 Good evening everyone. Today's our topic is directional coupler myself Piyusha Shedgar. These are the learning outcomes for this session. At the end of this session, students will be able to design microwave circuits using basic microwave components. And they will be able to analyze the use of these components in the design of microwave system. These are the contents. So what is directional coupler? Directional couplers are flanged built-in waveguide assemblies which can sample a small amount of microwave power for measurement purposes. Mainly, the directional couplers are designed to measure incident or reflected power. It is used to measure standing wave ratio values and also used to provide a signal path to a receiver. So there are the two types of the directional couplers. One is the unidirectional. That is, this type of directional coupler used to measure only the incident power. Whereas bi-directional coupler used to measure both incident as well as the reflected power. Now the directional coupler is having the four ports with the waveguide junction consisting of the primary waveguide or it is main waveguide and the secondary auxiliary waveguide. So from this figure you can observe that it having the four ports, port 1, port 2, port 3 and port 4. So at the port 1, P i indicates it is the incident power. At the port 2, P r is the received power whereas P f is the forward coupled power at port 4 and P b is the back power at port 3. So the main properties of the ideal directional couplers are given the points. First point, a portion of power travelling from port 1 to port 2 is coupled to port 4 but not to port 3. A portion of power travelling from port 2 to port 1 is coupled to port 3 but not to port 4. A portion of power incident on port 3 is coupled to port 2 but not to port 1. A portion of power incident on port 4 is coupled to port 1 but not to port 2. Port 1 and 3 are decoupled as port 2 and 4 are also decoupled. So the directional coupler having the three parameters coupling factor, directivity and the isolation. So how to define this coupling factor? It is the ratio of the incident power P i to the forward power P f which is measured in decibel value. So mathematically it is given by C equal to 10 log to the base train P i by P f in decibel. Similarly, directivity is also defined in decibel value. Directivity is defined as the ratio of forward power P f to the back power P b which is also measured in decibel value. And this directivity D is given by 10 log to the base train P f by P b in decibel value. Whereas the isolation is defined as the ratio of incident power P i to the back power P b which is also expressed in decibel. And mathematically it is given by 10 log to the base train P i by P b. Now before going to start the scattering matrix calculation you can pause video here and you can relate isolation with directivity and the coupling factor. How you can relate these parameters to each other? Yes, from the above definitions and the mathematical equation you can write isolation in decibel is equal to coupling factor plus directivity. Now what is two-hole directional coupler? It is one of the type of the directional coupler. So it is two-hole directional coupler that is two holes are provided here with the distance lambda g by 4. It also having the two waveguides it is the main waveguide and the secondary waveguide is also known as the auxiliary waveguide. And the two holes are provided to the main waveguide which is placed at a distance of lambda g by 4 where lambda g is the guide wavelength. So how to work the directional coupler? So here again consider the directional coupler which having the four ports port 1, 2, 3 and 4. Two holes are provided at this point hole number 1 and hole number 2. And these holes are separated by the distance lambda g by 4 where lambda g is the guide wavelength. Now the two leakages out of the holes 1 and 2 are in phase at this second hole. That is if the input is applied to port 1 that input is coming from the hole 1 and hole 2 at the hole 2 is in phase. Hence they add up here contributing to the forward power that is at port number 4. Whereas two leakages out of holes 1 and 2 are out of phase at the first hole. Hence they cancel each other making back power is equal to 0 which is true for the ideal case of the directional coupler. Now the directional couplers are also having the two holes or multi-hole. So depend on the number of holes there are the types of the directional couplers as shown in this figure. First figure shows the multi-hole directional coupler whereas the second figure shows the cross directional coupler. The magnitude of the power coming out of two holes depends on the dimension of the two holes. Now how to calculate the matrix for the directional coupler? Again that properties of directional coupler can be defined by its scattering matrix which is denoted with the letter s. Now first step is the directional coupler is a four-port network. Hence s matrix can be defined as with the order of 4 by 4 and these are called as the scattering coefficients or the scattering parameters as shown by the equation number a. In a directional coupler all four ports are perfectly matched to the junction that is there is no any reflection towards the source again and therefore the diagonal elements of the above matrix are 0. Thus you can equate the parameters s11, s22, s33 and s44 equal to 0. Use the next property as a symmetry property. sij equal to sji is used for the symmetry property that is the number of row for the first matrix equal to the number of columns for the second matrix. Thus you can write s12 equal to 21, s13 equal to 31, s14 equal to 41 and similarly for the other parameters. Now in ideal case for the directional coupler the back power is 0 that is there is no any coupling between port 1 and port 3 and therefore if the input is applied to port 3 output out of the port 1 is equal to 0. Similarly if the input is applied to port 1 output out of the port 3 is equal to 0. So in above property you can observe that s13 equal to 31 and therefore these two are equal to 0. Similar to port 1 and 3 there is no any coupling between port 2 and port 4 and therefore s24 equal to s42 also equal to 0. Now use all these above values in the given matrix equation A then the matrix becomes as shown in equation number 5. Here the all diagonal elements are 0 whereas these all elements are also 0. Thus we have the 4 unknown values s12, s23, s14 and s34. Now calculate all these values and by putting all these values in equation number 5 you are getting the scattering matrix for the directional coupler. So for that again use the unitary property. Unitary property is the scattering matrix multiplied with the complex conjugate of that scattering matrix equating it to unitary matrix. So this is the matrix take the complex conjugate of that matrix equal to unitary matrix. Now by multiplying this as you know that all the diagonal elements are equal to 0 by taking the combination r1, c1, r2, c2 and r3, c3 you are getting the equation 6, 7 and 8. Now take the additional combination of r1 and c3 and write the equation number 9. So by comparing equation 6 and 7 s12 square and this 11 is same and therefore you can equate s14 equal to s23. Similarly for equation number 7 and 8 s23 are same and the right hand is also same and therefore s12 can be written as equal to s34. Let us assume that s12 is real and the positive value with the letter p. s12 equal to s34 and it is also equal to s34 star that is complex conjugate of this s34 equal to p. Now from equation number 9 and 12 you can put this s12 as a p value and s34 as a p value. You are getting this equation equal to 0 taking p as a common p s23 star plus s23 equal to 0. As p is the real number it is not equal to 0 equate this bracket is equal to 0. So from this equation if you are considering s23 is equal to jy then the complex conjugate of that is equal to minus of jy that is s23 must be imaginary value. Now let s23 equal to s14 equal to jq therefore s12 equal to s34 equal to p and s23 equal to s14 equal to jq. Put all these values in the above matrix given by the equation number 5 then you are getting the matrix for the directional coupler shown below which having this real part as a p and the imaginary part as a jq. These are the references for this session. Thank you.