission line is modeled in terms of an inductance per unit length and its product with the length so L dz and then R dz and C dz and G dz and once we have modeled the transmission line in the form of this distributed parameter circuit then we have been able to calculate the propagation constant the characteristic impedance the loss etcetera alright. We shelved the question of determining these constants R L C G alright for the time being and now the time has come to discuss how one is going to calculate these constants which are also known as the primary constants. Using these primary constants the values that one gets the characteristic impedance the propagation constant those quantities become the secondary constants of the transmission line and of course once we know these primary constants then we know that we can solve the transmission line problem. We can predict its behavior or design it for various purposes. So the determination of the transmission line parameters is going to be extremely important that is the basis of the analysis of the transmission line. Before we take up the calculation of these transmission line parameters a small concept regarding the complex permittivity also needs to be introduced. The concept is best introduced by considering Ampere's law in its differential form as modified by Maxwell to include the displacement current term which reads as del cross H equal to del D by del T plus J where the term the second term on the right hand side is the conduction current term and the first term on the right hand side is the displacement current term alright. Now when we consider sinusoidally time varying signals and use phasor notation and make some simplifying assumptions which we shall discuss in a little more detail in the next lecture. We can write this as J omega epsilon E plus sigma E J equal to sigma E D equal to epsilon E the time derivative replaced by J omega alright which we can rewrite as J omega into epsilon plus sigma by J omega times the electric field or J omega times epsilon minus J sigma by omega times E and now one can conceptually say that if one replaces this term by a single term by a single symbol say the complex permittivity epsilon which has a real part and an imaginary part in particular the imaginary part being equal to sigma by omega then the right hand side can simply be written as J omega epsilon E where now this epsilon is the complex permittivity. What is the general utility of this concept? The general utility is that while in practice we are going to have lossy dielectric materials they will not be lossless analytically one could start with a lossless dielectric that is epsilon real obtain a solution which will involve epsilon in various places then if that epsilon real epsilon is replaced by the complex epsilon we will have taken into account any loss in the dielectric arising out of the finite conductivity of the dielectric material. So, the analysis for lossy dielectric materials will be simplified to that extent. So, that is how one in general utilizes the concept of complex permittivity and we are going to utilize this concept in calculating the loss in the dielectric alright. One could also say that epsilon r is epsilon r prime minus J epsilon r double prime where each quantity is divided by the free space permittivity epsilon not alright. So, having introduced these things now we can go on to the actual calculation of the transmission line parameters. Now, we require some defining relations for these parameters before we can calculate these. So, the commonly used definitions for the transmission line parameters L C R G there as follows you are familiar with these. How do we define the inductance L? Inductance L is the inductance associated with a urate length of the transmission line. And therefore, it can be obtained as the magnetic flux linkage it must be the flux linkage per unit length because we want to evaluate inductance per unit length divided by the current total current which is causing that magnetic flux alright. Similarly, one can put down the definition of the C capacitance per unit length as the total charge again per unit length not the total charge on one of the conductors of the transmission line charge per unit length divided by the voltage difference between the two conductors alright. Then the conductance G per unit length is defined as the total shunt current per unit length divided by the voltage difference between the two conductors. And finally, the resistance R per unit length would be the voltage drop per unit length divided by the total charge total current. Now, where we are just writing total current as will be clarified when we take up the illustrative example we have the series current in mind. There are two types of currents which are flowing one is the shunt current directly between the two conductors because of finite conductivity of the dielectric material. The other is the series current which is more or less uniform along the lengths of the transmission lines along the lengths of the along the lengths of the transmission line and is equal and opposite on the two conductors alright. And therefore, these this type of distinction one will have to keep in mind. We consider the example of a simple familiar transmission line and try and apply these definitions to show very clearly how these transmission line parameters can be calculated. We consider the example of a coaxial transmission line which is quite familiar to us and is quite simple also to analyze. The coaxial transmission line is composed of two conductors coaxial conductors. The cross section would look like this we can put down some coordinate system the Cartesian coordinate system. And also the cylindrical coordinate system where this becomes r and this angle is phi. The unit vector in the radial direction is this r cap and the unit vector in the phi direction is normal to this and is in the direction in which phi is increasing. And therefore, this is the phi cap direction for this particular value of phi. We can say that the radius of the inner conductor is A and the radius of the outer conductor is B and let the voltage difference between the two conductors be V naught. So, that the potential difference between the inner and the outer conductor is V naught. The inner conductor is at a higher and notionally positive voltage and the current that is flowing in the two conductors is I naught. Let us say in the z direction positive z direction in the inner conductor negative z direction in the outer conductor. This is the transmission line given to us and we attempt to calculate the transmission line parameters or the primary constants for this transmission line. We can take up the calculation of the capacitance C per unit length first and we will need the charge per unit length and in terms of that the voltage difference between the two conductors. Or in other words we need to determine the charge per unit length in terms of the potential difference or the voltage difference V naught. We start in the following manner. Let us say that Q is the charge per unit length on the inner conductor and then we apply Gauss's law to some cylinder with radius r where r has an intermediate value between a and b. So, that we get the displacement density d as Q by 2 pi r r cap. Through an application of Gauss's law we say that for r between a and b this is the displacement density vector. So, that the corresponding electric field is Q by 2 pi r and now we use the real part of the complex permittivity and say that it is again in the radial direction. Once we know the electric field of course Q is still unknown. We can determine the potential difference between the two conductors in terms of the electric field and say that V b a is minus e dot d l integral going from a to b. The potential of point b is higher with respect to the potential at point a by this amount. Therefore, it is equal to Q by 2 pi epsilon prime with a negative sign and then integral of 1 by r b r between the limits a and b giving us a value minus Q by 2 pi epsilon prime log of b by a. So, that V naught which is V a b becomes minus V b a and it is Q by 2 pi epsilon prime log of b by a. Consistent with the requirement that for a positive value of V naught Q should be positive. So, from here one can obtain a value of Q in terms of V naught, but as far as the calculation of C is concerned that is straight forward C is Q by V naught and therefore, it is twice pi epsilon prime by natural log of b by a I am sure a familiar expression. We might need a little later the value of Q in terms of V naught. So, we can write it here itself. From here Q is 2 pi epsilon prime V naught by log b by a. So that the electric field replacing Q by this new expression turns out to be V naught by r log b by a in the radial direction. Once again I am sure a familiar expression. So, we have evaluated one parameter that is the capacitance per unit length for this transmission. Now, it is available in terms of the geometry and the material that is utilized in the construction. Similarly, one can go on to the calculation of the inductance and I am afraid we will need to erase this. As far as the inductance is concerned we will have to worry about the magnetic field and the magnetic flux. For this simple geometry the magnetic field intensity between the two conductors is I naught by 2 pi r and what is the direction? Considering that I naught is in the positive z direction it can be taken to be phi cap. The corresponding magnetic flux density is going to be h multiplied by the permeability and since we normally use a dielectric material the permeability is that of the free space. But one could generalize it if necessary. Here we simply write that b is equal to mu naught h. So, it is mu naught I naught by 2 pi r once again in the phi cap direction. Now, the flux that is associated with this is going to be constant at any surface like this. No matter at what value of phi this surface is considered because of the symmetry of the problem. And therefore, one could consider any surface like this extending from a to b and in the z direction having a unit length. And calculate the total flux associated with this system total magnetic flux associated with this system. And therefore, phi which is b dot d a integral over the surface that we are considering it becomes mu naught I naught by 2 pi and then once again 1 by r d r integrated from a to b. And therefore, the value that we get is mu naught I naught by 2 pi and then log of b by a. So, that the inductance l per unit length because the flux has been calculated per unit length l which is phi by I naught is simply mu naught by 2 pi and then natural log of b by a. Some quantities of interest can be made out right here since l and c are available. The losses are usually small or if we assume that it is a lossless line then the characteristic impedance z naught which is square root of l by c can be seen to be mu naught by epsilon prime whole square root and then natural log of b by a by 2 pi. And therefore, we see that straight away in terms of the geometry and the material parameters the characteristic impedance is of a. This could have been a question that what determines the characteristic impedance of the transmission line. So, this is what determines the characteristic impedance of the transmission line. And therefore, you can design transmission lines with the required characteristic impedance right. Similarly, the phase velocity which we have seen is 1 by square root of l c comes out to be 1 by square root of mu naught epsilon prime. And depending on the real part of the relative permittivity the phase velocity will be that much different by square root of epsilon r prime compared to the free space velocity alright. Next we take up the calculation of the conductance g. If you recall the definition of the conductance g it was the total shunt current shunt current which is flowing between the two conductors directly between the in the dielectric space between the two conductors alright. That is what we have that is what we have to calculate total shunt current divided by the potential difference between the two conductors. Total shunt current let us denote it by i s is going to be the current density shunt current density j s integrated over the area through which the shunt current is flowing. Now, for this purpose one could consider some surface of radius r between the two conductors. Whatever is the total shunt current flowing through any such surface is the total shunt current through any other such surface from continuity of current alright. And therefore, this could be done at any point, but first we must identify what is j s. What is the basic cause of this shunt current? The basic cause is the shunt current is the electric field between the two conductors and the fact that the dielectric is not perfectly insulated. It has a finite conductivity and using the standard expression for the current density it is sigma times the electric field. Now, since there are two materials involved the conductor these two conductors and the dielectric both of which have some conductivity it is usually required to keep in mind what is the it is the conductivity of which material that we are talking about, but the context will make it amply clear which conductivity we have in mind. We could write a small subscript d here or when we are talking about the conductivity of the metal we could write a small subscript m there. We choose the second alternative and leave this sigma as it is understanding clearly that this is the dielectric conductivity. Now, sigma if you recall we had written as sigma by omega was epsilon prime the imaginary part of the complex permittivity and therefore, sigma is omega times epsilon prime and e is already known to us. Therefore, this current density can be put down it will turn out to be sigma which is omega epsilon prime and then v naught by r log of d by a and the direction automatically is the direction of the electric field the radial direction. Once the direction has specifically come into the expression it leaves no doubt about the nature of the shunt current. Now, this is not a function of phi in any case phi is not appearing as a parameter because the system is symmetric with respect to phi and as I said we could consider any surface. Now, what will be the surface area of that surface it will be this circumference multiplied by the unit length along the z direction. So, we need to simply multiply this by 2 pi r to get the total shunt current I s is going to be omega epsilon double prime v into 2 pi v naught by log natural log of d by a the total shunt current which flows between the two conductors. Therefore, the conductors g which is I s by v naught is simply omega epsilon prime into 2 pi by natural log of b by r log by a and one could compare the expressions for the conductance g per unit length and the capacitance c per unit length and one finds that this is simply omega epsilon double prime by epsilon prime times the capacitance c per unit length. Which relation is a general relation will hold good for transmission lines and as one can realize that both the shunt current and both the conductance and the capacitance are related to the dielectric and they are simply related through the two parts of the complex permittivity. Which parameter we have to calculate next it is the resistance r per unit length we can make some space here. Now, as far as r is concerned one could estimate the voltage drop per unit length and divide that by the total series current, but we have a simpler way out as far as the resistance is concerned since we already know the expressions in terms of which the total resistance can be obtained. In any case to estimate the voltage drop we will need that resistance per unit length. Therefore, we can straight away utilize the standard expressions for the resistance to calculate the resistance that we require here. Now, at high frequencies the calculation of resistance is complicated, but if we use the concept of skin effect the calculation can be simplified. What happens is that at high frequencies the current flows only over a limited depth in the conductors. For example, only over such a depth inside the conductor will the current be significant beyond this the current will drop to an insignificant value. This is the phenomenon of skin effect. Now, this depth within which the current flows can be calculated using the concepts of electromagnetic theory and it turns out to be 2 by omega mu sigma and now it is the metal conductivity that we talking about and therefore, we write a subscript m here. So, the skin depth is a function of the radiant frequency the permeability and the metal conductivity and one can straight away make out how this skin depth delta varies as these parameters change. This is the skin depth delta. Now, the concept of skin depth says that one could replace the entire thick metal by a thin metal the conductor which is thick which has a thickness delta equal to skin depth and assume that the current within this thickness is uniform whereas, actually in the say the infinitely thick conductor it is going to decay exponentially away from the surface. In fact, it is that phenomenon which makes the calculation of the resistance complicated, but using the concept of skin depth we simplify and we say that the actual thick metal is replaced by a thin conductor with thickness delta within which the current is assumed to be uniform over the entire surface. In which case the resistance calculation can be made very simple almost trivial. You would recall that the Ohm's law says that the resistance is equal to the resistivity rho times the length divided by the area of cross section of the entity of which we want to calculate the resistance in terms of conductivity it can be written as L by sigma A. In our case what is the conductivity? The conductivity is the metal conductivity what is L? L is the unit length along the z direction so it does not specifically enter the calculations and what is the area? Area will be this circumference multiplied by the skin depth delta. And therefore, this R comes out to be let us say if we consider the inner conductor or the resistance associated with the inner conductor it will be 1 by sigma M and then 1 upon delta into 2 pi A and of course, a similar term corresponding to the outer conductor. So, that it is 1 by sigma M delta into 2 pi B of course, assuming that the 2 conductors are made out of the same material which is going to be the practice. And therefore, it is 1 by sigma M delta into 1 by 2 pi into 1 upon delta into 2 pi A plus 1 upon B. The product sigma M delta has a value it is omega mu by what is it going to be 2 sigma M by omega mu. And it is given a new symbol 1 by R M with R M going to be omega mu by 2 sigma M whole square root and it is called the skin resistance of the material. In terms of this it is R M divided by 2 pi into 1 by A plus 1 by all right. Now, that completes the calculation of the transmission line parameters of the primary constants of the transmission line for this coaxial transmission line. Now, things were extremely simple here because of the symmetry of the transmission line symmetry of the structure, but depending on our application and our requirement we are not going to be restricted to such symmetric simple transmission lines. We are going to use any structure which serves our purpose well in which case this geometrical simplicity may not be maintained. And in such cases in such a general case it becomes difficult to apply these commonly used definitions in terms of voltage and current for these transmission line parameters. In any case we have been saying that field description is more accurate particularly at high frequencies. It so happens that for transmission lines we can still continue to use the voltage and current concepts. It is a simplified case of the general transmission structures that one uses at high frequencies. So, what would be the general definitions for these transmission line parameters which could be utilized even when the geometry is not so simple. Those general definitions will be in terms of the fields. And how do we arrive at those definitions that is the next point we consider. Field based definitions for these transmission line parameters. What would be the basis of these definitions? The basis would be either the energy stored the power loss. For example, let us consider the calculation of the capacitance based on such an energy stored definition. Then we realize that the energy stored in the capacitor in the simple situation in terms of voltage and current is half C v square. What is the corresponding expression for the energy stored in the form of the electric field? That is going to be half epsilon or epsilon prime and magnitude of e square which is the energy density. And therefore, it needs to be integrated over the volume over which the electric field is non-zero. And then we will have the energy stored in the form of the electric field. These will be the two alternative forms of the same thing. When we have sinusoidally time varying quantities voltage current electric field and magnetic field then you know that the average energy density is going to become 1 by 4 epsilon prime magnitude e square integrated over the volume. And so also will this become 1 by 4 C times magnitude of v square. And the comparison of these two terms becomes the basis for an alternative definition for the capacitance per unit length C. We can write that C is equal to epsilon prime by the V star using the phasor notation. And then within the integral we have e star d v integrated over the volume over which the electric field is non-zero. It is an alternative definition for C. It will not give us a new value for C. For example, for the structure that we just considered. And what value of C we get using this expression or in particular that we get the same value of C we will leave as an exercise. What would be the other parameters in terms of similar quantities? The L will be going through a similar argument similar procedure. It will be mu by or mu naught by i i star. If you like we could write v naught v naught star i naught i naught star. If we want to make it correspond to the example that we just considered. But that is not so important. And then here we have h dot h star where star stands for the complex conjugate of the quantity. E star is the complex conjugate of the electric field. V star is the complex conjugate of the voltage. Similarly considering the power lost in the dielectric we arrive at the following definition. It is omega epsilon double prime by v naught or v v star. And then we have within the integral E E star integrated over the cross section. Basically what we are doing is we are equating g v squared the power lost in the conductance g to the power lost in the dielectric like sigma magnitude E squared d v. Essentially that is what we are doing. Where sigma is the dielectric conductivity is equal to omega epsilon double prime. And therefore this should be quite easily acceptable. Similarly r can be written in terms of the ohmic loss or the power dissipated because of the resistance. And it will turn out to be r m by i i star where r m we have already indicated it is the skin resistance of the conducting material. And then here we have integral of h dot h star integrated over the peripheries of the conductors. And one can readily see the correspondence between this definition and the definition we have just used to calculate the resistance r. The derivation of this kind of a formula will be taken up later when we considered wave propagation in conducting medium. So this is where we will end this lecture. In today's lecture we first considered a generalization of the single stub tuner. And then we considered the calculation of the transmission line parameters which are also known as the primary constants. First using commonly used definitions and then introduce the definitions based on the fields concepts. Thank you.