Microwave rules of thumb

Updated April 6, 2013

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.

What we mean by a microwave rule-of-thumb 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 new-hires by microwave old farts for the last century. We know there are a lot of O.F. out there, so please send in your favorite rule of thumb and win a pocket knife (what O.F. could resist an offer like that?) We will acknowledge your contribution here (unless you prefer to remain anonymous). Attention, humor-disabled 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.

1. Keep your fat fingers out of expensive hybrid modules, or someone might break your thumb!
2. 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
   
3. The loss of a branchline coupler is reduced as the square-root of frequency, given that the same substrate and metallization is used. This is one outcome of the skin depth effect.
4. Five skin depths of a good conductor will keep your losses to a minimum in microstrip.
5. If you are using copper boards with half-ounce 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, current-hungry solid-state 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.
6. Electromagnetic energy such as microwave radiation travels one foot in one nanosecond in free space. In teflon-dielectric coax cables, it travels one foot in about 1.5 nanoseconds. In waveguide, speed is a function of frequency due to dispersion.
7. The return loss of a circulator is very nearly equal to its isolation.
8. The third-order intercept point of an amplifier is generally 10 dB higher than its one-dB compression point, when measured at the output. This corresponds to 9 dB higher when measured at the input.
9. 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.
10. For a given switch-arm 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.
11. For 1-mil 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 rule-of-thumb so that baseball fans can use it: 1 mil of bond wire is equal to 25 pico-Henries of inductance, or 40 mils of bondwire is equal to one nanohenry.
12. 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.
13. 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!
14. 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.
15. A good way to remember which is the E-plane and which is the H-plane in rectangular waveguide is when you bend it, bends in the E-plane are the "easy way", while bends in the H-plane are the "hard way".
16. 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 long-term reliable operation. For gallium nitride HEMT (GaN), 175 C is a good rule for maximum channel temperature.
17. In order to cutoff spurious modes, the width of a package should generally not exceed one-half of a wavelength in free space at the maximum operating frequency.
18. For microstrip and stripline curved lines, use a minimum radius of three line widths at X-band and below. At higher frequencies, use five line widths for minimum radius. Even better, use an optimum miter instead of a curve!
19. 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.
20. If you are trying to effect an RF short circuit using a quarter-wave stub, use a low impedance line, or better still, use a radial stub.
21. 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.
22. Two identical mismatches can be made to cancel each other by locating them approximately one-quarter (or perhaps three-quarters) wavelength apart. This rule is often used in PIN diode switch and limiter design. Note that shunt capacitive VSWRs require slightly less than one-quarter-wavelength to cancel (thanks, Mike!), while shunt inductive mismatches require slightly more.
23. Want to remember the correct order of Ku, K and Ka radar bands? K is the middle band (18-27 GHz), while Ku-band is lower in frequency (think K-"under") and Ka-band is higher in frequency (think K-"above").
24. A chip attenuator is good for at least 1/16 watt if it is mounted to a circuit card such as Duroid or FR-4, 1/4 watt if mounted to metal with conductive epoxy, and 1/2 watt if it is attached with solder to a metal heatsink.
25. 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.
26. When designing "split-block" waveguide sections, tell your mechanical engineer that you have to split the guide the hard way, cutting through the H-plane (along the broad wall). If you put a mechanical seam in the E-plane (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.
27. For N-way 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, wide-band solution.
28. For an impedance-matched 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.
29. 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!
30. 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. Self-biased amplifiers have much lower gain/temperature coefficients (less variation with temperature).
31. When it comes to capacitor materials, the ones with the highest dielectric (K) are most likely to have the worst variation over temperature.
32. The physical length of a Lange coupler is approximately equal to one quarter-wavelength at the center frequency on the host substrate. The combined width of the strips is comparable to the width of a fifty-ohm line on the host substrate.
33. 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.
34. If you are 2d²/ or farther from an antenna, you are in the far-field.
35. Via-hole inductance: on 2-mil (50 micron) GaAs, Lvia is about 10 pico-henries. For four-mil (100 um) GaAs, it is about 20 pico-Henries. Anyone have a ROT for alumina or PWB inductance?
36. When measuring high power with a microwave watt meter assemble your in-line 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 high-power source! -UE
37. 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!
38. 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!)
    
39. 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.
40. 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.
41. You should measure the return loss of a mixer's three ports at the recommended LO drive level, or you will get ugly results.
42. For the best LO to IF isolation in a double-balanced mixer, always tap off the IF from the RF balun, not the LO balun. You should get 20 dB better LO rejection this way.
43. When subscribing to trade journals, always give them a fake email address and phone number. Otherwise they will be bugging constantly!
44. 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).
45. 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.
46. 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.
47. The antenna pattern for a horn antenna can be approximated as
    
    P(dB)=-10x(/_-10dB)^2
    
    
48. Doppler shift at X-band is approximately 30 Hertz for 1 mile per hour. If you are traveling at 60 miles per hour, your Doppler frequency on police X-band radar will be approximately 1800 Hz.
49. 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.
50. The gain of a narrow-beam reflector antenna is approximately 27000/(12), where 1 and 2 are the 3 dB (half-power) beamwidths in the principal planes, measured in degrees (not radians).
51. Electromagnetic radiation at frequencies higher than light (such as x-rays) 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 non-ionizing 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 person’s pain threshold will induce cataracts over time. It should be noted that frequencies whose wavelength more closely match the size of the eye (X-band freqs with wavelengths in the 2-4 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
    
    
52. When measuring S-parameters 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...
53. 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.
54. 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.
55. Switch isolation is often limited by package isolation. If you design a 60 dB switch, you should think carefully about how to package it!
56. 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.
57. 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!)
58. 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.
59. 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.
60. 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!
61. 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!
62. For pure alumina (_R=9.8), the ratio of W/H for fifty-ohm 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 PTFE-based soft board materials _R=2.2), W/H to get fifty ohms is about 3. Remember these!
63. The accepted limits of operation for rectangular waveguide are (approximately) between 125% and 189% of the lower cutoff frequency. Thus for WR-90, the cut-off is 6.557 GHz, and the accepted band of operation is 8.2 to 12.4 GHz.
64. 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.
65. 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.
66. 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.
67. 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!
68. 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.
    
69. 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 3-6% higher impedance and you will end up with a better match.
    
70. 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.
71. The isolation of a Wilkinson is limited to 6 dB better that the return loss of the source match at its common port.
72. 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.
73. 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!
74. In order to use silicon as a substrate, you need resistivity at least 100 ohm-cm or the loss is going to eat your lunch.
75. 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 two-mil to four-mil GaAs.
76. 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.
77. 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 super-genius with IQ 250. You pick which one, then accuse them when they bring this up at their next peer review!
78. 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!
79. Different loss mechanisms have different behaviors over frequency. Metal loss is proportional to square-root frequency. Dielectric loss is proportion to frequency. Dielectric conduction loss is constant over frequency.
80. When considering the transmission line loss due to dielectric conductivity, if the resistivity of the dielectric is greater than 10,000 Ohm-cm, forget it! That pretty much rules out all substrates except silicon, which can be anywhere from 1 Ohm-cm (very lossy) to 10,000 Ohm-cm (very expensive float-zone silicon). PTFE is 1E18 Ohm cm!
81. 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 low-loss if the loss per wavelength is less than 0.1 dB. Waveguide will routinely be 10X better than this benchmark!
82. 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 PTFE-filled).
83. 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.
84. 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 co-worker'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 co-worker'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 one-of-a-kind artifact, but there's a good chance the picture will be an unwelcome surprise to the co-worker's spouse (why would the picture be buried in a drawer in the first place?) We speak from a near-miss experience a long time ago, when an alert friend of the co-worker pulled the "that's not his wife!" photo from the box just as it was heading to the shipping department!
85. Here's a freespace path loss rule of thumb, thanks to Stefan.
86. The typical isolation you can expect from a two channel receiver is on the order of 25 dB. For dual-channel MMICs, expect no more than 30 dB.
87. 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!
88. When a solid state amplifier is pulsed on for 100 microseconds or longer ("long" pulse), it reaches a quasi-steady-state 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).
89. 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).
90. 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.
91. The graceful degradation of gain in an N-way combiner used in a power amplifier decreases as [(N-X)/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 (N-X)/N. This assumes ideal conditions, at center frequency, and an isolated power splitter is used (such as a Wilkinson).
92. 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....
93. The number of elements required in an electronically-scanning 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!
94. 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!
95. 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.
96. 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.
97. If your circuit does funny things when you close up the box, it's oscillating. If you can't see it on the spec-an then it's above the instrument range.
98. 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.
99. 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.
100. 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.
101. 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 high-pass/low-pass bit.
102. 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.
103. 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.
     
104. 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!
105. 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 (alpha-c) is increased by 60%. If surface roughness is much more than one skin depth, the increase is 100% (2X ideal loss).
106. 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!
107. You should plan on the off-state resistance to somewhere between 5000 ohm-mm and 50,000 ohm-mm. In most designs you can just ignore it, but in this example, it's important.
108. 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).
109. 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 cross-sectional 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).
110. 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 two-way combiner, the peak error is twice the RMS error, so your amplifiers must be within 11.5 degrees transmission phase.
111. Following rule 110, suppose you pick amplifiers from the "Waffle Pak of Uniformly Distributed Phase", or WPUDP, spanning X degrees (more specifically, WPUDP-X). The expected RMS phase error is X divided by the square-root of twelve. You should recognize that term as the square root of the variance (σ, not σ² 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....
112. 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 zero-bias detector…. anyone want to comment?
113. In slow wave structures using alternate high and low impedance segments, the Bragg frequency occurs when the segments are 30 electrical degrees.
114. 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.
115. The Bragg frequency of an artificial transmission line occurs where the unit cell is 1/3 of a wavelength.