Updated December 31,
here to go to our page on basic concepts of microwaves
here to learn about a rule of thumb for measuring the length
here to go to our page on the Smith Chart
here to go to our page on Schiffman phase shifters
here to go to our page on Lange couplers
How do you turn a short circuit
into an open circuit, or a capacitor into an inductor? Here we will
describe some of the magic that happens when you use the distributed
properties of transmission lines, in particular when you use
lines of length one-quarter wavelength, or multiples of a quarter-wavelength.
Here is a clickable index to
our material on quarterwave transformers:
so special about a quarter-wavelength?
transformers (separate page)
flat transformers (new for November 2008!)
interference of two equal VSWRs (featuring more wisdom from
to use constructive interference when designing with PIN diodes
Quarter-wave coupled lines
so special about a quarter-wavelength?
Referring to a Smith chart, if
you are already at a matched impedance condition, any length of
transmission line at the system characteristic impedance Z0
does nothing to your input match. But if the reflection coefficient
of your network (S11 for example) is at some non-ideal impedance,
adding transmission line between the network and the reference plane
rotates the observed reflection coefficient clockwise about the
center of the Smith chart. Further, the rotation occurs at a fixed
radius (and VSWR or return loss magnitude) if the transmission line
has the same characteristic impedance as the source impedance Z0.
By the time you have added a quarter-wavelength, you have gone 180
degrees about the center of the Smith chart.
Suppose your network was a short
circuit, the left "corner" of the Smith Chart. Adding
one quarter-wavelength moves you 180 degrees to the right side of
the chart, to an open circuit. It's Opposite Day, Sponge Bob! That's
the magic of a quarter-wavelength But there's much more that you
can do, as you will see on this page.
Here's a frequently asked question:
if a quarter-wave is 90 degrees in phase length, why does
it transform you 180 degrees on a Smith chart? Consider that
here we are plotting reflection coefficients on the Smith chart.
Thus an imaginary signal that you send through a transmission line
of one quarter wavelength must travel fully half a wavelength, since
it travels down the quarter-wave line, gets reflected, then returns
down the quarter-wave line. So when you are plotting reflection
coefficients, moving in a complete "circle" means only
adding 180 degrees! Note that if you are plotting transmission coefficients
(such as S21 of a two-port), "90 degrees is 90 degrees".
Starting with an open circuit,
one quarter wavelength away you will "see" a short circuit.
Starting from a short circuit, one quarter wave away you have an
open circuit. Thus you can create an "RF open circuit"
that is a DC short circuit, and vice versa. These two properties
are used to create DC and/or RF grounds for circuits, bias tees,
and much more.
Below are ADS models of a open
circuit and a short circuit stub. The electrical length of each
stub is 90 degrees.
The following figure shows the
reflection coefficient of the above stubs, versus frequency, starting
at DC and sweeping up to the quarter-wave frequency (10 GHz, denoted
by markers M1 and M2.) The open circuit S11, plotted on the left)
sweeps from an open to a short, while the short circuit (S22, on
the right) sweeps from a short to an open. ADS wouldn't lie, would
it? Note that both responses sweep clockwise with frequency, and
would keep going around and around the Smith chart if you kept increasing
The open circuit stub trick (creating
an RF short circuit that is DC-open) is often done with lower impedance
lines than Z0. This effectively gives a wider bandwidth.
An even better trick is to use a microstrip radial stub; it provides
a low impedance, it doesn't suffer from a large distributed tee
junction that a constant-width low-impedance stub would, and it
just works better (trust us!)
The tradeoff you are making
is circuit size , because low-impedance microstrip lines are wider
than high-impedance lines. Below are ADS models of two microstrip
stubs on a 50 micron (2 mil) GaAs substrate (Er=12.9). We'll replace
the model with a layout one of these days, it will help illustrate
the point better.
What we are looking
for with an open-circuit stub is a high return loss, like -1 dB.
The plot below shows that the radial stub has better bandwidth.
Why is a radial; stub better? We think it's because it has a larger
fringing capacitance at the open end, which adds a "lumped-element"
quality to it. Who cares, it just is!
Note that whatever
EDA software you are using, you will need to take into account the
fringing capacitance at the end of the open stub. The ADS radial
stub model attempts to do this, but it would be better if you modeled
the stub using Momentum or another 2.5D E-M solver.
Time for another Microwaves101
rule of thumb:
Rule of thumb:
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.
Later we'll show you that if
you are trying to transform your RF short stub to an open circuit
(as you would in a bias tee), you should use a high impedance line.
There are other considerations, such as current carrying capacity,
that you'll need to consider.
A simple bias can be arranged
from quarter-wave lines as shown in the figure below (again we are
using 50-micron think GaAs as the substrate in our example). A bias
tee is a "diplexer", a type of filter that provides two
paths from a common node, the dominant path is dependent of the
frequency of the signal. Look this up in our section
In the simplest bias tee, an
open circuit stub (low impedance, here W=100 microns) is used to
create an RF choke in the DC arm, to allow DC bias to inject but
isolate RF from the bias network. A high-impedance line (here W=25
microns) is used to transform the RF short circuit to an RF open
circuit; this is necessary so that the bias arm does not load the
RF path. The lengths of the quarter-wave sections were chosen so
that the bias tee works best at about 25 GHz (1000 microns is about
a quarter-wave at that frequency).
The most important
S-parameters of the simple bias tee are shown below. S21 represents
the transmission coefficient along the RF path, it is ideally 0
dB at the RF frequency, but we would forgive the bias tee if it
had a few tenths of a dB of RF loss. S31 represents the isolation
of the DC connection to the RF path, it is ideally a small magnitude
(perhaps 30 dB). The reflection coefficient S11 is also plotted
separately, this gives an idea of how much the bias arm is loading
the RF path. A good value for S11 is better than 14 dB (VSWR of
better than 1.5:1). Studying the graphs, it is apparent that this
simple circuit performs well over 20% bandwidth.
This topic has moved to a separate
Constructive and destructive
interference occur in many physical sciences involving wave phenomenon,
including microwaves. Is it time for another sad Microwaves101 story
in which Wally enlightens his coworkers on this subject? You betcha!
Bert and Ernesto are in the lab
(any resemblance to any of these characters and real microwave people
is just too bad!), testing an integrated circuit in their new two-port
test fixture. Its performance stinks compared to the manufacturer's
data sheet. Let's listen in...
Bert: Better go get Evita,
it's time to scrape off another one of these mixer chips. I guess
the whole waffle-pak stinks.
Ernesto: Hey, here comes Wally,
let's ask him what's going on.
Wally: The boss-man woke me
up from my nap and said to come out and fix your latest mess.
Did you measure the test fixture back-to-back like I told you?
Ernesto: Sure, we did that,
the return loss at 28 GHz was better than 20 dB. Like everything
I design on HFSS, the connector transition is just about perfect!
Wally: That don't mean squat.
What's the worst case return loss you have between 0 and 40 GHz?
Bert: We didn't have time for
all those frequencies, so we just measured it at a single point.
Who cares about performance outside of the band, O Great One?
Wally: Tarnation, didn't ITT
Tech teach you hippie fools about constructive addition of VSWRs?
Now get busy and measure the fixture back-to-back over as much
bandwidth as you can, and report back to me the WORST CASE return
loss that is nearest to 30 GHz. I gotta go read Microwave Journal
and see how much better our competitors must be doing...
Bert: Wally, we see about 4
dB return loss at 20 GHz. So what?
Wally: I'll tell you what,
you longhaired freak, that means each connector probably has no
better than 9 dB return loss in your band, which explains why
Evita has been so crabby lately since you've had her scraping
off perfectly good MMICs all week for no reason. If it wasn't
lunch time I'd go have a talk with the Big Man and get me a piece
of your next raise!
Bert: I don't understand how
he did that calculation in his head, but we'd better go have a
look at that connector transition you designed on HFSS.
Ernesto: Sure, but I'm late
for the "Midget Transvestite Hispanic Engineer of the Year"
Bert: Good luck! Diversity
is a beautiful thing. Don't worry if you don't win, everyone at
the meeting gets a nice wooden plaque that says "Participant Award!"
Evita: Von't von of you dahlinks
pleeeeze rub my neck?
Ernesto: Yikes! That's harassment,
I'm telling HR!
Here's what Wally knows. When
you measure a two-port fixture back-to-back, the return loss you
measure is really the composite return loss of the two connectors,
with a pretty good fifty-ohm transmission line separating them. By "composite" we mean the additive effects of the two mismatches. As you go up in frequency,
the connectors become separated by a quarter-wavelength a some frequency,
and if they have identical reflection coefficients, a null occurs
in the "composite" return loss. At a higher frequency
the connectors are separated by 3/4 of a wavelength, and a null
occurs again. This effect is plotted below:
Indeed, if you measured the above
part at 28 GHz, you might conclude that it has good connectors.
But you'd be wrong. The relationship between composite return loss
and individual return loss in the constructive condition of identical
mismatches is that the composite VSWR is the the square of the individual
VSWRs. For one example, if you have two 2:1 mismatches separated by some distance, the worst case at some frequencies will look like 4:1 VSWR.
You could spend two minutes converting what we just said
to dB, but we will save you the trouble:
In the above
plot you can see that for high return losses, the additive return
loss approaches 6 dB worse than the individual return losses. As
your mismatch gets worse and worse, eventually the additive return
loss equals the individual return loss (in the case of infinite
VSWR, they are equal). So it is time to present another Microwaves101
rule of thumb:
Rule of thumb: due to constructive
interference, the individual return loss of identical mismatches
are usually about 6 dB better than the worst case observed return
loss of two mismatches measured together.
to use constructive interference when designing with PIN diodes
Let's beat this topic to death,
showing how you can use it to your advantage. Suppose you are building
a limiter or switch that requires a PIN diode in shunt across a
transmission line. In the "low-loss" state it behaves
like a capacitor. It's impedance, equal to jwC, puts you slightly
northwest of fifty ohms on the Smith chart. As you go higher in
frequency, your VSWR degrades more and more, until it is unusable
. A very simple solution is to use two identical diodes, separated
by slightly more than a quarter-wavelength, to cancel out each other's
mismatch. Below is an example, using PIN diodes with off-capacitance
equal to 0.1 pF.
Below we've plotted the return
loss of a single 0.1 pF capacitor (blue), and the combined return
loss of the pair of 0.1 pF capacitors separated by a 50-ohm line
of 135 degrees length at 10 GHz (red, don't ask why we chose 135
degrees, we need to change that because it is confusing!) Nulls
occur in the reflection coefficient at 1/4 wave, 3/4 wave, 5/4 wave
frequencies, and these are nulls you can use! This leads to our
third rule of thumb on this page...
Rule of thumb: 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. The next two figures help drive home this
point. Shunt capacitive cancellation is illustrated below;
remember that a quarter wavelength line would move you 180 degrees
on the Smith chart.
cancellation is shown below:
here to go to our page on coupled line couplers.