Category: microwave Engineering

Microwave Engineering – Example Problems In this chapter, let us have some fun by solving a few numerical problems related to microwaves. Problem 1 A transmission system using a $TE_{10}$ mode waveguide of dimensions $a = 5cm, b = 3cm$ is operating at 10GHz. The distance measured between two minimum power points is 1mm on a slotted line. Calculate the VSWR of the system. Solution Given that $f = 10GHz; a = 5cm; b = 3cm$ For $TE_{10}$ mode waveguide, $$lambda_c = 2a = 2 times 5 = 10 cm$$ $$lambda_0 = frac{c}{f} = frac{3times10^{10}}{10times10^9} = 3cm$$ $$d_2-d_1 = 1mm = 10^{-1}cm$$ We know $$lambda_g = frac{lambda_0}{1-({lambda_0}/{lambda_c})^2} = frac{3}{sqrt{1-({3}/{10})^2}} = 3.144cm$$ For double minimum method VSWR is given by $$VSWR = frac{lambda_g}{pi(d_2-d_1)} = frac{3.144}{pi(1times10^{-1})} = 10.003 = 10$$ Hence, the VSWR value for the given transmission system is 10. Problem 2 In a setup for measuring impedance of a reflectometer, what is the reflection coefficient when the outputs of two couplers are 2mw and 0.5mw respectively? Solution Given that $$frac{P_i}{100} = 2mw quad and quad frac{P_r}{100} = 0.5mw$$ $$P_i = 2 times 100mw = 200mw$$ $$P_r = 0.5 times 100mw = 50mw$$ $$rho = sqrt{frac{P_r}{P_i}} = sqrt{frac{50mw}{200mw}} = sqrt{0.25} = 0.5$$ Hence, the reflection coefficient $rho$ of the given set up is 0.5. Problem 3 When two identical couplers are used in a waveguide to sample the incident power as 3mw and reflected power as 0.25mw, then find the value of $VSWR$. Solution We know that $$rho = sqrt{frac{P_r}{P_i}} = sqrt{frac{0.25}{3}} = sqrt{0.0833} = 0.288$$ $$VSWR = S = frac{1+rho}{1-rho} = frac{1+0.288}{1-0.288} = frac{1.288}{0.712} = 1.80$$ Hence, the $VSWR$ value for the above system is 1.80 Problem 4 Two identical 30dB directional couplers are used to sample incident and reflected power in a waveguide. The value of VSWR is 6 and the output of the coupler sampling incident power is 5mw. What is the value of the reflected power? Solution We know that $$VSWR = S = frac{1+rho}{1-rho} = 6$$ $$(1+rho) = 6(1-rho) = 6 – 6rho$$ $$7rho = 5$$ $$rho = frac{5}{7} = 0.174$$ To get the value of reflected power, we have $$rho = sqrt{frac{{P_r}/{10^3}}{{P_i}/{10^3}}} = sqrt{frac{P_r}{P_i}}$$ $$or quad rho^2 = frac{P_r}{P_i}$$ $$P_r = rho^2.P_i = (0.714)^2.5 = 0.510 times 5 = 2.55$$ Hence, the reflected power in this waveguide is 2.55mW. Learning working make money

Measurement Devices Among the Microwave measurement devices, a setup of Microwave bench, which consists of Microwave devices has a prominent place. This whole setup, with few alternations, is able to measure many values like guide wavelength, free space wavelength, cut-off wavelength, impedance, frequency, VSWR, Klystron characteristics, Gunn diode characteristics, power measurements, etc. The output produced by microwaves, in determining power is generally of a little value. They vary with the position in a transmission line. There should be an equipment to measure the Microwave power, which in general will be a Microwave bench setup. Microwave Bench General Measurement Setup This setup is a combination of different parts which can be observed in detail. The following figure clearly explains the setup. Signal Generator As the name implies, it generates a microwave signal, in the order of a few milliwatts. This uses velocity modulation technique to transfer continuous wave beam into milliwatt power. A Gunn diode oscillator or a Reflex Klystron tube could be an example for this microwave signal generator. Precision Attenuator This is the attenuator which selects the desired frequency and confines the output around 0 to 50db. This is variable and can be adjusted according to the requirement. Variable Attenuator This attenuator sets the amount of attenuation. It can be understood as a fine adjustment of values, where the readings are checked against the values of Precision Attenuator. Isolator This removes the signal that is not required to reach the detector mount. Isolator allows the signal to pass through the waveguide only in one direction. Frequency Meter This is the device which measures the frequency of the signal. With this frequency meter, the signal can be adjusted to its resonance frequency. It also gives provision to couple the signal to waveguide. Crystal Detector A crystal detector probe and crystal detector mount are indicated in the above figure, where the detector is connected through a probe to the mount. This is used to demodulate the signals. Standing Wave Indicator The standing wave voltmeter provides the reading of standing wave ratio in dB. The waveguide is slotted by some gap to adjust the clock cycles of the signal. Signals transmitted by waveguide are forwarded through BNC cable to VSWR or CRO to measure its characteristics. A microwave bench set up in real-time application would look as follows − Now, let us take a look at the important part of this microwave bench, the slotted line. Slotted Line In a microwave transmission line or waveguide, the electromagnetic field is considered as the sum of incident wave from the generator and the reflected wave to the generator. The reflections indicate a mismatch or a discontinuity. The magnitude and phase of the reflected wave depends upon the amplitude and phase of the reflecting impedance. The standing waves obtained are measured to know the transmission line imperfections which is necessary to have a knowledge on impedance mismatch for effective transmission. This slotted line helps in measuring the standing wave ratio of a microwave device. Construction The slotted line consists of a slotted section of a transmission line, where the measurement has to be done. It has a travelling probe carriage, to let the probe get connected wherever necessary, and the facility for attaching and detecting the instrument. In a waveguide, a slot is made at the center of the broad side, axially. A movable probe connected to a crystal detector is inserted into the slot of the waveguide. Operation The output of the crystal detector is proportional to the square of the input voltage applied. The movable probe permits convenient and accurate measurement at its position. But, as the probe is moved along, its output is proportional to the standing wave pattern, which is formed inside the waveguide. A variable attenuator is employed here to obtain accurate results. The output VSWR can be obtained by $$VSWR = sqrt{frac{V_{max}}{V_{min}}}$$ Where, $V$ is the output voltage. The following figure shows the different parts of a slotted line labelled. The parts labelled in the above figure indicate the following. Launcher − Invites the signal. Smaller section of the waveguide. Isolator − Prevents reflections to the source. Rotary variable attenuator − For fine adjustments. Slotted section − To measure the signal. Probe depth adjustment. Tuning adjustments − To obtain accuracy. Crystal detector − Detects the signal. Matched load − Absorbs the power exited. Short circuit − Provision to get replaced by a load. Rotary knob − To adjust while measuring. Vernier gauge − For accurate results. In order to obtain a low frequency modulated signal on an oscilloscope, a slotted line with a tunable detector is employed. A slotted line carriage with a tunable detector can be used to measure the following. VSWR (Voltage Standing Wave Ratio) Standing wave pattern Impedance Reflection coefficient Return loss Frequency of the generator used Tunable Detector The tunable detector is a detector mount which is used to detect the low frequency square wave modulated microwave signals. The following figure gives an idea of a tunable detector mount. The following image represents the practical application of this device. It is terminated at the end and has an opening at the other end just as the above one. To provide a match between the Microwave transmission system and the detector mount, a tunable stub is often used. There are three different types of tunable stubs. Tunable waveguide detector Tunable co-axial detector Tunable probe detector Also, there are fixed stubs like − Fixed broad band tuned probe Fixed waveguide matched detector mount The detector mount is the final stage on a Microwave bench which is terminated at the end. Learning working make money

Microwave Engineering – Useful Resources The following resources contain additional information on Microwave Engineering. Please use them to get more in-depth knowledge on this topic. Useful Video Courses 108 Lectures 14.5 hours Best Seller 173 Lectures 23 hours 248 Lectures 11 hours Featured 35 Lectures 2.5 hours 124 Lectures 12 hours 88 Lectures 6 hours Learning working make money

Microwave Engineering – Measurements In the field of Microwave engineering, there occurs many applications, as already stated in first chapter. Hence, while using different applications, we often come across the need of measuring different values such as Power, Attenuation, Phase shift, VSWR, Impedance, etc. for the effective usage. In this chapter, let us take a look at the different measurement techniques. Measurement of Power The Microwave Power measured is the average power at any position in waveguide. Power measurement can be of three types. Measurement of Low power (0.01mW to 10mW) Example − Bolometric technique Measurement of Medium power (10mW to 1W) Example − Calorimeter technique Measurement of High power (>10W) Example − Calorimeter Watt meter Let us go through them in detail. Measurement of Low Power The measurement of Microwave power around 0.01mW to 10mW, can be understood as the measurement of low power. Bolometer is a device which is used for low Microwave power measurements. The element used in bolometer could be of positive or negative temperature coefficient. For example, a barrater has a positive temperature coefficient whose resistance increases with the increase in temperature. Thermistor has negative temperature coefficient whose resistance decreases with the increase in temperature. Any of them can be used in the bolometer, but the change in resistance is proportional to Microwave power applied for measurement. This bolometer is used in a bridge of the arms as one so that any imbalance caused, affects the output. A typical example of a bridge circuit using a bolometer is as shown in the following figure. The milliammeter here, gives the value of the current flowing. The battery is variable, which is varied to obtain balance, when an imbalance is caused by the behavior of the bolometer. This adjustment which is made in DC battery voltage is proportional to the Microwave power. The power handling capacity of this circuit is limited. Measurement of Medium Power The measurement of Microwave power around 10mW to 1W, can be understood as the measurement of medium power. A special load is employed, which usually maintains a certain value of specific heat. The power to be measured, is applied at its input which proportionally changes the output temperature of the load that it already maintains. The difference in temperature rise, specifies the input Microwave power to the load. The bridge balance technique is used here to get the output. The heat transfer method is used for the measurement of power, which is a Calorimetric technique. Measurement of High Power The measurement of Microwave power around 10W to 50KW, can be understood as the measurement of high power. The High Microwave power is normally measured by Calorimetric watt meters, which can be of dry and flow type. The dry type is named so as it uses a coaxial cable which is filled with di-electric of high hysteresis loss, whereas the flow type is named so as it uses water or oil or some liquid which is a good absorber of microwaves. The change in temperature of the liquid before and after entering the load, is taken for the calibration of values. The limitations in this method are like flow determination, calibration and thermal inertia, etc. Measurement of Attenuation In practice, Microwave components and devices often provide some attenuation. The amount of attenuation offered can be measured in two ways. They are − Power ratio method and RF substitution method. Attenuation is the ratio of input power to the output power and is normally expressed in decibels. $$Attenuation : in : dBs = 10 : logfrac{P_{in}}{P_{out}}$$ Where $P_{in}$ = Input power and $P_{out}$ = Output power Power Ratio Method In this method, the measurement of attenuation takes place in two steps. Step 1 − The input and output power of the whole Microwave bench is done without the device whose attenuation has to be calculated. Step 2 − The input and output power of the whole Microwave bench is done with the device whose attenuation has to be calculated. The ratio of these powers when compared, gives the value of attenuation. The following figures are the two setups which explain this. Drawback − The power and the attenuation measurements may not be accurate, when the input power is low and attenuation of the network is large. RF Substitution Method In this method, the measurement of attenuation takes place in three steps. Step 1 − The output power of the whole Microwave bench is measured with the network whose attenuation has to be calculated. Step 2 − The output power of the whole Microwave bench is measured by replacing the network with a precision calibrated attenuator. Step 3 − Now, this attenuator is adjusted to obtain the same power as measured with the network. The following figures are the two setups which explain this. The adjusted value on the attenuator gives the attenuation of the network directly. The drawback in the above method is avoided here and hence this is a better procedure to measure the attenuation. Measurement of Phase Shift In practical working conditions, there might occur a phase change in the signal from the actual signal. To measure such phase shift, we use a comparison technique, by which we can calibrate the phase shift. The setup to calculate the phase shift is shown in the following figure. Here, after the microwave source generates the signal, it is passed through an H-plane Tee junction from which one port is connected to the network whose phase shift is to be measured and the other port is connected to an adjustable precision phase shifter. The demodulated output is a 1 KHz sine wave, which is observed in the CRO connected. This phase shifter is adjusted such that its output of 1 KHz sine wave also matches the above. After the matching is done by observing in the dual mode CRO, this precision phase shifter gives us the reading of phase shift. This is clearly understood by the following figure. This procedure is the mostly used

Microwave Engineering – Introduction Electromagnetic Spectrum consists of entire range of electromagnetic radiation. Radiation is the energy that travels and spreads out as it propagates. The types of electromagnetic radiation that makes the electromagnetic spectrum is depicted in the following screenshot. Let us now take a look at the properties of Microwaves. Properties of Microwaves Following are the main properties of Microwaves. Microwaves are the waves that radiate electromagnetic energy with shorter wavelength. Microwaves are not reflected by Ionosphere. Microwaves travel in a straight line and are reflected by the conducting surfaces. Microwaves are easily attenuated within shorter distances. Microwave currents can flow through a thin layer of a cable. Advantages of Microwaves There are many advantages of Microwaves such as the following − Supports larger bandwidth and hence more information is transmitted. For this reason, microwaves are used for point-to-point communications. More antenna gain is possible. Higher data rates are transmitted as the bandwidth is more. Antenna size gets reduced, as the frequencies are higher. Low power consumption as the signals are of higher frequencies. Effect of fading gets reduced by using line of sight propagation. Provides effective reflection area in the radar systems. Satellite and terrestrial communications with high capacities are possible. Low-cost miniature microwave components can be developed. Effective spectrum usage with wide variety of applications in all available frequency ranges of operation. Disadvantages of Microwaves There are a few disadvantages of Microwaves such as the following − Cost of equipment or installation cost is high. They are hefty and occupy more space. Electromagnetic interference may occur. Variations in dielectric properties with temperatures may occur. Inherent inefficiency of electric power. Applications of Microwaves There are a wide variety of applications for Microwaves, which are not possible for other radiations. They are − Wireless Communications For long distance telephone calls Bluetooth WIMAX operations Outdoor broadcasting transmissions Broadcast auxiliary services Remote pickup unit Studio/transmitter link Direct Broadcast Satellite (DBS) Personal Communication Systems (PCSs) Wireless Local Area Networks (WLANs) Cellular Video (CV) systems Automobile collision avoidance system Electronics Fast jitter-free switches Phase shifters HF generation Tuning elements ECM/ECCM (Electronic Counter Measure) systems Spread spectrum systems Commercial Uses Burglar alarms Garage door openers Police speed detectors Identification by non-contact methods Cell phones, pagers, wireless LANs Satellite television, XM radio Motion detectors Remote sensing Navigation Global navigation satellite systems Global Positioning System (GPS) Military and Radar Radars to detect the range and speed of the target. SONAR applications Air traffic control Weather forecasting Navigation of ships Minesweeping applications Speed limit enforcement Military uses microwave frequencies for communications and for the above mentioned applications. Research Applications Atomic resonances Nuclear resonances Radio Astronomy Mark cosmic microwave background radiation Detection of powerful waves in the universe Detection of many radiations in the universe and earth’s atmosphere Food Industry Microwave ovens used for reheating and cooking Food processing applications Pre-heating applications Pre-cooking Roasting food grains/beans Drying potato chips Moisture levelling Absorbing water molecules Industrial Uses Vulcanizing rubber Analytical chemistry applications Drying and reaction processes Processing ceramics Polymer matrix Surface modification Chemical vapor processing Powder processing Sterilizing pharmaceuticals Chemical synthesis Waste remediation Power transmission Tunnel boring Breaking rock/concrete Breaking up coal seams Curing of cement RF Lighting Fusion reactors Active denial systems Semiconductor Processing Techniques Reactive ion etching Chemical vapor deposition Spectroscopy Electron Paramagnetic Resonance (EPR or ESR) Spectroscopy To know about unpaired electrons in chemicals To know the free radicals in materials Electron chemistry Medical Applications Monitoring heartbeat Lung water detection Tumor detection Regional hyperthermia Therapeutic applications Local heating Angioplasty Microwave tomography Microwave Acoustic imaging For any wave to propagate, there is the need of a medium. The transmission lines, which are of different types, are used for the propagation of Microwaves. Let us learn about them in the next chapter. Learning working make money

Microwave Engineering – Magnetrons Unlike the tubes discussed so far, Magnetrons are the cross-field tubes in which the electric and magnetic fields cross, i.e. run perpendicular to each other. In TWT, it was observed that electrons when made to interact with RF, for a longer time, than in Klystron, resulted in higher efficiency. The same technique is followed in Magnetrons. Types of Magnetrons There are three main types of Magnetrons. Negative Resistance Type The negative resistance between two anode segments, is used. They have low efficiency. They are used at low frequencies (< 500 MHz). Cyclotron Frequency Magnetrons The synchronism between the electric component and oscillating electrons is considered. Useful for frequencies higher than 100MHz. Travelling Wave or Cavity Type The interaction between electrons and rotating EM field is taken into account. High peak power oscillations are provided. Useful in radar applications. Cavity Magnetron The Magnetron is called as Cavity Magnetron because the anode is made into resonant cavities and a permanent magnet is used to produce a strong magnetic field, where the action of both of these make the device work. Construction of Cavity Magnetron A thick cylindrical cathode is present at the center and a cylindrical block of copper, is fixed axially, which acts as an anode. This anode block is made of a number of slots that acts as resonant anode cavities. The space present between the anode and cathode is called as Interaction space. The electric field is present radially while the magnetic field is present axially in the cavity magnetron. This magnetic field is produced by a permanent magnet, which is placed such that the magnetic lines are parallel to cathode and perpendicular to the electric field present between the anode and the cathode. The following figures show the constructional details of a cavity magnetron and the magnetic lines of flux present, axially. This Cavity Magnetron has 8 cavities tightly coupled to each other. An N-cavity magnetron has $N$ modes of operations. These operations depend upon the frequency and the phase of oscillations. The total phase shift around the ring of this cavity resonators should be $2npi$ where $n$ is an integer. If $phi_v$ represents the relative phase change of the AC electric field across adjacent cavities, then $$phi_v = frac{2 pi n}{N}$$ Where $n = 0, : pm1,: pm2,: pm : (frac{N}{2} -1), : pm frac{N}{2}$ Which means that $frac{N}{2}$ mode of resonance can exist if $N$ is an even number. If, $$n = frac{N}{2} quad then quad phi_v = pi$$ This mode of resonance is called as $pi-mode$. $$n = 0 quad then quad phi_v = 0$$ This is called as the Zero mode, because there will be no RF electric field between the anode and the cathode. This is also called as Fringing Field and this mode is not used in magnetrons. Operation of Cavity Magnetron When the Cavity Klystron is under operation, we have different cases to consider. Let us go through them in detail. Case 1 If the magnetic field is absent, i.e. B = 0, then the behavior of electrons can be observed in the following figure. Considering an example, where electron a directly goes to anode under radial electric force. Case 2 If there is an increase in the magnetic field, a lateral force acts on the electrons. This can be observed in the following figure, considering electron b which takes a curved path, while both forces are acting on it. Radius of this path is calculated as $$R = frac{mv}{eB}$$ It varies proportionally with the velocity of the electron and it is inversely proportional to the magnetic field strength. Case 3 If the magnetic field B is further increased, the electron follows a path such as the electron c, just grazing the anode surface and making the anode current zero. This is called as “Critical magnetic field” $(B_c)$, which is the cut-off magnetic field. Refer the following figure for better understanding. Case 4 If the magnetic field is made greater than the critical field, $$B > B_c$$ Then the electrons follow a path as electron d, where the electron jumps back to the cathode, without going to the anode. This causes “back heating” of the cathode. Refer the following figure. This is achieved by cutting off the electric supply once the oscillation begins. If this is continued, the emitting efficiency of the cathode gets affected. Operation of Cavity Magnetron with Active RF Field We have discussed so far the operation of cavity magnetron where the RF field is absent in the cavities of the magnetron (static case). Let us now discuss its operation when we have an active RF field. As in TWT, let us assume that initial RF oscillations are present, due to some noise transient. The oscillations are sustained by the operation of the device. There are three kinds of electrons emitted in this process, whose actions are understood as electrons a, b and c, in three different cases. Case 1 When oscillations are present, an electron a, slows down transferring energy to oscillate. Such electrons that transfer their energy to the oscillations are called as favored electrons. These electrons are responsible for bunching effect. Case 2 In this case, another electron, say b, takes energy from the oscillations and increases its velocity. As and when this is done, It bends more sharply. It spends little time in interaction space. It returns to the cathode. These electrons are called as unfavored electrons. They don”t participate in the bunching effect. Also, these electrons are harmful as they cause “back heating”. Case 3 In this case, electron c, which is emitted a little later, moves faster. It tries to catch up with electron a. The next emitted electron d, tries to step with a. As a result, the favored electrons a, c and d form electron bunches or electron clouds. It called as “Phase focusing effect”. This whole process is understood better by taking a look at the following figure. Figure A shows the electron movements

Microwave Engineering – H-Plane Tee An H-Plane Tee junction is formed by attaching a simple waveguide to a rectangular waveguide which already has two ports. The arms of rectangular waveguides make two ports called collinear ports i.e., Port1 and Port2, while the new one, Port3 is called as Side arm or H-arm. This H-plane Tee is also called as Shunt Tee. As the axis of the side arm is parallel to the magnetic field, this junction is called H-Plane Tee junction. This is also called as Current junction, as the magnetic field divides itself into arms. The cross-sectional details of H-plane tee can be understood by the following figure. The following figure shows the connection made by the sidearm to the bi-directional waveguide to form the serial port. Properties of H-Plane Tee The properties of H-Plane Tee can be defined by its $left [ S right ]_{3times 3}$ matrix. It is a 3×3 matrix as there are 3 possible inputs and 3 possible outputs. $[S] = begin{bmatrix} S_{11}& S_{12}& S_{13}\ S_{21}& S_{22}& S_{23}\ S_{31}& S_{32}& S_{33} end{bmatrix}$ …….. Equation 1 Scattering coefficients $S_{13}$ and $S_{23}$ are equal here as the junction is symmetrical in plane. From the symmetric property, $S_{ij} = S_{ji}$ $S_{12} = S_{21} : : S_{23} = S_{32} = S_{13} : : S_{13} = S_{31}$ The port is perfectly matched $S_{33} = 0$ Now, the $[S]$ matrix can be written as, $[S] = begin{bmatrix} S_{11}& S_{12}& S_{13}\ S_{12}& S_{22}& S_{13}\ S_{13}& S_{13}& 0 end{bmatrix}$ …….. Equation 2 We can say that we have four unknowns, considering the symmetry property. From the Unitary property $$[S][S]ast = [I]$$ $$begin{bmatrix} S_{11}& S_{12}& S_{13}\ S_{12}& S_{22}& S_{13}\ S_{13}& S_{13}& 0 end{bmatrix} : begin{bmatrix} S_{11}^{*}& S_{12}^{*}& S_{13}^{*}\ S_{12}^{*}& S_{22}^{*}& S_{13}^{*}\ S_{13}^{*}& S_{13}^{*}& 0 end{bmatrix} = begin{bmatrix} 1& 0& 0\ 0& 1& 0\ 0& 0& 1 end{bmatrix}$$ Multiplying we get, (Noting R as row and C as column) $R_1C_1 : S_{11}S_{11}^{*} + S_{12}S_{12}^{*} + S_{13}S_{13}^{*} = 1$ $left | S_{11} right |^2 + left | S_{12} right |^2 + left | S_{13} right |^2 = 1$ …….. Equation 3 $R_2C_2 : left | S_{12} right |^2 + left | S_{22} right |^2 + left | S_{13} right |^2 = 1$ ……… Equation 4 $R_3C_3 : left | S_{13} right |^2 + left | S_{13} right |^2 = 1$ ……… Equation 5 $R_3C_1 : S_{13}S_{11}^{*} – S_{13}S_{12}^{*} = 0$ ……… Equation 6 $2left | S_{13} right |^2 = 1 quad or quad S_{13} = frac{1}{sqrt{2}}$ ……… Equation 7 $left | S_{11} right |^2 = left | S_{22} right |^2$ $S_{11} = S_{22}$ ……… Equation 8 From the Equation 6,$S_{13}left ( S_{11}^{*} + S_{12}^{*} right ) = 0$ Since, $S_{13} neq 0, S_{11}^{*} + S_{12}^{*} = 0, : or : S_{11}^{*} = -S_{12}^{*}$ Or $S_{11} = -S_{12} :: or :: S_{12} = -S_{11}$……… Equation 9 Using these in equation 3, Since, $S_{13} neq 0, S_{11}^{*} + S_{12}^{*} = 0, : or : S_{11}^{*} = -S_{12}^{*}$ $left | S_{11} right |^2 + left | S_{11} right |^2 + frac{1}{2} = 1 quad or quad 2left | S_{11} right |^2 = frac{1}{2} quad or quad S_{11} = frac{1}{2}$….. Equation 10 From equation 8 and 9, $S_{12} = -frac{1}{2}$……… Equation 11 $S_{22} = frac{1}{2}$……… Equation 12 Substituting for $S_{13}$, $S_{11}$, $S_{12}$ and $S_{22}$ from equation 7 and 10, 11 and 12 in equation 2, We get, $$left [ S right ] = begin{bmatrix} frac{1}{2}& -frac{1}{2}& frac{1}{sqrt{2}}\ -frac{1}{2}& frac{1}{2}& frac{1}{sqrt{2}}\ frac{1}{sqrt{2}}& frac{1}{sqrt{2}}& 0 end{bmatrix}$$ We know that $[b]$ = $[s] [a]$ $$begin{bmatrix}b_1 \ b_2 \ b_3 end{bmatrix} = begin{bmatrix} frac{1}{2}& -frac{1}{2}& frac{1}{sqrt{2}}\ -frac{1}{2}& frac{1}{2}& frac{1}{sqrt{2}}\ frac{1}{sqrt{2}}& frac{1}{sqrt{2}}& 0 end{bmatrix} begin{bmatrix} a_1\ a_2\ a_3 end{bmatrix}$$ This is the scattering matrix for H-Plane Tee, which explains its scattering properties. Learning working make money

Travelling Wave Tube Travelling wave tubes are broadband microwave devices which have no cavity resonators like Klystrons. Amplification is done through the prolonged interaction between an electron beam and Radio Frequency (RF) field. Construction of Travelling Wave Tube Travelling wave tube is a cylindrical structure which contains an electron gun from a cathode tube. It has anode plates, helix and a collector. RF input is sent to one end of the helix and the output is drawn from the other end of the helix. An electron gun focusses an electron beam with the velocity of light. A magnetic field guides the beam to focus, without scattering. The RF field also propagates with the velocity of light which is retarded by a helix. Helix acts as a slow wave structure. Applied RF field propagated in helix, produces an electric field at the center of the helix. The resultant electric field due to applied RF signal, travels with the velocity of light multiplied by the ratio of helix pitch to helix circumference. The velocity of electron beam, travelling through the helix, induces energy to the RF waves on the helix. The following figure explains the constructional features of a travelling wave tube. Thus, the amplified output is obtained at the output of TWT. The axial phase velocity $V_p$ is represented as $$V_p = V_c left ( {Pitch}/{2pi r} right )$$ Where r is the radius of the helix. As the helix provides least change in $V_p$ phase velocity, it is preferred over other slow wave structures for TWT. In TWT, the electron gun focuses the electron beam, in the gap between the anode plates, to the helix, which is then collected at the collector. The following figure explains the electrode arrangements in a travelling wave tube. Operation of Travelling Wave Tube The anode plates, when at zero potential, which means when the axial electric field is at a node, the electron beam velocity remains unaffected. When the wave on the axial electric field is at positive antinode, the electron from the electron beam moves in the opposite direction. This electron being accelerated, tries to catch up with the late electron, which encounters the node of the RF axial field. At the point, where the RF axial field is at negative antinode, the electron referred earlier, tries to overtake due to the negative field effect. The electrons receive modulated velocity. As a cumulative result, a second wave is induced in the helix. The output becomes larger than the input and results in amplification. Applications of Travelling Wave Tube There are many applications of a travelling wave tube. TWT is used in microwave receivers as a low noise RF amplifier. TWTs are also used in wide-band communication links and co-axial cables as repeater amplifiers or intermediate amplifiers to amplify low signals. TWTs have a long tube life, due to which they are used as power output tubes in communication satellites. Continuous wave high power TWTs are used in Troposcatter links, because of large power and large bandwidths, to scatter to large distances. TWTs are used in high power pulsed radars and ground based radars. Learning working make money

Modes of Propagation A wave has both electric and magnetic fields. All transverse components of electric and magnetic fields are determined from the axial components of electric and magnetic field, in the z direction. This allows mode formations, such as TE, TM, TEM and Hybrid in microwaves. Let us have a look at the types of modes. The direction of the electric and the magnetic field components along three mutually perpendicular directions x, y, and z are as shown in the following figure. Types of Modes The modes of propagation of microwaves are − TEM (Transverse Electromagnetic Wave) In this mode, both the electric and magnetic fields are purely transverse to the direction of propagation. There are no components in $”Z”$ direction. $$E_z = 0 : and : H_z = 0$$ TE (Transverse Electric Wave) In this mode, the electric field is purely transverse to the direction of propagation, whereas the magnetic field is not. $$E_z = 0 : and : H_z ne 0$$ TM (Transverse Magnetic Wave) In this mode, the magnetic field is purely transverse to the direction of propagation, whereas the electric field is not. $$E_z ne 0 : and : H_z = 0$$ HE (Hybrid Wave) In this mode, neither the electric nor the magnetic field is purely transverse to the direction of propagation. $$E_z ne 0 : and : H_z ne 0$$ Multi conductor lines normally support TEM mode of propagation, as the theory of transmission lines is applicable to only those system of conductors that have a go and return path, i.e., those which can support a TEM wave. Waveguides are single conductor lines that allow TE and TM modes but not TEM mode. Open conductor guides support Hybrid waves. The types of transmission lines are discussed in the next chapter. Learning working make money

Microwave Engineering – E-H Plane Tee An E-H Plane Tee junction is formed by attaching two simple waveguides one parallel and the other series, to a rectangular waveguide which already has two ports. This is also called as Magic Tee, or Hybrid or 3dB coupler. The arms of rectangular waveguides make two ports called collinear ports i.e., Port 1 and Port 2, while the Port 3 is called as H-Arm or Sum port or Parallel port. Port 4 is called as E-Arm or Difference port or Series port. The cross-sectional details of Magic Tee can be understood by the following figure. The following figure shows the connection made by the side arms to the bi-directional waveguide to form both parallel and serial ports. Characteristics of E-H Plane Tee If a signal of equal phase and magnitude is sent to port 1 and port 2, then the output at port 4 is zero and the output at port 3 will be the additive of both the ports 1 and 2. If a signal is sent to port 4, (E-arm) then the power is divided between port 1 and 2 equally but in opposite phase, while there would be no output at port 3. Hence, $S_{34}$ = 0. If a signal is fed at port 3, then the power is divided between port 1 and 2 equally, while there would be no output at port 4. Hence, $S_{43}$ = 0. If a signal is fed at one of the collinear ports, then there appears no output at the other collinear port, as the E-arm produces a phase delay and the H-arm produces a phase advance. So, $S_{12}$ = $S_{21}$ = 0. Properties of E-H Plane Tee The properties of E-H Plane Tee can be defined by its $left [ S right ]_{4times 4}$ matrix. It is a 4×4 matrix as there are 4 possible inputs and 4 possible outputs. $[S] = begin{bmatrix} S_{11}& S_{12}& S_{13}& S_{14}\ S_{21}& S_{22}& S_{23}& S_{24}\ S_{31}& S_{32}& S_{33}& S_{34}\ S_{41}& S_{42}& S_{43}& S_{44} end{bmatrix}$ …….. Equation 1 As it has H-Plane Tee section $S_{23} = S_{13}$…….. Equation 2 As it has E-Plane Tee section $S_{24} = -S_{14}$…….. Equation 3 The E-Arm port and H-Arm port are so isolated that the other won”t deliver an output, if an input is applied at one of them. Hence, this can be noted as $S_{34} = S_{43} = 0$…….. Equation 4 From the symmetry property, we have $S_{ij} = S_{ji}$ $S_{12} = S_{21}, S_{13} = S_{31}, S_{14} = S_{41}$ $S_{23} = S_{32}, S_{24} = S_{42}, S_{34} = S_{43}$…….. Equation 5 If the ports 3 and 4 are perfectly matched to the junction, then $S_{33} = S_{44} = 0$…….. Equation 6 Substituting all the above equations in equation 1, to obtain the $[S]$ matrix, $[S] = begin{bmatrix} S_{11}& S_{12}& S_{13}& S_{14}\ S_{12}& S_{22}& S_{13}& -S_{14}\ S_{13}& S_{13}& 0& 0\ S_{14}& -S_{14}& 0& 0 end{bmatrix}$…….. Equation 7 From Unitary property, $[S][S]^ast = [I]$ $begin{bmatrix} S_{11}& S_{12}& S_{13}& S_{14}\ S_{12}& S_{22}& S_{13}& -S_{14}\ S_{13}& S_{13}& 0& 0\ S_{14}& -S_{14}& 0& 0 end{bmatrix} begin{bmatrix} S_{11}^{*}& S_{12}^{*}& S_{13}^{*}& S_{14}^{*}\ S_{12}^{*}& S_{22}^{*}& S_{13}^{*}& -S_{14}^{*}\ S_{13}& S_{13}& 0& 0\ S_{14}& -S_{14}& 0& 0 end{bmatrix}$ $ = begin{bmatrix} 1& 0& 0& 0\ 0& 1& 0& 0\ 0& 0& 1& 0\ 0& 0& 0& 1 end{bmatrix}$ $R_1C_1 : left | S_{11} right |^2 + left | S_{12} right |^2 + left | S_{13} right |^2 = 1 + left | S_{14} right |^2 = 1$ ……… Equation 8 $R_2C_2 : left | S_{12} right |^2 + left | S_{22} right |^2 + left | S_{13} right |^2 = 1 + left | S_{14} right |^2 = 1$ ……… Equation 9 $R_3C_3 : left | S_{13} right |^2 + left | S_{13} right |^2 = 1$ ……… Equation 10 $R_4C_4 : left | S_{14} right |^2 + left | S_{14} right |^2 = 1$ ……… Equation 11 From the equations 10 and 11, we get $S_{13} = frac{1}{sqrt{2}}$…….. Equation 12 $S_{14} = frac{1}{sqrt{2}}$…….. Equation 13 Comparing the equations 8 and 9, we have $S_{11} = S_{22}$ ……… Equation 14 Using these values from the equations 12 and 13, we get $left | S_{11} right |^2 + left | S_{12} right |^2 + frac{1}{2} + frac{1}{2} = 1$ $left | S_{11} right |^2 + left | S_{12} right |^2 = 0$ $S_{11} = S_{22} = 0$ ……… Equation 15 From equation 9, we get $S_{22} = 0$ ……… Equation 16 Now we understand that ports 1 and 2 are perfectly matched to the junction. As this is a 4 port junction, whenever two ports are perfectly matched, the other two ports are also perfectly matched to the junction. The junction where all the four ports are perfectly matched is called as Magic Tee Junction. By substituting the equations from 12 to 16, in the $[S]$ matrix of equation 7, we obtain the scattering matrix of Magic Tee as $$[S] = begin{bmatrix} 0& 0& frac{1}{2}& frac{1}{sqrt{2}}\ 0& 0& frac{1}{2}& -frac{1}{sqrt{2}}\ frac{1}{sqrt{2}}& frac{1}{sqrt{2}}& 0& 0\ frac{1}{sqrt{2}}& -frac{1}{sqrt{2}}& 0& 0 end{bmatrix}$$ We already know that, $[b]$ = $[S] [a]$ Rewriting the above, we get $$begin{vmatrix} b_1\ b_2\ b_3\ b_4 end{vmatrix} = begin{bmatrix} 0& 0& frac{1}{2}& frac{1}{sqrt{2}}\ 0& 0& frac{1}{2}& -frac{1}{sqrt{2}}\ frac{1}{sqrt{2}}& frac{1}{sqrt{2}}& 0& 0\ frac{1}{sqrt{2}}& -frac{1}{sqrt{2}}& 0& 0 end{bmatrix} begin{vmatrix} a_1\ a_2\ a_3\ a_4 end{vmatrix}$$ Applications of E-H Plane Tee Some of the most common applications of E-H Plane Tee are as follows − E-H Plane junction is used to measure the impedance − A null detector is connected to E-Arm port while the Microwave source is connected to H-Arm port. The collinear ports together with these ports make a bridge and the impedance measurement is done by balancing the bridge. E-H Plane Tee is used as a duplexer − A duplexer is a circuit which works as both the transmitter and the receiver, using a single antenna for both purposes. Port 1 and 2 are used as receiver and transmitter where they are isolated and hence will not interfere. Antenna is connected to