
Microwave Rules
of Thumb
Updated April
18, 2014
The origin of the phrase "rule
of thumb" is debatable; some say it was once a man's right
to beat his wife with a stick no wider in diameter than his thumb. Sadly, in some countries today women have to deal with even worse treatment.
What we mean by a microwave ruleofthumb
could be an inexact but notable relationship of one or more design
parameters with performance, or it could just be an easy way to
remember something that other lesser people often mix up. Obviously,
you must use some discretion when you apply these rules, exact results
can vary widely depending on influences you haven't considered,
such as the phase of the moon.
Microwave rules of thumb have
been handed down to newhires by microwave old farts for the last
century. We know there are a lot of OFs out there, so please send
in your favorite rule of thumb and win a
pocket knife (what OF could resist an offer like that?) We
will acknowledge your contribution here (unless you prefer to remain
anonymous). Attention, humordisabled readers... any tired references
to Murphy's law will never make it to this page, so please don't
feel the need to share any of this boring crap with us!
We will keep compiling microwave
rules of thumb on this page, in no particular order, and we don't
guarantee that we will reorder the rules in the future. These rules
are scattered about the web site in appropriate places as well.
We try to cross reference this section with other parts of
the Microwaves101 encyclopedia so you can learn more about any subject that you
are interested in.
 Keep your fat fingers out
of expensive hybrid modules, or someone might break your thumb! Seriously, how many times in your life have you seen an idiot point to something in a module and crush a dozen wirebond?
 The minimum noise figure of
a FET varies linearly with
frequency, up until Fmax. This related rule came from John, who
also supplied a reference (thanks!) The minimum noise figure of
a BJT varies quadratically with frequency, up until Fmax.
This rule was quoted from Bahl, I. and Bhartia, P. 2003, Microwave
Solid State Circuit Design, 2 Ed, John Wiley & Sons, New Jersey,
p.377
 The loss of a branchline coupler
is reduced as the squareroot of frequency, given that
the same substrate and metallization is used. This is one outcome
of the skin depth effect.
 Five skin
depths of a good conductor will keep your losses to a minimum
in microstrip.
 If you are using copper
boards with halfounce or thicker copper, you don't have to
worry about skin depth problems unless you are working below 200
MHz. To clarify that obtuse statement (thanks to Sylvia), if you are working at microwave frequencies, using more than ½ ounce copper does not improve loss as you can reached the maximum surface conductivity. But if you are working on the bias circuits for high power, currenthungry solidstate amplifiers down at DC, then adding more copper can decrease the loss, as ALL of the copper is used in conduction. The “skin depth problem” is that you didn't achieve at least three (preferably five) skin depths so you didn't do your best to minimize loss.
 Electromagnetic energy such
as microwave radiation travels one
foot in one nanosecond in free space. In teflondielectric
coax cables, it travels one foot in about 1.5 nanoseconds. In
waveguide, speed is a function of frequency due to dispersion.
 The return
loss of a circulator is very nearly equal to its isolation.
 The thirdorder intercept
point of an amplifier is generally 10 dB higher than its onedB
compression point, when measured at the output. This corresponds
to 9 dB higher when measured at the input.
 The isolation resistor on
a quadrature coupler (such as a Lange) on the output of a power
amp should be able to handle 25% of the total power if you want
the amplifier to still (sort of) work if one amp blows up. Otherwise
10% of the total power for a tuned hybrid, or 5% of the power
if the entire amp is on a MMIC.
 For a given switcharm design,
a SPDT switch will have 6 dB
more isolation than a comparable SPST switch, as long as the
"through" arm of the switch is properly terminated.
 For 1mil gold wirebonds,
inductance of the bond wire
in nanohenries is roughly equal to its length in millimeters...
an advantage of the Metric System that was brought to our attention
by a French engineer named Yves (merci!) Let's restate this ruleofthumb
so that baseball fans can use it: 1 mil of bond wire is equal
to 25 picoHenries of inductance, or 40 mils of bondwire is equal
to one nanohenry.
 To be considered a "lumped
element", no feature of a structure can exceed 1/10 of
a wavelength at the maximum frequency of its usage.
 To be a useful substrate,
the height of a microstrip board should never exceed 1/10 of a
wavelength at the maximum frequency of it usage. We've made a
table for you on this subject!
 How do you know what WR
number a waveguide is just by looking at it? The WR number
is simply the dimension of the broad wall in mils, divided by
10.
 A good way to remember which
is the Eplane and which is the
Hplane in rectangular waveguide is when you bend it, bends
in the Eplane are the "easy way", while bends in the
Hplane are the "hard way".
 For silicon or SiGe, 110 degrees
C is the maximum junction temperature for reliable operation (1,000,000
hours is typical median time to failure criteria). With the exception
of silicon LDMOS, which can operate up to 175C for 800 years,
according to Leonard who works for a major LDMOS supplier (thanks!)
GaAs FET (or HEMT) channel temperature should not exceed 150 C
for longterm reliable operation. For gallium nitride HEMT (GaN),
175 C is a good rule for maximum channel temperature.
 In order to cutoff spurious
modes, the width of a package should generally not exceed onehalf
of a wavelength in free space at the maximum operating frequency.
 For microstrip and stripline
curved lines, use a minimum radius of three line widths at Xband
and below. At higher frequencies, use five line widths for minimum
radius. Even better, use an optimum
miter instead of a curve!
 The 10% to 90% rise time of
a pulsed signal, in nanoseconds, will be approximately equal to
0.35 divided by the bandwidth of the network, in GHz.
 If you are trying to effect
an RF short circuit using a quarterwave stub, use a low impedance
line, or better still, use a radial
stub.
 Due to constructive interference,
the individual return loss of two identical mismatches is 6
dB better than the worst case observed return loss of the
two mismatches measured together.
 Two identical mismatches can
be made to cancel each other
by locating them approximately onequarter (or perhaps threequarters)
wavelength apart. This rule is often used in PIN diode switch
and limiter design. Note that shunt capacitive VSWRs require slightly
less than onequarterwavelength to cancel (thanks, Mike!),
while shunt inductive mismatches require slightly more.
 Want to remember the correct
order of Ku, K and Ka radar
bands? K is the middle band (1827 GHz), while Kuband is
lower in frequency (think K"under") and Kaband is
higher in frequency (think K"above").
 A chip attenuator is good
for at least 1/16 watt if it is mounted to a circuit card such
as Duroid or FR4, 1/4 watt if mounted to metal with conductive
epoxy, and 1/2 watt if it is attached with solder to a metal heatsink.
 For a ten dB pad (90% power
dissipation), size the input
resistor to handle 1/2 of the maximum intended input power.
For a 20 dB pad (99% power dissipation), size it for 80% of the
max input power. For higher attenuation values, size the input
resistor for the full RF input power.
 When designing "splitblock"
waveguide sections, tell your mechanical engineer that you
have to split the guide the hard way, cutting through the Hplane (along the broad wall). If you put
a mechanical seam in the Eplane (along the short wall), you are looking for trouble,
because the guide needs to pass RF current through the seam, and
very high signal losses and VSWRs can result.
 For Nway
resistor power dividers, power is transferred as (1/N)^2.
Compare this to a lossless power divider, which transfers power
at (1/N), and you see that resistive dividers are extremely inefficient
(and get worse and worse the more arms you add), but for some
applications, they offer a cheap, wideband solution.
 For an impedancematched amplifier,
the impedance match it sees on one port will not affect
the impedance match it provides on the opposite port, provided
that its ratio of S21 to S12 is down
by at least 20 dB. Example: you are designing a receiver in
which your mixer has a very bad match at the IF port, say 3:1
VSWR, or 6 dB. The mixer is followed by a GaAs HBT amplifier,
where S21 (gain) is 23 dB, and S12 (reverse isolation) is 25
dB. You are in trouble, because the "round trip" through
the amp is only 2 dB, so your receiver IF output match will be
only 2 dB better than the 3:1 match of the mixer, or 8 dB.
 The
P1dB point of a mixer at its RF input is often about 6 dB
less than its LO drive level. We've seen references that show
P1dB can be between 10 dB and 0 dB from the LO power level, so
consult your mixer's data sheet!
 The gain temperature coefficient
of a MESFET or PHEMT amplifier is often approximately 0.007 dB/degree
C/stage, if the gate bias voltage is fixed. Selfbiased amplifiers
have much lower gain/temperature coefficients (less variation
with temperature).
 When it comes to capacitor
materials, the ones with the highest dielectric (K) are most
likely to have the worst variation over temperature.
 The physical length of a
Lange coupler is approximately equal to one quarterwavelength
at the center frequency on the host substrate. The combined width
of the strips is comparable to the width of a fiftyohm line on
the host substrate.
 If you forget to build image
rejection into your receiver design, you might be adding three
dB to your receiver's noise figure. Approximately 20 dB image
rejection will all but eliminate image noise foldover.
 If you are 2d^{2}/
or farther from an antenna, you are in the
farfield.
 Viahole
inductance: on 2mil (50 micron) GaAs, Lvia is about 10 picohenries.
For fourmil (100 um) GaAs, it is about 20 picoHenries. Anyone
have a ROT for alumina or PWB inductance?
 When measuring high power
with a microwave watt meter assemble your inline attenuators
with three dB pad nearest to the power source, next six dB and
so on. That way the power is dissipated in a fashion to cause
least heating of the pads. Thanks to Paul, retired USCG! And always
check the power rating of every component that you screw onto
the output of a highpower source! UE
 If your getting high SWRs
in a system with a wattmeter (i.e. Bird directional wattmeter)
and your antenna and coax haven't had a lighting hit or physical
damage, use a spectrum analyzer to check the output of your transmitter.
Maybe what your seeing is harmonics from a bad transmitter not
a bad coax or antenna. Also thanks to Paul, retired USCG!
 The group
delay of a filter is nearly proportional to its order. Also,
filter group delay is inversely proportional to filter bandwidth
(small percentage bandwidth filters have large group delay. This
"corollary" came from Chip but we haven't had time to
test it out: The insertion loss at the band edge of a filter is
equal to the insertion loss at the band center times the ratio
of the group delay at the band edge to the group delay at the
band center. (i.e., the insertion loss is proportional to how
long the signal is in the filter!)
 The effective
dielectric constant for CPW is merely the average of the dielectric
constant of the substrate, and that of free space. If you are
using GaAs, Er=12.9, the effective dielectric constant would be
(12.9+1)/2=6.95.
 The noise
figure of a mixer is generally equal to the magnitude of its
conversion loss, or maybe just a little bit less. A mixer with
6 dB conversion loss may have a noise figure of 5.5 dB.
 You should measure
the return loss of a mixer's three ports at the recommended
LO drive level, or you will get ugly results.
 For the best LO to IF isolation
in a doublebalanced mixer,
always tap off the IF from the RF balun, not the LO balun. You
should get 20 dB better LO rejection this way.
 When subscribing to trade
journals, always give them a fake email address and phone
number. Otherwise they will be bugging constantly!
 To compute wavelengths in
free space in your head, remember that 30 divided by frequency
in GHz will give you wavelength in centimeters. Thus 10 GHz is
3 centimeters wavelength, and 30 GHz is one centimeter wavelength
(the "break point" where millimeter wavelengths start).
 The beam width of an antenna
of fixed area is proportional to its wavelength. Thus a 40 GHz
signal can be focussed to one quarter of the beam width of a 10
GHz signal.
 The coupled port on a microstrip
or stripline directional
coupler is closest to the input port because it is a backward
wave coupler. On a waveguide broadwall directional coupler,
the coupled port is closest to the output port because it is a
forward wave coupler.
 The antenna pattern for a
horn antenna can be approximated
as
P(dB)=10x(/_{10dB})^2
 Doppler shift at Xband is
approximately 30 Hertz for 1 mile per hour. If you are traveling
at 60 miles per hour, your Doppler frequency on police Xband
radar will be approximately 1800 Hz.
 Let's call this a "proposed"
rule of thumb, because we don't have any supporting data yet.
For finite groundplane microstrip, you'll need at least five times
either the substrate height, or the microstrip width, as your
groundplane width, whichever is MORE.
 The gain of a narrowbeam
reflector antenna is approximately
27000/(12),
where 1
and 2
are the 3 dB (halfpower) beamwidths in the principal planes,
measured in degrees (not radians).
 Electromagnetic radiation
at frequencies higher than light (such as xrays) can cause
cell damage (ionizing radiation). EM radiation below light
(such as microwaves) don't damage cells, they only cause heating
(which can cause injury as well, but is easy to avoid because
it causes pain!)
This additional info came from John (thanks!)
I would like to add some important information pertaining to
microwave rule of thumb 51. Specifically the rule states that
nonionizing radiation injury is easy to avoid because the heat
it generates causes pain.
Exposure to microwave radiation at power levels below the pain
threshold does cause heating in the lens of the eye producing
cataracts. Heating denatures proteins in the crystalline lens
of the eye (in the same way that heat turns egg whites white and
opaque) faster than the lens can be cooled by surrounding structures.
The lens and cornea of the eye are especially vulnerable because
they contain no blood vessels that can carry away heat. The damage
is accumulative and over time degrades vision. High power levels
will produce discomfort that includes irritation of the eye however;
levels of power well below the average personâ€™s pain threshold
will induce cataracts over time. It should be noted that frequencies
whose wavelength more closely match the size of the eye (Xband
freqs with wavelengths in the 24 centimeter range) are particularly
dangerous.
Also, there has been some infomation in the news on terahertz
waves, which are purported to unzip
DNA molecules. Stay tuned!  UE
 When measuring Sparameters
to get group delay, you should pick
the frequency interval to achieve about 10 degrees S21 phase difference
between frequency points. Less than this will make the measurement
jumpy, greater than 10 degrees might mask some real problems in
group delay flatness. How do you know in advance what frequency
interval to pick? Excuse us while we go think up a formula for
this...
 If you divide the switch
element Figure of Merit by 100 (FOM=(1/(2RonxCoff)),
you will arrive at the highest frequency that the device can be
made to perform as a switch. Thus MESFET switches will work up
to about 26 GHz, PHEMTs will work up to 40 GHz, and PIN diodes
will work up to 180 GHz.
 When you are counting the
number of squares in a meandering
resistor to determine its value, the squares at each bend should
be counted as 1/2 square.
 Switch
isolation is often limited by package isolation. If you design
a 60 dB switch, you should think carefully about how to package
it!
 Linear passive devices have
noise figure equal to their loss. Expressed in dB, the NF
is equal to S21(dB). Something with one dB loss has one dB noise
figure. But wait, as Gene points out, there is more to consider!
This statement is true only if the passive linear device is at
room temperature. You'd best analyze the problem using noise
temperature.
 If you have 20 dB gain in
your LNA or receiver, the noise
figure contribution of the subsequent stage will be small
(unless the noise figure of the next stage is horrendous!)
 Twenty dB of image rejection
is about all you need before you can neglect
image noise foldover. Worst case, image noise foldover can
degrade receiver noise figure by 3 dB.
 The minimum width for a stripline
that is encased by metal on the edges is 5
times the line width, in order for the impedance to calculate
with the "normal" closed form equations.
 The angular beam width of
a parabolic reflector can
be estimated from the diameter of the dish and the frequency of
operation as: angular beam width (degrees)=70 degrees/(D/lambda).
Corrected thnks to Vincenzo!
 If you are so fed up with
your job that you are going to quit, line up another one first,
unless you like clipping coupons. As a corollary, don't burn every
bridge on your way out of town, you never know when you might
be desperate enough to come back!
 For pure alumina (_{R}=9.8),
the ratio of W/H for fiftyohm
microstrip is about 95%. That means on ten mil (254 micron)
alumina, the width for fifty ohm microstrip will be about 9.5
mils (241 microns). On GaAs (_{R}=12.9),
the W/H ratio for fifty ohms is about 75%. Therefore on four mil
(100 micron) GaAs, fifty ohm microstrip will have a width of about
3 mils (75 microns). On PTFEbased soft board materials _{R}=2.2),
W/H to get fifty ohms is about 3. Remember these!
 The accepted limits of operation
for rectangular waveguide
are (approximately) between 125% and 189% of the lower cutoff
frequency. Thus for WR90, the cutoff is 6.557 GHz, and the accepted
band of operation is 8.2 to 12.4 GHz.
 There is considerable overlap
between waveguide standards, you can almost always find two types
that will work at one frequency. In order to get the lowest
loss, choose the waveguide that has the largest dimensions.
 For a given frequency, waveguide
will give the lowest loss
per unit length. Coax loss will be about 10X higher (in dB).
Transmission line loss on MMICs (microstrip or coplanar waveguide)
is about 10X worse than coax, or 100X that of waveguide (but the
lengths of the transmission lines are really small!) Stripline,
depending on its geometry, usually will be slightly higher in
loss than coax.
 This one came from Scott!
The wavelength in air, measured inches, is 11.803 divided by the
frequency in GHz. Throw in that the wavelength in or on dielectric
is the wavelength in air divided by the square root of the effective
(close to actual for low dielectrics) relative dielectric constant.
 Whenever you
bend a transmission line, to model the length of the line
you should simply ignore the extra length that is added by the
bend. We'll cover our butts by saying this is just an approximation,
if the effective length of a line is critical to the design success,
you'd better simulate it in Sonnet!
 If you use a radius greater
than three times the line width,
you will have a transmission line that is almost indistinguishable
in impedance characteristics from a straight section. According
to Chip: radiused bends are a waste of valuable real estate. Stick
with well compensated right angles.
 Coax line impedance is not
a strong function of the eccentricity
of the center conductor. You can be off by a full 50% and
the impedance will decrease on the order of only 10%! And remember,
impedance can only decrease if the center conductor is off
center, it will never increase! Another suggestion of Chip:
When designing a coaxial structure you will never end up perfectly
concentric. Therefore, always design coaxial structures with 36%
higher impedance and you will end up with a better match.
 For coax and stripline 50
ohm transmission lines that employ PTFE dielectric (or any dielectric
material with dielectric constant=~2), the inductance
per foot is approximately 70 nH, and the capacitance per foot
is about 30 pF.
 The
isolation of a Wilkinson is limited to 6 dB better that the
return loss of the source match at its common port.
 The split
port return loss of a Wilkinson is no better than the return
loss that is seen by the Wilkinson at its common port.
 An acceptable voltage droop
for a power amplifier during pulsed operation is 5%, which will
drop the power by a similar amount (5%, or about a quarter of
a dB). So for a PHEMT amp operating at 8 volts, you allow a voltage
droop of 0.4 volts. Use this rule when you calculate charge
storage capacitance!
 In order to use silicon
as a substrate, you need resistivity at least 100 ohmcm or
the loss is going to eat your lunch.
 For
microstrip, you can (approximately) cut metal losses in half
by doubling the dielectric thickness. For example, going from
10 mil to 20 mil alumina, or twomil to fourmil GaAs.
 Any microwave
semiconductor house that doesn't invest in new technology,
is going to go out of business in the long run. By long run, we
mean five years.
 Anyone who designs complex
microwave circuits and claims they don't use the optimization
function in their EDA software is one of these three things:
a liar, an idiot, or a supergenius with IQ 250. You pick which
one, then accuse them when they bring this up at their next peer
review!
 When laying out the top layer
of a microstrip board many of us do a ground fill. The question
is how close to get to the microstrip lines – especially
since the ground fill function is automated in many layout programs.
The answer is to keep >3 line widths away. This insures minimal
additional loss and impact to line impedance. Contributed by Tom!
 Different loss mechanisms
have different behaviors
over frequency. Metal loss is proportional to squareroot
frequency. Dielectric loss is proportion to frequency. Dielectric
conduction loss is constant over frequency.
 When considering the transmission
line loss due to dielectric conductivity, if the resistivity
of the dielectric is greater than 10,000 Ohmcm, forget it! That
pretty much rules out all substrates except silicon, which can
be anywhere from 1 Ohmcm (very lossy) to 10,000 Ohmcm (very
expensive floatzone silicon). PTFE is 1E18 Ohm cm!
 Let's just call this a proposed
rule of thumb (your comments are appreciated!) A transmission
line (coax, microstrip CPW, stripline but NOT waveguide) can be
considered lowloss if the loss per wavelength is
less than 0.1 dB. Waveguide will routinely be 10X better than
this benchmark!
 For stripline and microstrip,
the attenuation factor
always decreases when characteristic impedance is reduced.
It's almost proportional; if you can live with 25 ohm transmission
lines instead of 50 ohms, you can cut your losses nearly in half!
This is a different result than coax, which has a sweet spot on
the attenuation/impedance curve (77 ohms for air coax, 52 ohms
for PTFEfilled).
 This rule of thumb has its
own page! You can electrically measure
the approximate length of a cable (or any long transmission line)
by noting the frequency separation between the dips in VSWR (S11)
and doing some simple math.
 This rule of thumb has nothing
to do with microwaves. At some point in your career you might
be asked to assist in the task of boxing up a coworker's stuff,
either because he or she died or otherwise became incapacitated.
Here's the rule: if you happen to find a framed picture of a woman
(man) tucked into a desk drawer, and you don't know what the coworker's
spouse looks like, just throw out the picture, maybe save the
frame for yourself if it's a nice one. There's little chance you
are discarding a priceless oneofakind artifact, but there's
a good chance the picture will be an unwelcome surprise to the
coworker's spouse (why would the picture be buried in a drawer
in the first place?) We speak from a nearmiss experience a long
time ago, when an alert friend of the coworker pulled the "that's
not his wife!" photo from the box just as it was heading
to the shipping department!
 Here's a
freespace path loss rule of thumb, thanks to Stefan.
 The typical isolation you
can expect from a two channel receiver
is on the order of 25 dB. For dualchannel MMICs, expect no more
than 30 dB.
 This rule came from Cheryl...
if you don't want to worry about the metal cover of a module pulling
the impedances of the microstrip
circuits you designed, make sure it is at a minimium height of
of 5X the substrate thickness and 5X the maximum line width, whichever
is more. Thanks!
 When a solid
state amplifier is pulsed on for 100 microseconds or longer
("long" pulse), it reaches a quasisteadystate junction
or channel temperature, so for thermal and reliability analysis,
this case can be considered the same as continuous wave. Under
the same operating conditions, to get any reliability benefits
from pulsed operation you need to operate with pulses of 10 us
or less ("short" pulse).
 When calculating the
peak power handling of a transmission line due to dielectric
breakdown (arcing), you need to derate by 6 dB for conditions
where the network might see a very high VSWR (like an accidental
open or short).
 For high altitude flights,
you should derate the peak power handling of circuitry where air
is the "dielectric" by as much
as 10 dB, if the electronics are exposed to atmospheric pressure.
 The graceful
degradation of gain in an Nway combiner used in a power amplifier decreases as [(NX)/N]^2
where X is the number of failures. Half the gain is lost on the input, and half on the output. If you can provide the equivalent increase in input power, the output power will drop as (NX)/N. This assumes ideal conditions, at center frequency, and an isolated power splitter is used (such as a Wilkinson).
 This came from JC... the only
thing that HBTs have been good for is being cheap and having lower
phase noise for VCOs. Otherwise short gate length pHEMTs are better
in every other respect....
 The number of elements required
in an electronicallyscanning phased
array antenna can be estimated by the gain it must provide.
A 30 dB gain array needs about 1000 elements and a 20 dB gain
array needs about 100. Thanks to Glenn!
 Numbers 94 to 100 are from
Tom. Thanks for putting us over the top! On Microstrip layers,
keep ground fill at least 3 line widths away from the microstrip
to maintained the originally designed impedance. This means if
the line width is 10mils keep the ground on that layer at least
30mils away or you'll have a mismatched coplanar waveguide!
 Third Order Intercept can
generally be estimated to be 10dB higher than P1dB (except for
the latest PHEMT devices, but I really doubt they're that good).
So if your amp starts to compress around +20dBm then the TOI is
probably around +30dBm.
 Guesstimating wavelength in
free space is always a race to see who can think faster. Just
remember that 300MHz is 1 meter and ratio your way from there.
So 1GHz is approximately 3 times 300MHz so the wavelength is approximately
1/3 or 30cm OR 100MHz is 3 meters.
 If your circuit does funny
things when you close up the box, it's oscillating. If you can't
see it on the specan then it's above the instrument range.
 If your circuit is oscillating
when the cover is on, stick 1 square inch of absorber material
(the adhesive backed stuff) stripe down the middle of the inside
roof of the cover for every 3 sq. inches of cover. It doesn't much
matter what the thickness is although you can adjust that later
when you write the ECO.
 Tom's Law: in any broadband
(>1octave) design, the overall gain/attenuation will be about
0.75dB/GHz worse than expected by design calculation or simulation.
You've been warned so plan ahead.
 Gain in a microwave chain
is like a gun. Better to have it and not need it than to need
it and not have it.
 Switched
filter phase shifter bits, either high pass, or low pass,
are not useful above 90 degrees of phase shift. For a 180 bit,
you must cascade two 90s, or use an alternate structure. The preferred
structure is a highpass/lowpass bit.
 For a given microstrip or
stripline geometry, the filling
factor is very nearly a constant versus the value of the dielectric
constant of the substrate. The inductance per length does not
change versus the dielectric constant of the substrate, only the
capacitance/length does.
 For ideal coplanar
waveguide (with very thick substrate and no ground plane on
the back side, thin, perfect conductors), filling
factor is 50%. Therefore the Keff is equal to:
Keff=(ER+1)/2
This is the average value of air (ER=1) and the substrate.
 The side dimension of a cube
corner reflector is ideally greater than 10 wavelengths of
the signal you are trying to reflect. Any comments are appreciated!
 The effect
of surface roughness on microstrip lines is a gradual degradation
in attenuation due to conductor loss. If RMS roughness is on the
order of one skin depth, conductor attenuation (alphac) is increased
by 60%. If surface roughness is much more than one skin depth,
the increase is 100% (2X ideal loss).
 We finally have a rule of
thumb for the equivalent electrical length of a microstrip mitered
bend. The "extra" length is equal to half the width
of the line. It is explained
here. Thanks to Kevin!
 You should plan on the offstate
resistance to somewhere between 5000 ohmmm and 50,000 ohmmm.
In most designs you can just ignore it, but in this
example, it's important.
 The minimum size for a gate
choke resistor is on
the order of 500 ohms. Many designers use thousands of ohms,
this merely slows down the switch. The gate is already (at least
partially) decoupled from the RF without the resistor! Some day
we'll add an analysis to back up this bold statement... For very
high speed, you can eliminate the gate choke resistor by using
a low impedance bias network (quarterwave stub terminated in a
capacitor for example).
 If you want to simulate rectangular
coax with a linear simulator such as Microwave Office or ADS,
you can model it quite accurately as a round coax line, subject
to one condition described
here. Just compute the equivalent crosssectional area of
the center conductor and divide it by pi to get the diameter of
an equivalent coax. Set the outer conductor to whatever you need
to get the impedance you want (generally 50 ohms).
 Regarding RMS transmission phase errors in a solid state power amp (SSPA), the "phase efficiency" is approximated simply as the cosine squared of the RMS phase error of the amplifiers. Thus if your RMS phase error is 45 degrees (which really sucks) your phase efficiency is 50% and you are losing half of your power into isolation loads (3.01 dB below what is possible). In order to hit 99% phase efficiency your amplifiers need to be phased to 5.74 degrees RMS. In the case of a twoway combiner, the peak error is twice the RMS error, so your amplifiers must be within 11.5 degrees transmission phase.
 Following rule 110, suppose you pick amplifiers from the "Waffle Pak of Uniformly Distributed Phase", or WPUDP, spanning X degrees (more specifically, WPUDPX). The expected RMS phase error is X divided by the squareroot of twelve. You should recognize that term as the square root of the variance (σ, not σ^{2} which is the variance) of a uniform distribution, something that should be covered in Six Sigma training but is usually omitted to make the course so easy that an imbecile can become a "black belt." So, if your amplifiers are uniformly spread over 90 degrees, your RMS error is expected to be 26.0 degrees, and you should see 80.8% phase efficiency (0.93 dB below what is possible). Of course, basing this calculation on just the waffle pak distribution ignores phase tolerance contributions of wirebonds and combiners, you are on your own to estimate these and recalculate the phase distribution. If you don't understand (and memorize) rules 110 and 111, you might want to pick a different career other than SSPAs....
 To be considered in "small signal operation", a signal should have less than 5% effect on the voltage bias point of an amplifier or diode or whatever nonlinear element is first affected. Not so sure how this would apply this to a zerobias detector…. anyone want to comment?

In slow wave structures using alternate high and low impedance segments, the Bragg frequency occurs when the segments are 30 electrical degrees.
 In slow wave structures using alternate high and low impedance segments, the line segments should be 5 electrical degrees or less at your maximum operating frequency.
 The Bragg frequency of an artificial transmission line occurs where the unit cell is 1/3 of a wavelength.
 TEM transmission line, loss tangent of 0.01 (which is pretty high) results in almost exactly 1 dB/cm loss at 110 GHz, before you scale it by SQRT(dielectric constant). Since it is linear with frequency, you should be able to scale loss tangent attenuation in your head. You can approximate attenuation in microstrip or CPW if you scale by the effective dielectric constant.
 The 90% rule: coax is never specified to operate beyond 90% of its TE11 cutoff frequency.

In order to achieve perfect cancellation of identical poorlymatched amplifiers, there are two necessary and sufficient conditions:
 The coupler must provide perfect amplitude balance
 The coupler must provide perfect 90 degree coupling
Perhaps more interesting is this: the termination resistor VSWR does not matter when you are considering only the VSWR of the combined amplifier.

To convert Nepers to decibels, multiply by (approximately) 8.68.
More accurately,
dB/Np = 20/ln(10)=8.68588

