noc19-ee21 Lecture 33-Friis Transmission Formula

Welcome to NPTEL MOOC on electromagnetic waves in guided and wireless media. In this module, we will look at what is called as freeze transmission formula, which tells us how the power that you receive at the receiver will vary and how does it depend on the transmit and receive antennas, okay. So we have already set up the basic ideas about the received power. So when there is an electromagnetic wave impinging on the antenna, what you have to do is to look at the pointing power density and then multiply that pointing power density with the effective aperture. Now obviously, the effective aperture may be oriented in a different direction than the actual aperture, meaning that you have to take into account the possibility that the field and the aperture, if they are perpendicular to each other, then there would not be any power that is intercepted by the receiving antenna. On the other hand, if the orientation is complete, meaning that if the EM wave is propagating in this direction and the aperture is also in the same, you know, orientation so that you can grab the maximum amount of power, then you will essentially obtain maximum power. So this concept is also related to the effective length, meaning that the orientation, the vector nature of the electromagnetic wave means that unless the effective length or the effective aperture are oriented appropriately parallel to the direction of propagation or rather perpendicular to the direction of propagation and you would not have maximum or you would not have the maximum possible power to be grabbed by the receiving antenna. And we express this effective aperture in terms of the effective length in the previous module and we also related what would be the relationship between effective aperture and directivity. Although I did not derive it, I gave you the equation because that is very important for us in the next step of development and you can see the important relationships that we developed in the last module being summarized here. Effective aperture scales as lambda square at least for the electric short dipole antenna. It may not be true for other antennas, so you have to be careful as to what antenna is being used and, you know, use the effective aperture of that particular antenna. And then the power that you receive will be proportional to the incident power density that would be there on the antenna, the receiver antenna times the effective aperture that you can have. And then you have this gain of the antenna being related to the directivity in this particular manner, that is efficiency times D. Usually you take efficiency to be equal to 1 in order to simplify the calculations, but in practice efficiency is something that you have to experimentally calculate. And of course, when I write G and D, you have to remember that G and D are dependent on theta and phi. So, these are the patterns that you are actually looking at and the pattern is dependent on theta and phi. However, many times we only use the maximum value of D because we do want to orient or receive antennas in such a way that we are obtaining the maximum power. So, for that, you will have to orient it in the direction where the beam energy is maximum. But please note here one very important thing, when I say maximum power, I only mean the maximum power that one can extract from the freely propagating electromagnetic wave by putting up an antenna. So, the antenna in the vicinity of the antenna, there is an electromagnetic wave propagating and you put your antenna in order to grab as much power from that as possible. But that is not the whole story because that power would effectively show up as a voltage across the antenna terminals as we have seen. And that voltage along with the antenna impedance, which we called as ZA, will then determine how much voltage is actually transmitted to the load that gets connected to the receiving antenna. So, if for example, you have an RF block, the RF receiver chain connected to the antenna and antenna is the front end, usually the next block would be a low noise amplifier followed by a filter and so on. So, the input impedance of the low noise amplifier is the one that will interact with the antenna impedance and the antenna voltage that would appear at the antenna output terminals. So, I am not talking about the power, maximizing power transmission there, I am maximizing the power transmission in here. I am orienting my antenna in such a way that I grab as much as energy from the electromagnetic wave in order to maximize the antenna terminal voltage VA. From there I have to, you know, like I can do whatever I want to do in order to maximize the power transfer, that is a separate thing. And of course, when I do a conjugate matching, you know, with ZL equal to ZA conjugate, then I can extract maximum power from the equivalent circuit of the antenna that we have already seen. And that is another maximized power. And we have already seen that, sorry, we have already seen that that power that you are going to obtain at the load will always be less than that maximum power that you can actually obtain. And the maximum load power that can be delivered by the antenna to the load that is connected is given by this expression, right. So, you have this expression that we derived in the last class. In fact, from this expression is what we derived the effective aperture and utilizing the value of radiation resistance we found out for the short dipole or for a dipole with, sorry, delta Z of lambda by 4, we found that A effective was roughly 0.12 times lambda square. So, these relationships and where I am maximizing the powers are actually very important for us to know. Now, that we have these relationships, we can complete the derivation that we were looking for and understand how much is the power that you would obtain at the antenna terminals when you consider both the transmit antenna as well as the receiving antenna. So, that is what we are going to do. So, we have a transmit antenna. I am just representing this transmit antenna in the form of a, you know, this horn, but of course that is not the antenna that we are considering, it is just a pictorial representation of the antenna, right. Now, suppose I connect this antenna to a source which delivers to the transmit antenna the amount of power Pt, okay. So, I am delivering this power Pt to the antenna and if the antenna has a gain of Gt, which of course would be proportional to theta and phi, remember, this would be a function of, so not proportional, it will be a function of theta and phi. So, you have this gain, which is the pattern, right, the power pattern of the antenna. The actual power that is radiated onto the free space or the connection between transmitter and antenna, medium of connection between antenna and transmitter antenna would be, the radiated power will be Pt times Gt. In the given direction of theta and phi, it would be Pt times Gt. I will suppress the notation of theta and phi in these expressions just for simplification, but you should remember that technically if I change theta and phi, that is if I move from one position to another position, the receiving antenna or the transmit antenna, then the power that is received at the receiver will also change, okay. Anyway, at this point, we do not really have any receiving antenna, we have just taken the transmit antenna with its gain pattern and then supplied an amount of power Pt, so that power will be distributed by the antenna in space depending on the direction pattern or the antenna pattern, right, whatever the power pattern that you have, that energy will be distributed in that way. So, if for example, the antenna power pattern is in this manner, okay, this is a 3D picture that I am showing, so with two additional side lobes, what it means is that the power is distributed in this direction, okay, in this way, the power is distributed in the maximal way and some amount of power is also distributed here, which if you do not keep an antenna, you are only going to lose out that power, okay. Anyway, so that was just a pictorial way of representing this radiated power. Now, that radiated power is okay, if you imagine that you are at a distance R from the transmit antenna, you can find out what would be the power density, okay, the power density will be the power being radiated or power being carried by the antenna divided by 4 pi R square, where R is the distance from the transmit antenna, okay, I am using small case R, customarily this is actually written as a capital R indicating the radius of the sphere that of course, there is no actual sphere, it is an hypothetical sphere around the antenna whose radius is R and this is the power density, right. Now, if this power density were to appear at the input terminals of the antenna, okay, if this were to appear at the input terminals of an antenna, then what would be the power that the antenna would receive? The power that the antenna would receive will be whatever the effective aperture of the receiver is, right, times the power density. So, if I call power of the receiver P rx, that would be the radiated power density that is present or its value at the antenna input terminals times the effective aperture, but I know that effective aperture can be rewritten in terms of D. So, I can rewrite in also I can substitute for P or the radiated power is as product PtGt. So, I am going to do that PtGt divided by 4 pi R square and effective aperture is basically lambda square by 4 pi times D of the receiver, so I am going to write this as Dr, okay. And what is Dr and Gr related to? Dr is equal to Gr divided by efficiency of the antenna, assuming that this efficiency is maximum, that is it is taken to be unity, then I can simply replace Dr by Gr, okay. So, let us collect all these terms together, so I have PtGt, which of course is oriented with respect to theta and phi and Gr is also a quantity that is dependent on theta and phi. So, you can see that it is not only that receiver antenna, you know, maximizes the power that is delivered to it, it is both the transmit antenna as well as the receive antenna. So, if the transmit antenna is radiating this way and the receive antenna is kept here, then you will only intercept a part of it. So, if the receive antenna is kept here, you will not receive any power because the power is kind of travelling in a different direction, so it is important to match the beam directions of both the transmit as well as the receiving antenna, this perhaps is a very commonsensical thing, but that is kind of mathematically included in these expressions, right. So, you have PtGtGr divided by, so there is a 4 pi here and another 4 pi here, there is a lambda in the numerator, so I can rewrite this one as lambda by 4 pi R whole square, okay. So, this is the power that I am going to receive at the input terminals of the antenna and this power would essentially generate certain antenna, certain voltage at the antenna output terminals, which when you connect through the antenna impedance ZA, which is usually complex to the load impedance ZL and conjugate match, right. So, if you conjugate match ZL to ZA, then you will extract maximum power at the load, okay. Of course, in practice, none of these conditions can be met ideally, meaning that you can extract maximum possible, the overall efficiency of the link will always be less in the sense that it will be less than what one can achieve in a maximum way, okay. But if you take care as much as possible to orient the antenna in the direction of the same beam, maximized beam direction of the receiving antenna or the receiving antenna oriented in the transmitting antenna direction and maximize the, I mean maximize the power transfer to the load by conjugate matching, then you can increase the efficiency of the system to a very high level. Of course, please also note that it is not the power that delivers that finally tells you how much is the information content. Information content depends on the voltages because you are not coding them in terms of power. Usually the, if you are following some kind of a coherent coding, coherent modulation scheme, then information may be sitting in the phase and getting the phase information is more crucial than getting the power information. Because if you are only modulating the phase, information is in phase and you have modulated the phase of the carrier, then it becomes important to extract the phase, at which point you may find maximizing power by conjugate matching to be useful or not, but that depends on the problem at hand, okay. I am just telling you that only maximizing the power is not going to give you the full benefit of the link. You have to understand the problem and see where information resides. Of course, if the information resides in the form of power, like you know on-off power keying systems, then maximize the power by all means, okay. So anyway, this formula that we have written for the received power in terms of the transmit antenna, in terms of the input power to the transmit antenna and the antenna pattern is actually very important relationship, okay. In fact, we call this term, which is getting multiplied to PtGtGr, this term that is there in the numerator, we can rearrange this term to show up in the denominator. We can write this as PtGtGr, suppressing theta and phi, divided by 4 pi r by lambda whole square, okay. So you can write it in this particular manner and call this factor 1 by 4 pi r lambda square as the path, sorry, Lp as 4 pi by lambda whole square times r square as the path loss, okay. And you can see that the path loss is kind of going as 20 dB per decade, if you actually convert all of this Lp in terms of dB by taking 10 log Lp and I am taking 10 log because this is power, not voltage, right. So this expression of freeze transmission formula is relating the power, not the voltage. Therefore, I take 10 log of this and express the path loss then and remember Lp comes in the denominator. So you have PtGtGr divided by Lp and of course, Lp is a function of r, so maybe better we will write it as Lp of r and for the ideal free space transmission, this turns out to be 20 dB per decade, okay. So if you plot as a function of the distance, right, r, what is the loss that you are going to get or rather the total power that is received, then this power steadily decreases with a slope of about 20 dB per decade. So in decade does not really mean that you are changing in frequency, here it means that it is a distance that changes. So you take r equal to 1 and let us say the value of the received power is 110 dBm, okay, this is extremely large value, I am just giving you some numbers to tell you what actually happens and if you now go to r equal to 10, right, then you have, you know, 10 meters. So say r equal to 1 meter and r equal to 10 meter, then the power that you are going to measure here would have dropped by 20, meaning from 110 or rather let us keep it realistic. So we will keep it some minus 70 dBm. So what you are going to get will be minus 90 dBm of power, okay, that is what it meant by 20 dB. Let me also tell you that this is the best possible scenario that you can have, okay. You have in the entire universe nothing else except a transmit antenna and a receive antenna, okay. So that the power loss that happens between these two is directly proportional to, I mean is actually the best case scenario of 20 dB per decade, okay. Now this happens because the amplitude scales as 1 by r, meaning that as you move away from the transmit antenna, the amplitude goes as 1 by r and because power is related to amplitude square, the power goes as 1 by r square, okay. This is very important. You are looking at voltage which is going as 1 over r and power is going as 1 over r square, okay. What happens if r, instead of the input power decaying as 1 over r, it starts to decay as 1 over r to the power 1.5, then the power would go as 1 over r cube because you are going to take this and square it up, right, so you are going to get 1 by r cube and now you can immediately see that path loss in dB actually goes as 30 dB per decade, okay. In the other case, if r, if the amplitude itself falls as 1 over r square, then the power will go as 1 over r power 4 and Lp will go as 40 dB per decade, okay. You may ask, where are these 1.5 factor and 2 factor for the amplitude coming from? They did not exist in the, you know, equations so far, even the spherical wave that we considered was actually e power minus jk r by r. So you do not understand where this amplitude going, I mean, growing faster than 1 by r coming from. In fact, this path loss exponent that you will see with an ideal value of 1 or you know, if you are looking at the power in the path loss exponent, then that would be the power going as r2, so 1 over r amplitude, power goes as 1 over r square. So this is the path loss exponent, okay. If the path loss exponent, the minimum value is 2, it cannot be less than that, at least in the far fields that we are considering. Then r cube means that the path loss exponent is 3, here r4 means the path loss exponent is 4. Steadily as your path loss exponent changes, the losses increase rapidly, okay. And this is the situation that you will find yourself when you are working with real-world scenarios. Like in real world, you may put up your transmit antenna, okay, on some building you may put it up and or maybe close to the building, let us say, and then you have a receiver antenna somewhere else. So if you are imagining the wireless scenario, you have a base station and the base station antenna is present, you know, put some on a tower, you locate the base station antenna and you are a user who is, you know, walking around nicely, talking to someone and we do not of course expect that between the base station antenna and yourself, there is absolutely nothing else in the world. Unfortunately, there will be buildings, there will be trees, the atmosphere may not be nicely, you know, weathered condition, it may be raining, it may be foggy, it may be snow, all these things essentially contribute to what is called as a clutter and they will increase the losses and this path loss exponent is one way of representing those increased losses, okay. So if the expected power is not falling as 20 dB per decade, but it is falling as 40 dB per decade, then you can, I mean, then you can actually go back and change the, you know, value of the path loss exponent from 2 to 4 or rather 2, yeah, 2 to 4 and that is your attributing the extra losses into an effective path loss exponent, okay. So that is what you are doing. Of course, we will have lot more to say about these buildings, trees and other things that come around and reduce your signal levels and, you know, when we start talking about wireless channel model in the upcoming modules, but for now, please keep in mind that power loss exponent of 2 and hence a loss of 20 dB per decade is kind of the ideal that you can push. In practice, you do not get such an ideal scenario. The losses are either 30 dB per decade or in worst case, in urban, semi-urban, there are also again variations with respect to what environment you are looking at. In a forest, that will be even drastically low, right, because there is lot of trees in the forest, I hope, right. So all these attributes or all these losses can be attributed to an increased path loss exponent and you can tweak the equation of freeze transmission formula to reflect these additional losses, okay. Life would have been very easy if it was just the addition of this or rather increase of these losses. But what actually happens as people have figured out by measurements is that the path loss variation, whether it is 20 dB or 30 dB per decade depending on the loss thing, is actually not the final loss profile, okay. It would not be like this single line. The actual path that you are going to see would involve some local variations, okay. These local variations are called as shadowing or shadow losses. Shadowing means that if the electromagnetic wave is coming in, there is an opaque screen somewhere in between, right. So a tree suddenly comes up, the tree is already there, but because you moved in a direction that you are now seeing a tree, that tree will act like an obstacle or let us say you are receiving a call in your home, but for some reason you come out of the home, right, then there is a building behind you, that is an obstacle, right. Yes, electromagnetic waves go around these obstacles, suppose you are travelling in near mountainous region, the electromagnetic waves actually go around the mountains also, but those actually happen at very low frequencies, okay. The reason why that happens at low frequencies and what this phenomenon is called, we will discuss it later on in the other modules, but what it actually does is that sometimes these obstacles are not so bad, they can actually give you slightly higher loss, sometimes they can give you lower loss. But those things actually depend on how many obstacles are there, what is their position, what is the base station position, what is the receiving antenna position and so on. So it is kind of rather difficult to predict. So people rely on many empirical formulas and if you know the, and especially in the indoor wireless channels, you know where the obstacles are. For example, if you are sitting in your room, you know where your computer screen is, you know where you hang the clothes, you know where the television is, you know, kept or you have a Almera. So you know those locations. So in that case, because you know all the locations perfectly, you can do what is called as ray tracing and then estimate what is the total loss, okay. It will also show a large scale reduction. So if your source happens to be at the top of your head, and then you are in one corner of the room, that length will be different, okay, and that would be the direct path that you are getting. But because of the obstructions and obstacles, you may also get additional paths, okay. And sometimes those signals can add up, sometimes those signals do not add up and you get increased or a decreased loss, okay. In addition to this shadowing effect, technically the shadowing effect occurs because of the obstacles blocking EM waves and to understand that correctly, you need to know what is called as diffraction and we are going to look at diffraction in the upcoming modules. Diffraction is also closely related to interference. Interference means two waves can interfere with each other and can either add up or subtract, okay, depending on how they add up in terms of their phase. So if they are in phase, then they can add up, if they are out of phase, then they do not add up, okay. In fact, the point of, you know, interference will also be taken up in the next module when we discuss antenna arrays or at least two element antenna array, there you will see that sometimes the signal from two antennas can add up, sometimes the signal do not add up. In fact, it can be out of phase and you will get it 0. But coming back to the wireless channel model or the channel model that we were considering, the black line would represent the, you know, free space path loss that you are going to get or the large scale path loss and LP may not be, you know, 20 dB per decade, it may be more than that, okay. In addition to shadowing, which can increase or decrease the losses, but this is also depending on the environment that you are working in, there are very, very small scale channel variations, okay. So there are these small scale channel variations and the scale of these channel variations is roughly lambda by 2, okay. It is less than lambda by 2, but this is approximately lambda by 2 and as what it actually means is that suppose you are at this position, you know, suppose you are at this arc, now if you move to R, which is about lambda by 2, you will see that the power that you have, sorry, I will use this green one, you see that the power that you get is lesser than what the power that you got previously, okay. If the time scales that you are looking at are so sensitive to the received power, then this kind of variations that you are going to get around the large scale variations is called as small scale variations or small scale fading, okay. You may have heard of what is called as Rayleigh fading and Rayleigh fading is one such type of fading that occurs because of the small scale variation. So you have actually 3 types of path losses that you can expect in a wireless channel. One there is a large scale path loss that depends on, you know, like that is the basic path loss that you are going to get and that is simply because transmit and receive antennas are kept separately, okay. They have to fall off as 1 over R amplitude or 1 over R square as power. Typically they fall off slightly higher than that and those effects can be taken into the large scale path loss. Then there is shadowing effect because there are additional obstacles and EMF can be reflected from those additional obstacles or diffracted from the additional obstacles, increasing or decreasing loss around this basic variation. And finally, at the very micro level, you know, as you move in terms of lambda, then there are variations in the received power and those small scale variations are called as small scale fading or Rayleigh fading, okay. In general, wireless channel is what is called as a multipath channel, meaning that there could be more than 1 path and in fact, there is a transition from 1 path to multipath channels and the corresponding losses that you are going to get, multipath channels are characterized by small scale fading and large scale path loss, okay. Large scale path loss is usually represented by square root g and small scale path loss is represented by small h, normalized to unity, this is in amplitude, so power wise it would be g. The only surprising thing is that losses are being represented by g, where g is usually used for gain, but in the literature for channel modeling, unfortunately, someone back in time chose these losses to be called as path gains, okay, they are not really gains, they are only losses, you do not have an amplifier in between, they are all losses, but someone labeled it as gain. So, now we are stuck with that terminology, so we call the large scale path gain or rather large scale path loss as large scale path gain, although we know that it is loss and we denote that one by square root g, okay. The small scale path loss for the amplitude is denoted by square root h and in general, the wireless channel will have a path loss of square root g into h and if you have transmitted an information which is represented by x of t, the received signal y of t can be written in this particular manner, okay. So, we will come back to these, you know, losses and other things after we have covered one additional topic in antennas and that is of arrays, okay. Thank you very much.