Why TEM doesnot exist in waveguide while TE and TM are possible

This post again discusses the propagation in a waveguide. A waveguide is a single conductor with a dielectric inside it (Air is also a dielectric). TEM mode is characterised by electric fields and magnetic fields perpendicular to one another and perpendicular to the direction of propagation. Inorder to have such a configuration, there has to be a source of electric field at the centre from where E fields originate and terminate on the outer conductor and there has to be a current source which can generate magnetic fields. For TEM mode the current flow should be along the axis of the waveguide, which creates rotating magnetic fields which are normal to the electric fields generated due to the (moving) charges in the current carrying conductor. Since such a current source is absent and waveguide being a single conductor configuration, TEM mode cannot exist inside a waveguide. Also it is evident from the above explanation that for TEM mode to exist, presence of atleast two conductors is compulsory.

Now the question is what are the other field configurations possible?

We so far talked about conduction currents along the axis of the waveguide whose absence prevented TEM mode. But there are another class of currents called as Displacement Currents, whose existence is proposed to explain flow of currents in a capacitor. Displacement current is formed by time varying Displacement Field (D). Now imagine time varying Electric Field to be present along the axis of the wave guide. This causes a time varying Displacement Field in the same direction. This Displacement field in turn results in Displacement Currents which can serve as the source of magnetic fields. The direction of Magnetic fields can be found from amperes law and deduced to be in the transverse plane. Thus "Electric field along the axis and Magnetic field in the transverse plane" is a non-vanishing combination inside a waveguide. Such a mode is called Transverse Magnetic(TM) mode.

If a axial time varying magnetic field is present in the waveguide, it results in Electric Fields in the transverse plane from Faraday's law. Thus "Magnetic field in the axial direction and Electric Fields in the transverse direction" is another possible non-vanishing combination. This mode is called Transverse Electric(TE) mode.

Interaction of electric fields with conductors

An electric conductor has very low resistance. It's conductivity is extremely high. Since current density is product of conductivity and electric field inside a conductor, electric fields tend to zero to avoid the condition of infinite current density. Also, for such conductors, the current flows in a very thin layer on its surface. Since fields cannot exist inside the conductor, all of the incident energy is reflected in the case of a perfect conductor. This reflection can be seen as the following steps

1. Incident fields induce currents over a thin area on its surface.
2. The currents re-radiate fields into space.

Thus a perfect conductor behaves as a perfect reflector of electromagnetic energy.

The Curl of a Vector field

Essentially, curl is a property exhibited by certain kinds of vector fields. Curl quantifies the net rotational effect of the field. Imagine a fluid flowing in a two dimensional space with certain velocity. Consider a tube that bends back onto itself to be placed in the flow. Now if you imagine that all the fluid except the one present in the tube is frozen, you would see the fluid still under motion due to its inertia. If it happens that the velocity vector of the fluid had non-zero curl, then we would be seeing water gushing around with net momentum inside the tube.

Generally speaking, curl of a vector indicates the presence of tangential components of the vector field that can form a closed loop.

Curl= average tangential component times circumference of the closed loop under consideration.

Standing Wave Ratio explained

Imagine water flowing through a flexible pipe. Assume that the water is being pumped by a motor and the other end of the pipe is with us. Now the amount of water coming out of the other end can at best be equal to the amount of water sucked by the pump. Now consider there is a mild twist in the pipe. What does this mean to flowing water? Water sees a portion on pipe that is resisting flow and eventually sending back a portion of water coming towards the twist/bend. This is electrical terms can be told "Water faces a high resistance." In other words, until the bend the water was facing very low resistance and all of a sudden, at the bend it begins to face a much higher value of resistance. This mismatch in resistance is causing a portion of water coming in to flow back eventually causing what is known as "reflections" of water. In a nutshell, resistance mismatch on the line( caused by the bend) resulted in reflections and in the process it is worth to note that not all water sucked by the pump is reaching the other end.

Let us now migrate to transmission line parlance and connect the above scenario with that world. Energy sent down the transmission line is guaranteed to reach the other end "maximum" only when there is no change in the impedance as seen through out the line. When there is a impedance change on the line, there is said to be an impedance mismatch. And when there is an impedance mismatch, RF energy is partially or completely reflected back. A way to quantify these reflections is Standing wave ratio. Reflected waves interact with incident-forward travelling waves to form what is known as standing wave in which the positions of maximum and minimum are constant. For a symmetrical wave sent down, the ratio of maximum and minimum should be the same in magnitude. This ratio of maximum to minimum is called as standing wave ratio. Lower the standing wave ratio, better is the matching and higher is the power delivered. Higher the standing wave ratio, worse is the matching, higher are the reflections and lesser is the power delivered.

Waveguides

In this write up, I attempt to demystify propagation of energy in waveguides without touching upon complex mathematics which anyway is found in literature. Traditionally, at lower frequencies upto VHF and UHF, coaxial cable has been obvious choice for transmission of energy. However, skin effect severely degrades the performance of such lines at high frequencies and this presses us to look at an alternative that could let us transmit energy not in the form of currents but in the form of fields. Simply using an antenna to radiate fields in to free space is not viable due to the inherent path losses involved. We look for a setup that could 'guide' energy in the form of fields from one point to another.

It is well known that the reflection coefficient of a surface made out of pure conductor is -1. Which means that an EM wave incident on a metallic surface is turned back completely with its phase inverted by 180 degrees. If we could find an arrangement such that the EM waves are completely reflected back into a region from 'all' sides and the arrangement is such that the waves are also carried forward, what we arranged is a waveguide.

This is precisely what a waveguide does. It is made out of a nearly perfect conductor to make the reflection coefficient as close to -1 as possible. This would mean little penetration of fields into the thickness of the conductor and hence little losses in the walls.

Having established that wave propagation is indeed possible by multiple reflections from all sides, Iam rather tempted to assert at this point that, not all frequencies are willing to go down the waveguide of given dimensions.

In my coming posts, I will attempt to explain further as what kind of arrangement accommodates a given frequency. I hope you enjoyed reading this. A rather lucid explanation is found in the book "Electronic communication systems" by John.F.Kennedy.

What is MIMO?

World as we all know is growing smarter by the day. We see smart phones, 3G, 4G, WLANs becoming integral part of our lives. At the crux of all progress lies the fact that we desperately seek high data rates. We can't wait any longer than few seconds for a video to be flashed across our screens. To add to this ever increasing thirst for speed, humans also seek quality. We migrated from gramophones to tapes to optical disks to digital signal processors for high fidelity. Quality comes only with higher volume of data. Bitter but true! Compare a real media filesize and its quality with mp3 or mp4  filesize and its quality.

A MIMO or a Multiple Input Multiple Output system is a clever solution to achieve high data rates. It is a technique which combines tweaking with antennas and DSP to get blazing speeds. We all know that when a signal is fed to antenna, it takes several paths before it reaches a receiver. Also if one manages to keep several receivers-spaced few wavelengths apart- in the vicinity of a transmitter, they all receive and receive signals of different magnitudes and phases which is primarily attributed to multipath propagation of signals before reaching each receiver.

MIMO cashes on this thing-multipath effect. A data stream to be transmitted is scrambled, encoded and interleaved only to be split into several parallel data streams. These parallel data streams are fed to equal number of transmitters. Effectively, what the design has achieved is that data has been divided into chunks with each chunk been thrown to the receiver with a separate transmitter.  

Remember that each of the transmitter throws data to the receiver, with the receiver getting it from multiple paths. Here lies the key to MIMO. If one has a fixed spatial receiver and transmitter configuration, the receiver always gets a uniquely weighed combination of the transmitted signals. In other words, each receiver receives a unique pattern from all the transmitters-the pattern being decided by the paths taken by the signals coming in. If one employs more that just a single receiver, all of them receive all the transmitted signals but each of the receiver gets a uniquely weighed version of the parallel data streams.

With the help of Digital Signal Processing Algorithms, it is possible to analyze and extract all the transmitted parallel data streams. These data streams are descrambled and decoded to get back the original transmitted message. Below figure illustrates the operation of MIMO configuration.

 
This is how MIMO quenches thirst for high data rates.Transmitting two or more data streams in the same bandwidth multiplies the data rate by the number of streams used.

Hello World!

The intent of creating this blog is to share my understanding of the revered fields-Electromagnetics, Antennas, Microwaves. Though there have been attempts by me previously to maintain such blogs on the Internet, they have largely been derailed by vicious schedules of myself. Here I come again with a new avatar this time and I hope this blog will stay alive for the longest possible time.