# Communication Systems/Microwave Systems

Microwaves are high frequency radio waves (\> 1 GHz) and they propagate
much like any other electromagnetic phenomenon in free space. Because
they are electromagnetic waves and not electrons, they do not propagate
well down copper cables. instead, the waves are guided down hollow
conductors.

## Waveguides

Two boundary conditions must be met in a waveguide for a wave to travel
down the guide:

:   • The electric field must terminate normally on the conductor (the
    tangential component of the electric field must be zero).
:   • The magnetic field must lie entirely tangent along the wall
    surface. (the normal component of the magnetic field must be zero).

Consequently, TEM waves cannot be conducted in a waveguide. (Recall that
in a TEM wave, the electric and magnetic fields are at right angles to
each other and the direction of propagation.)

The most common wave used in waveguides is the TE~10~, meaning
transverse electric. The subscript TE~ab~ denotes the number of half
cycles which appear in the *a* and *b* dimensions of the waveguide.

:   ![](Waveguide.gif "fig:Waveguide.gif")

There are many other waveguide modes. A TE~10~ wave may be formed when
two TEM wave intersect:

:   ![](TE10a.gif "fig:TE10a.gif")

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:   ![](TE10b.gif "fig:TE10b.gif")

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:   

    :   The velocity of a TEM wave in air is: $c=\lambda f$

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:   

    :   The phase velocity (the apparent wave velocity along the guide
        wall) of a TE wave: $v_{p}=\lambda _{g}f$

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:   

    :   Therefor the phase or guide velocity is:
        $v_{p}=\frac{\lambda _{g}}{\lambda }c$

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:   

    :   From the right angle triangle BCD, we obtain:
        $\cos \alpha =\frac{\lambda }{2a}$

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:   

    :   From the right angle triangle ABD we obtain:
        $\sin \alpha =\frac{\lambda }{\lambda _{g}}$

Since

$$\text{sin}^{2}\alpha \text{ + cos}^{2}\alpha \text{ = 1}$$

:   

    :   We obtain:
        $$\left( \frac{\lambda }{\lambda _{g}} \right)^{2}+\left( \frac{\lambda }{2a} \right)^{2}=1$$

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:   

    :   or
        $$\frac{1}{\lambda _{g}^{2}}=\frac{1}{\lambda ^{2}}-\frac{1}{\left( 2a \right)^{2}}$$

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:   

    :   where
        $$\lambda _{g}=$$ guide wavelength for the TE~10~ mode

        $$\lambda =$$ free space wavelength of the TEM wave

        $$a=$$ broad dimension of the waveguide

When $\lambda =2a$, the guide wavelength becomes infinite. This
corresponds to a TEM wave bouncing from side to side in the guide with
no velocity component along the guide. This is cutoff wavelength
$\left( \lambda _{c} \right)$and represents the lowest frequency that
can be propagated.

**Phase & Group Velocity**


:   ![](TE10c.gif "fig:TE10c.gif")

The equations above can be manipulated to obtain:

$$\lambda _{g}=\frac{\lambda }{\sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}}$$

This means that the phase velocity is always greater than the speed of
light. The TEM components zigzag through the guide at the speed of
light, but they convey power at the group velocity. The group velocity
is always less than the speed of light.

**Analogy**

Nothing physical can exceed the speed of light, but it is possible for
some aspect of a wave to exceed light speed. For example, the splash
that occurs when a wave washes up on the beach can travel faster than
the wave, if it strikes at some angle other than 90^0^.

:   ![](Microwave_Systems_splash.gif "fig:Microwave_Systems_splash.gif")

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:   **Example**
:   A rectangular waveguide measuring 0.9 by 0.45 inches is fed with a
    10 GHz carrier. Determine if a TE~10~ wave will propagate, and if
    so, determine its guide wavelength, group velocity and phase
    velocity.

$$a=0.9\ \text{in}\text{.}\ =2.28\ cm$$

$$\lambda _{c}=2\times 2.28=4.58\ cm$$

$$\lambda =\frac{3\times 10^{8}}{10^{10}}=3\ cm$$

:   Since $\lambda <\lambda _{c}$, a TE~10~ wave will propagate.

$$\lambda _{g}=\frac{\lambda }{\sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}}=\frac{3}{\sqrt{1-\left( \frac{3}{4.58} \right)^{2}}}=3.97\ cm$$

$$v_{p}=c\frac{\lambda _{g}}{\lambda }=3\times 10^{8}\frac{3.97}{3}=4.4\times 10^{8}\ m/\sec$$

$$v_{g}=c\frac{\lambda }{\lambda _{g}}=3\times 10^{8}\frac{3}{3.97}=2.27\times 10^{8}\ m/\sec$$

### Wave Impedance

The ratio of the transverse electric and magnetic fields constitutes an
impedance. This should not be considered the same as a resistance,
because it does not dissipate power.

The TE~10~ wave consists of two equal TEM waves where the electric field
is transverse along the *b* dimension but the magnetic field is along
the a dimension is described by: $H_{a}=H\sin \alpha$. Consequently, the
wave impedance is:

$$Z_{w}=\frac{E_{b}}{H_{a}}=\frac{E}{H\sin \alpha }$$

:   For a TEM wave: $\frac{E}{H}=120\pi$

Therefore:

$$Z_{w}=\frac{120\pi }{\sin \alpha }=120\pi \frac{\lambda _{g}}{\lambda }\quad \Omega$$

:   or

$$\frac{120\pi }{\sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}}$$
for TE waves.

$$120\pi \sqrt{1-\left( \frac{\lambda }{\lambda _{c}} \right)^{2}}$$ for
TM waves.

A TE wave has a characteristic impedance $\Omega >120\pi$. As the cutoff
frequency is approached, the impedance approaches infinity. These modes
are generated by means of a vertical probe antenna.

A TM wave has a characteristic impedance $\Omega <120\pi$. As the cutoff
frequency is approached, the impedance approaches zero. These modes are
generated by means of a horizontal probe antenna.

A waveguide with a dielectric other than air will have its cutoff
wavelength increased by the square root of the dielectric constant.

Waveguide attenuation characteristics are quite complex and are
generally derived empirically. Attenuation varies with interior wall
coating, guide dimensions, and operating frequency Attenuation can occur
because the induced wall currents are electrons interacting with the
guide.

There is a limit to the amount of power which can be conveyed in a
guide. If this limit is exceeded, electrical arcing will occur and
severe attenuation results. The maximum power handling capability of an
air filled rectangular waveguide operating in the TE mode is:

$$P_{\max }=6.63\times 10^{-4}E_{\max }^{2}ab\frac{\lambda }{\lambda _{g}}$$

:   where *E~max~* = maximum voltage gradient in v/cm

### Standing Waves

Standing waves act much like the plucked string on a guitar. The signal
magnitude changes but its location does not. One of the most useful
places to create a standing wave is at a radiating antenna, but it may
also be useful to create one in a guide.

:   

    :   ![](Standing_waves.gif "fig:Standing_waves.gif")

**Wall Currents**

Traveling waves are time invariant. The pattern appears to keep its
shape as it moves down the guide. However, as they propagate down the
line, they induce currents in the guide walls, at right angles to the
magnetic field.

:   

    :   ![](Wall_currents.gif "fig:Wall_currents.gif")

Slots that do not noticeably interrupt wall current, do not radiate.

### Microwave Components

**Coupling Energy into a Waveguide**

Probe or capacitive coupling acts like a 1/4$\lambda$ Marconi antenna.
The probe, which might be the center conductor of a coaxial cable, is
positioned 1/4 $\lambda$ from the end of the guide. Since a total
inversion occurs when the wave reflects, it adds in phase with the
coupled signal.

:   

    :   ![](Microwave_probe.gif "fig:Microwave_probe.gif")

The probe can be tapered to handle wideband signals. TE modes are
generated if the probe is placed in the wide dimension of the guide,
while TM modes are produced when the probe is placed in the narrow
sidewall.

Loop or magnetic coupling creates magnetic fields that are launched from
the antenna. The loop may be placed anywhere the magnetic field exists.

:   

    :   ![](Loop_coupling.gif "fig:Loop_coupling.gif")

Aperture or slot coupling occurs when a slot is cut in the maximum of
the electric or magnetic field area

**Bends & Twists**

:   

    :   ![](Bends_twists.gif "fig:Bends_twists.gif")

The twist section is used to change between horizontal and vertical
polarization.

**Tees**

:   

    :   ![](Tees.gif "fig:Tees.gif")

Shunt tee - The signal at port C is the vector sum of ports A and B if
they are used as inputs. If port C is used as the input, the power is
evenly split between A and B.

Series Tee - Using the top port as the input, the power will b evenly
split between A and B however, they will be anti phase.

Hybrid Tee - If A and B are used as the inputs, port C will be the
vector sum and port D will be the vector difference. If C is the input,
the energy will be evenly split between A and B. This device is often
used to connect a transmitter and receiver to the same antenna. The
antenna is at port B, the transmitter at C, the receiver at D, and a
matched load at A.

**Tuners**

:   

    :   ![](Tuners.gif "fig:Tuners.gif")

If the reactive stub is \< 1/4$\lambda$ in the waveguide, it acts like a
capacitor. If it is \> 1/4$\lambda$ in the waveguide, it acts like an
inductor. If it is 1/4$\lambda$ in the waveguide, it acts like an LC
resonant circuit where the Q is proportional to the post diameter.

Tuning is sometimes done to minimize the amount of signal reflected from
a waveguide component. This has a significant effect on the reflection
coefficient and VSWR (voltage standing wave ratio). The effect of these
things can be plotted and observed on a Smith Chart.

**Terminators**

All of the energy travelling down a waveguide must be absorbed in order
to prevent reflections. This is accomplished by placing a resistor in
the center of the guide where the electric field intensity is the
greatest.

:   

    :   ![](Resistive_terminator.gif "fig:Resistive_terminator.gif")

A short circuit is used if reflections are required.

A variable attenuator can be constructed by repositioning the absorbing
material in the electric field.

**Directional Couplers**

:   

    :   ![](Directional_coupler.gif "fig:Directional_coupler.gif")

### Microwave Cavities

The dimensions of a microwave cavity must be some multiple of 1/2 l, and
the Q may be as high as 100,000.

$$Q=\frac{\text{cavity}\ \text{volume}}{\text{interior}\ \text{area}}$$

Energy is coupled into the cavity by means of a probe, loop, slot, or
electron beam.

Conventional tubes have constant velocity beams and vary the number of
electrons emitted to modulate the beam intensity. Microwave tubes emit a
constant stream of electrons and vary their velocity to modulate the
beam intensity.

**Two Cavity Klystron**

An electron beam is created at the cathode and passed by the buncher
cavity. The RF energy coupled into the cavity causes the beam velocity
to vary, thus causing the electrons to bunch. Energy is coupled to the
catcher cavity when the beam passes its port. The net result is that the
RF energy may be amplified by a factor of 1000.

:   

    :   ![](Two_cavity_klystron.gif "fig:Two_cavity_klystron.gif")

The klystron acts as a high Q narrowband amplifier. It may have more
bunching cavities to increase the gain or the cavities may be staggered
tuned to increase the bandwidth.

**Reflex Klystron**

The reflex klystron has a single cavity that acts as both buncher and
catcher. The beam is passed by the cavity, bunched, repelled and passed
by the cavity a second time, and the absorbed in the sidewalls. It is
necessary to turn on the repeller voltage prior to turning on the anode
supply, otherwise the tube will self-destruct.

The reflex klystron is used as a low power, low efficiency, variable
frequency oscillator. The operating frequency is adjusted by changing
the cavity volume.

**Magnetron**

The magnetron is a high power microwave oscillator that has found
widespread use in radar, radio beacon, and microwave oven applications.

The cylindrical cathode is surrounded by the anode, which has several
resonant cavities. The entire assembly is placed in an intense right
magnetic field running parallel to the cathode axis. This field causes
an electron emitted from the cathode to follow a curved path on its way
to the anode. However, the electron passes cavities on the route and so
looses energy, causing it to fall back towards the cathode. The cathode
repels the falling electron forcing it back towards the anode. This
process continues ad infinitum. Electron bunching occurs each time the
beam passes a cavity.

:   

    :   ![](Magnetron.gif "fig:Magnetron.gif")

The critical magnetic field is that which allows the electron to just
graze the anode. If the field intensity falls below this value,
oscillation will cease. The dominant or π mode occurs when the polarity
on adjacent poles is 180o apart. This causes the phase focusing effect
where electrons are bunched in adjacent cavities. The beam shift around
the interior is some integer multiple of the wavelength.

To prevent the bunched electrons to skip some cavities thus creating
spurious modes, every second pole is strapped together. The output is
taken from one of the cavities via a probe antenna. The magnetron can be
mechanically tuned by about 5%.

The continuous power output can be as high as 25 kW, but with low duty
cycle pulsing \[.001\] the peak output can be in the mega watt range.

**TWT**

Traveling wave tubes have a higher bandwidth but lower Q than klystrons.
The RF signal traveling a helical path around the beam causes the beam
to bunch. This in turn reinforces the RF signal causing amplification.

It can be used as a low power \[30 mW\], low noise amplifier with about
a bandwidth of about one octave. Medium power devices operate at about
25 watts. High-powered devices can be pulsed at about 100 kW.

TWTs are used as broadband amplifiers and repeaters in TV, radar and
satellite applications. It may be frequency or amplitude modulated.

### Microwave Semiconductors

**Gunn Diodes**

The Gunn diode is not a diode in the truest sense because it does not
contain a semiconductor junction. It is simply a piece of N-type gallium
arsenide.

Normally the mobility of electrons increases as the electric field
increases in a semiconductor. In this device, there is a range where the
mobility of the electrons actually decreases as the electric field
increases. Essentially the current falls as the voltage rises. This
means that for a narrow range, the device exhibits a negative
resistance. This is a very unstable situation and the device has a
tendency to oscillate when biased in this region. The actual frequency
of oscillation is dependent on the physical size of the semiconductor
and occurs when the electron transit time is equal to one oscillation
period.

**Pin Diodes**

PIN diodes are used as high frequency switches. The have very little
junction capacitance and a very low forward voltage drop.

### Microwave Measurements

Reflections in a waveguide are undesirable. They send power back to the
source, which may ultimately damage the signal source, and they create
standing waves.

**Standing Waves**

If waves are allowed to reflect from a microwave device, the incident
and reflected waves will interact to create a standing wave. This is
somewhat similar to the vibration of a stringed instrument.

The reflection coefficient is defined as the ratio of the reflected and
incident voltage.

$$\Gamma =\frac{E_{r}}{E_{i}}$$

The ratio of maximum to minimum voltage of a standing wave is known as
the VSWR (voltage standing wave ratio).

$$VSWR=\frac{\left| E_{i} \right|+\left| E_{r} \right|}{\left| E_{i} \right|-\left| E_{r} \right|}=\frac{E_{\max }}{E_{\min }}=\frac{1+\left| \Gamma  \right|}{1-\left| \Gamma  \right|}$$

To eliminate the constant use of absolute magnitude bars, the reflection
coefficient is often written as:

$$\rho =\left| \Gamma  \right|$$

And since voltage is generally assumed to be the measured parameter:

$$SWR=\frac{1+\rho }{1-\rho }$$ or $\rho =\frac{SWR-1}{SWR+1}$

The physical distance between two consecutive minimums corresponds to
one-half the guide wavelength $\frac{\lambda _{2}}{2}$

The VSWR can take on any value between 1 and ∞.

If VSWR = ∞, total reflection occurs. Ideally for a matched load, VSWR =
1 and there are no reflections. In practice, it is generally not cost
effective to try to reduce the VSWR below 1.1.

**Smith Chart**

All values on the Smith chart are normalized to the characteristic
impedance of the line.

$$Z_{norm}=\frac{Z}{Z_{0}}$$

Impedances are described by circles on a Smith chart.

:   

    :   ![](Smith_chart_loci.gif "fig:Smith_chart_loci.gif")

This peculiar plotting method has a number of benefits. For one thing,
the center of the chart corresponds to 1+1j. Consequently, it can be
used to characterize any normalized transmission environment.

Furthermore, the SWR appears as a circle in the chart.

:   

    :   ![](SWR_circle.gif "fig:SWR_circle.gif")

The normalized impedance is located at the intersection of SWR circle
and reflection coefficient phase angle.

**Review Questions**

Quick Quiz

:   1\. The magnetic field in a waveguide must terminate \[normally,
    tangentially\] on the conductor.
:   2\. The \[phase, guide\] velocity in a waveguide can exceed the
    speed of light.
:   3\. In a waveguide the electric field must terminate \[normally,
    tangentially\] on the conductor.
:   4\. TM waves can be generated by placing a probe antenna in the
    \[narrow, broad\] side of a waveguide.
:   5\. A \[shunt, series, hybrid\] tee is typically used to connect a
    transmitter and receiver to the same antenna.
:   6\. When capacitively coupling energy into a waveguide, the probe is
    placed \[1/4, 1/2, 1\] $\lambda$ away from the end plate.
:   7\. A reactive stub acts like a capacitor when it is inserted
    \[more, less\] than 1/4 $\lambda$ into the guide.
:   8\. A TE wave has an impedance \[greater, less\] than 120 π ohms.
:   9\. \[TE, TM, TEM\] waves cannot propagate in a waveguide.
:   10\. The subscript in TEab denotes the number of \[quarter, half,
    complete\] cycles which appear in the a and b dimensions of the
    waveguide.
:   11\. The dominant waveguide mode is \[TE01, TE10, TE11\].
:   12\. The speed at which energy travels down a waveguide is known as
    the \[group, phase\] velocity.
:   13\. The cutoff wavelength represents the \[highest, lowest\]
    frequency that can propagate down a waveguide.
:   14\. The \[shunt, series, hybrid\] tee evenly splits power between
    two branches, but the outputs are phase inverted with respect to
    each other.
:   15\. The \[shunt, series, hybrid\] tee is used to couple a
    transmitter and receiver to the same antenna.
:   16\. The \[magnetron, TWT\] is a microwave oscillator.
:   17\. The \[reflex klystron, TWT\] is a microwave amplifier.
:   18\. The magnetron \[is, is not\] a tube.
:   19\. The klystron can be used as either an amplifier or an
    oscillator. \[True, False\]
:   20\. The microwave tubes modulate the electron beam \[magnitude,
    velocity\].
:   21\. TWTs use buncher cavities to modulate the beam. \[True, False\]

Analytical Questions

:   1\. A rectangular waveguide has the following characteristics:

Internal dimensions: 10.16 x 22.86 mm Feed: 10.5 GHz gunn diode

`       Determine if:`

:   a\) If a TE~20~ wave will propagate
:   b\) If a TE~10~ wave will propagate
:   c\) The dominant mode guide wavelength
:   d\) The dominant mode group velocity
:   e\) The dominant mode phase velocity
:   f\) The dominant mode wave impedance
:   g\) Explain how you would measure the guide wavelength.

Composition Questions

:   1\. Sketch the cross section of a two-hole directional coupler and
    discuss its operation.
:   2\. Sketch the cross section of a magnetron and discuss its
    operation.
:   3\. Sketch the TE10 mode in a rectangular waveguide, and show the
    position of all of the radiating and non-radiating slots.
