Microwave Integrated Circuits Professor Jayanta Mukherjee Department of Eectrica Engineering Indian Intitute of Technoogy Bombay Modue 1 Lecture No 2 Refection Coefficient, SWR, Smith Chart. Heo wecome to another modue of this course, microwave integrated circuits. In the previous modue, we saw the properties of microwave circuits, the wave nature and the distributed and then we aso found the soution of transmission ines equation is for deaing with distributed circuit eements. In this ecture et us continue with the discussion.so the 1 st so one more concept that is associated with microwave circuits is the concept of impedance. (Refer Side Time: :55) L C ( z) intrinsic I The impedance can be defined in various ways. One is that characteristic impedance associated with distributed circuits that which we found out to be L and C. Then aso is defined as and I as we saw in the case of distributed circuits. We can have and I soutions so then we can aso have. of course is a function of z the sma z the distance from the source. Then there is aso another kind of impedance that is defined which is caed intrinsic which is reated to the properties of the materia. For exampe for air this ratio comes out to be 377 ohms. So this has nothing to do, this definition of impedance has nothing to do with votages and currents. Sti this hods. In fact for microwave engineering as we saw for singe conductor waveguide. You know that in a singe conductor waveguide we cannot have a unique definition of votage and current. But we can have a definition of impedance. Either from this reationship that is we have an eectromagnetic wave passing through some media. Then you can find out intrinsic from this reation.
(Refer Side Time: 2:34) L C ( z) intrinsic I Or is aso yet another definition of is ratio of eectric to magnetic fied. So these are the various definitions of impedance. (Refer Side Time: 2:57) x ( x) e e x I( x) e e x ( d) j tan( d) ( d) I( d) j tan( d) x Now et us see et us come back to our definition come back to our transmission ines where we had found out the soution for x or. This is equa to this is equa to L. Where at a particuar and I at a particuar. o is the characteristic impedance. Now soving this equation you can obtain the soution for d given by d/id where d is the distance from the oad end that is here d is equa to and here d is equa to L. And this comes out to be equa to o(l+jo). We see that this definition of d means d is periodic in d i.e. when every time d increases by ambda/2 we get the same vaue of d. Or when theeta given by beta d increases by pi we aso get the same vaue of d. In fact from this we can derive the expressions of input impedance i.e. d for some specia types of transmission ines. So we have obtained soution for the input refection coefficient of input impedance. I beg your pardon. We have not come across the concept of refection coefficient yet. We have found that
the input impedance of transmission ine oaded with an impedance. Now et us see what happens if we consider some specia cases. So the specia cases are: (Refer Side Time: 6:29) ( d) j tan( d) ( d) j cot( d) 1 st is when L is equa to. So when L equa to. So this is our transmission ine once again, L the oad. Then, d this is for a short-circuited ine. So short-circuited ine wi have this kind of an input impedance. For an open circuited ine this wi be sorry this wi be -jo quat of beta d. (Refer Side Time: 7:46) ( d) for a d And for the case when L is equa to o, d wi be simpy equa to o for a d. Means as if the transmission ine is never present. If we pot a curve the vaue of d for the shorted transmission ine, it appears to be something ike this. (Refer Side Time:8:16)
So we can see a shorted transmission ine can take on both positive vaues as we as negative vaues for different vaues of d. For exampe, when d is equa to, it has a. So it is ike a short. When d is between ambda/2 and 3ambda/4, it is positive. d is positive for a shorted ine. So this is for a shorted ine. d is positive. So it acts as an inductor. When d is between ambda/4 and ambda/2, d is negative and hence the shorted transmission ine acts as capacitor. A specia type of transmission ine which is very commony used is what is caed as a quarter wave Transformer. (Refer Side Time: 1:8) R in G ( d ) 4 R R R 2 L L G 2 For quarter wave Transformer is a transmission ine which is quarter waveength ong i.e. ambda/for ong. It is aso terminated by a oad L. Now, in pace of d I can aso write this as ind. So in wi be equa to d equa to ambda/4 and this comes out to using the equation that I just gave if we put in the equation that I had previousy stated that if this equation
(Refer Side Time: 11:7) x ( x) e e x I( x) e e x ( d) j tan( d) ( d) I( d) j tan( d) x this equation if we substitute in pace of d as ambda/4 and beta as 2pi/ambda then we get this particuar reationship. (Refer Side Time: 11:21) in ( d ) 4 2 So what we see is that the input impedance is proportiona to the inverse of the oad impedance. If the oad impedance L shorted, input impedance wi appear to be open. If the oad is open, the input wi appear to be shorted.
(Refer Side Time: 11:46) R in G ( d ) 4 R R R 2 L L G 2 Conversey, if you want to convert any impedance say you want to convert resistance RG sorry resistance RL connected at the oad to some vaue RG then you choose a transmission ine which a characteristic impedance square root of RL RG and using this, you wi get RG is equa to o square upon RL. So we see, by connecting our RL this is converted to RG when we have this transmission ine with a o given by this vaue. This is one of the that is why it is caed a Transformer.A transfer from one vaue of oad impedance to another vaue of source impedance. Now coming to yet another important concept that is frequenty associate it with transmission ines that is caed as the refection coefficient.
(Refer Side Time: 12:43) e ( x) ( x) e ( x) x x 2x 2 ( d ) e e So once again if we have our transmission ine with a oad L connected ike this and suppose the votage as I said there are 2 concepts, one is + and -then the refection coefficient at a particuar ength x is given by - or E+ gama into upon (+-gama x). Now this actuay, the refection coefficient is given as simpy +x upon -x. We know we can write +x in terms of the votage at the source, o ike this. So from here we get this is nothing but o-the refected wave at the source upon o+2gama x. If we do a coordinate transformation where we write X is equa to L-d then this becomes so then now if you take out this E raise to +2gamaL then this o-upon o+ is converted to L- upon L+. Let me write it on a different sheet.
(Refer Side Time: 14:52) e 2 x 2d 2 x 2d () e e e This is converted to L L+E raised to -2gama d. This is nothing but the ratio of the refected wave to the incident wave at the oad end. That is at X equa to L and this is nothing but the refection coefficient at the oad end and this is simpy 2 gamma d. Gamma at a particuar point X in terms of the source votages, refected and incident votages is given by this and in terms of the oad parameters, it is given by this. So we have both key, you can express gamma at any point in terms of both the source parameters as we as the oad parameters. (Refer Side Time: 16:13) Now one interesting thing that we observed if I write the gamma X vaue now assuming this is a non-dispersive media gamma is equa to jbeta then we can write gamma X is equa to gamma L raised to -2jbeta d. Now what we see is that we had our transmission ine at a particuar point X or d. So d is away from the oad. This is our gamma L and this is we are going away from the oad. Now as we go away from the oad, the phase of +d or -d changes by jbeta d ony. Pus or minus jbeta d. But the phase of this refection coefficient changes twice. So if we can draw
this, as d increases or we are going away from the oad, our phase ange keeps on decreasing and phase ange decreasing means we are moving in a cockwise direction. The other thing that we note is that the magnitude of the refection coefficient as we go away from the oad remains constant and if we are going towards the oad, then we move in an anticockwise direction. If we are moving in a cockwise direction if we are moving towards the oad, if we are going away from the oad, we are moving in a cockwise direction. If we are moving towards the oad, we are going in an anticockwise direction. And the phase transit is twice the phase transit of as of that what we encountered whie going aong the transmission ine.so this is one important concept associated with transmission ines. The refection coefficient is changes twice faced change for the votage for the incident and refected waves. (Refer Side Time: 19:31) P P P P (1 ) 2 Now we had aready come across this formua for the tota rea power that is deivered to the oad and that was discussed in the previous modue ike this. Where say, + and - are the votages are the incident and refected votages at the oad. Now, if we represent this incident power so this I can ca incident power and this I can ca the refected power. If I ca denote it by the term P+ then PL can be written as P+-P-which in turn can be written as P+ if I take this + square upon o common 1 - gamma L whoe square. Because - upon + is nothing but gamma L. And so, we see that for tota power transmission that if we want the entire incident power to go to the oad gamma L shoud be equa to. Because for any other vaues of this gamma L some component of power wi be refected back and ony when gamma L is equa to we have compete transmission of the incident power to the oad. No see one thing about this gamma L is that this is a constant quantity. Gamma L does not change. It ony depends on the oad. Whereas the other components are not fixed. For exampe, what is the input refection coefficient or what is the refection coefficient at any other point aong the transmission ine, gamma L is a quantity that depends purey on the oad property. So can we then derive a formua of gamma L in terms of the oad impedance?
(Refer Side Time: 21:57) z z ( z) e e ( z) I( z) z e e z The impedance at any point or X s whatever. We can take X aso in pace of is given of sma is equa to of upon I of. If I write the votage and current reationship at, that is now this can be simpified to this reationship. (Refer Side Time: 22:51) ( z) ( z ) 1 ( z) 1 ( z) 1 1 1 1 So then the impedance or the input impedance at equa to L is nothing but o upon 1+ gamma L upon 1-gamma L. So then L is nothing but o upon 1+ gamma L upon 1 - gamma L. From here, we can write if we want to find gamma L in terms of L that wi come out to So this is the vaue of oad refection coefficient. (Refer Side Time: 24:17) ( z) ( z ) 1 ( z) 1 ( z) 1 1 1 1
Now for tota power transmission, we saw that gamma L shoud be which aso impies L shoud be equa to o. And such a condition where the entire incident power is transmitted to the oad is caed matching, impedance matching. So impedance matching is that condition where the entire incident power is transmitted to the oad and that happens when my oad impedance is equa to the characteristic impedance. Now this concept of oad impedance this refection coefficient, we saw the reationship that reates the 2, gamma L and L. The reationship is nothing but a biinear transformation from one coordinate to another. So once again if I write the question for gamma L (Refer Side Time: 25:7) 1 1 z jx This is a biinear transformation of. A specia chart which does this transformation for a impedances is what is known as a Smith chart.
(Refer Side Time: 25:36) 1 1 z jx So if we want to draw 1 st of a the definition of Smith chart is ike this or this can aso be written as this sma where this sma is equa to the normaised what you ca as normaised impedance. This is in terms of the normaised resistance and reactance. Sma can be written ike this. (Refer Side Time: 26:36) Now, if we pot this transformation, this particuar transformation for various vaues of, the curve that we get is something ike this. Now this ine as you can see, this red and bue ine is what is caed as a constant resistance and constant a constant reactance ine and upon doing this biinear transformation, they are transformed to these circes. So is the R equa to ine, this is the R equa to.5 ine. Sorry, this is X equa to ine, this is X equa to +.5 ine, this is X equa to -.5 ine: this is X equa to-1 ine, this is X equa to 1 ine. These ines are ike the R equa to one ine, R equa to.5 ine. And this is the R equa to ine. This is the way conversion is done. Now 1 st thing we have to know is why do we do this conversion? We do this conversion because the entire right haf of the pain this is our pain and this is our gamma pain. The entire right
haf of our pain is now confined within this entire circe. The circe have a radius 1. This heps us to visuaise a passive devices because for a passive devices, R is greater than and that is a convenient way of expressing it. (Refer Side Time: 3:1) So we can write this as suppose this is a unit circe, the circe have been unit radius and within the circe, we have gamma L moduus esser than 1 and outside the circe, we have gamma L moduus greater than1. So this corresponds to a active devices and this corresponds to a passive devices. In the same way that we obtained the curve from vaue, we shoud aso be abe to get the curve, Smith chart form the Y vaues and such a Smith chart is caed as Y Smith chart.
(Refer Side Time: 3:52) Now suppose this is our Smith chart, then the Y Smith chart wi be 18 oriented from the Smith chart and ike this. This wi join. (Refer Side Time: 31:32) So if we can go back to the sides the monitor for a moment, these are the various configurations of the impedance and admittance Smith charts. Say this is our impedance Smith chart, then the top haf we wi have inductive i.e for the top haf, the reactance X wi be positive and for the bottom haf, reactance X wi be negative.
For the Y Smith chart, it is just the opposite. Top haf is capacitive, bottom haf is inductive. For the Smith chart, the right-hand side is open. Open means infinite impedance. And the eft hand side is short or impedance. Now, a combination of the 2 means -Y Smith chart as we were discussing as shown here. And a point that is potted for the Smith chart, wi have the same sense for the Y Smith chart. That particuar point on this -Y Smith chart wi have the same sense for the Y Smith chart. Now one fina concept that I want to introduce is what is caed as the votage standing wave ratio. (Refer Side Time: 32:44) votage standing wave ratio (SWR) z e e e e j z j z j z j 2 z ( ) (1 ) max min SWR (1 ) (1 ) 1 max 1 min Maybe we can go to the sides on the monitor itsef. So we saw that the votage at any point is on a transmission ine is given by this reationship which in turn is aso reated to the refection coefficient. The maximum votage at any point and the minimum votage at any point, maximum means the maximum that can be achieved over time, minimum aso means minimum that can be achieved over time is given by this ratio and by this reationship the ratio of these 2 vaues is what you caed as the votage standing wave ratio. Now, SWR was a very important concept in the oden days when we coud not directy measure the refection coefficient. It is something that can be directy measured with a motor because it is the ratio of the maximum to the minimum votages. So, usuay for good microwave matching, we require SWR of ess than 2. SWR ess than 2 is considered a good matching or as we saw from the concept of impedance matching that when most of the incident power is
transmitted to the oad. So in terms of refection coefficient, that transates is to gamma L equa to. In terms of SWR that transates to SWR esser than 2. So in this course, in this modue, we covered some of the more standard parameters associated with microwave circuits. In the next modue we sha be further covering the more advanced concepts. Thank you