Lumped
element filters
Updated April 18,
2014
Click here
to go to our main page on filters
Click
here to learn more about the definition of lumped elements
Click
here to go to our page on mismatch loss
Click
here to go to our discussion on inductors
Click
here to go to our discussion on capacitors
Click
here to go to our download area and get the lumpedelement filter
calculator
Click
here to learn how our lumped filter calculator works
On this page we will discuss
the topologies for lowpass, highpass and bandpass lumped element
filters. We will also introduce you to a Microwaves101 download
which calculates all of the inductors and capacitors for threepole,
fourpole and five pole Chebyshev lumpedelement filters, including
lowpass, highpass and bandpass. This is something that Agilent
ADS can't do for you, and Eagleware would charge you big $$$ for!
Let's also point out that in the download area you can find "Vlad's Filter Calculator
Update, January 2005...
we've revised the lumpedelement filter download into something
much more useful. It now plots the frequency response of Chebyshev
filters in real time, while you vary the passband and ripple requirements!
And we now have a page of instructions
and examples of using this spreadsheet. All of the examples
presented on both pages were synthesized and plotted using our download
filter calculator
Here is the usual Microwaves101
clickable index to our material on lumped element filters:
Frequency
limitations of lumped element filters
Tee versus
pi networks
The relationship
between VSWR and ripple for a filter
Lowpass
lumped element filters
Highpass
lumped element filters
Bandpass
lumped element filters
Free
download for Chebyshev lumped element filters (separate page)
Lumped
filter suppliers
Lumped
element filter frequency limitations
Before we get any further, the
analysis below and the free download are all for ideal lumped elements.
As you go up in frequency, lumped elements become less and less
ideal until you can't make a filter this way at all. If you intend
to base a microwave design on lumped elements you will have to resimulate
the filter including all parasitic elements such as parallel resonances
of inductors and series resonances of capacitors, and of course
all resistive contributions. If you are building an RF filter at
50 MHz you will get good results. The use of lumpedelement surface
mount filters above Lband can be tricky.
When we are discussing lumped
elements, their are two broad categories of inductors
and capacitors. The first is surface
mount parts, which are suitable for use on a microwave printed circuit
board. The second category is thinfilm lumped elements, which are
used on microwave integrated circuits (MICs) on alumina or other
"hard" substrates, as
well as in MMIC implementations. Surface
mount capacitors can be useful up to Kuband, while the inductors
generally have selfresonances below Xband. Thinfilm capacitors
are used routinely up to 100 GHz in MMICs. Lumped inductors on thinfilms
(and MMICs) take the shape of spiral inductors, and are limited
to frequencies Kuband and lower. Inductors are the problem stepchild
of lumped element filters and other microwave circuits!!
You can obtain 10%, 100% or more
bandwidth with lumped element filters. The attraction to lumped
element filters is that the filter can often be very compact compared
to a filter based on halfwave resonant structures.
Tee versus
pi networks
The terms "tee" and
"pi" are used to describe lumped element filters, as well
as attenuators and other networks.
A tee element starts with a series element, while a pi network starts
with a shunt element as shown below. The "tee" resembles
a letter T while the "pi" resembles a Greek letter pi.
The figures below are for threepole networks. Odd numbers of poles
still resemble the tee and pi letters somewhat, but for even numbers
the distinction is nearly lost since a tee network would start with
a series element and end with a shunt element. We recommend that
you use odd numbers of poles in your lumped element filters because
the elements are symmetrical, that is, for a fivepole network,
C1=C5, C2=C4, L1=L5 and L2=L4. This will cut down on your bill of
materials and there will be fewer possibilities of assembly mistakes.
Tee network highpass
filter

Pi network lowpass filter

Below are pictures of N=3, N=4
and N=5 filters, pi and tee, BPF, LPF and HPF. Some things to note
when selecting pi or tee, which will be apparent when you look over
the figures.
 All LPFs will pass DC from
input to output
 All HPFs block DC from input
to output.
 In certain cases, filters
will present a short circuit to ground on one or both inputs.
In other cases the filter will look like an open circuit to DC.
For N=even, the two ends of the filter are different.
The DC properties of a filter
are important to understand. Quite often in microwave design, transmission
lines are used to provide DC bias voltages and currents to active
devices.
The
relationship between VSWR and ripple for a filter
The topic of mismatch
loss rate its own page, here we will
discuss it in the context of filter design.
Using ideal, lossless
elements, there is a specific relationship between the passband
ripple and the return loss of a filter. That is because the passband
ripple is 100% due to mismatch loss. The ripple is equal to the
worstcase insertion loss, which occurs at the same frequency as
the worst case VSWR (the worst case mismatch loss).
Mismatch
loss is calculated from the reflection coefficient rho:
mismatch loss=(1rho^{2})
(% power reflected)
mismatch loss (dB) = 10*log(1rho^{2})
Note that in this
case we use 10*log because the quantity (1rho^{2}) is in
units of power, not voltage (which would use 20*log). The mismatch
of a filter is often specified as a maximum VSWR. Rho is calculated
from VSWR as:
rho=(VSWR1)/(VSWR+1)
So the relationship
between VSWR and mismatch loss is then:
Ripple (dB) = mismatch loss
(dB) = 10*log[1((VSWR1)/(VSWR+1))^{2}]
Let's make a little
table:
VSWR

rho

Return
loss (dB)

Ripple
(dB)

2

0.333

9.542

0.512

1.9

0.310

10.163

0.440

1.8

0.286

10.881

0.370

1.7

0.259

11.725

0.302

1.6

0.231

12.736

0.238

1.5

0.200

13.979

0.177

1.4

0.167

15.563

0.122

1.3

0.130

17.692

0.075

1.2

0.091

20.828

0.036

1.1

0.048

26.444

0.010

Note that we have a VSWR calculator that
will allow you to convert between VSWR, mismatch loss and return
loss!
Lowpass
lumped element filters
The lowpass filter often is
a natural choice for lumped element filters. This is because the
parasitics of lumped elements tend to kill the frequency response
as you go higher in frequency. For example, you might want to consider
adding a lumpedelement LPF at the output of your downconvertor
design. If you are mixing Xband down to 50 MHz, your output will
be clean of RF and LO leakage and you'll be styling! Lumped element
lowpass filters are quite often used as "cleanup" stages
for planarresonator filters, to defeat reentrant modes.
In the figure below we compare
low=pass designs for 0.5 dB ripple, cutoff frequency 5 GHz. Note
that the return loss is directly related to the insertion loss S21.
For 0.5 dB ripple, you get about 10 dB return loss. The other thing
to note is that the skirt on the N=5 filter falls off far faster
than for N=3. In this example, if 8 GHz was a critical frequency
you need to block, you might not get enough rejection with N=3 (about
12 dB), but with N=5 you might satisfy your requirement because
you would have 30 dB rejection.
Highpass
lumped element filters
Highpass filters are tricky.
Because of parasitics your highpass filter will actually end up
more like a bandpass filter, its frequency response will eventually
die out. But you can certainly design a filter that will reject
low frequencies using lumped elements. A DC block is a simple example.
We'll add some examples, check back later!
Bandpass
lumped element filters
Used in bandpass filters, lumped
elements can really help your designs. Not only are the resulting
filters compact, but they have no "natural" reentrant
modes that you will encounter with other filter technologies such
as planar resonator structures.
In each resonator of a lumpedelement
bandpass filter you will find both an inductor and a capacitor.
The series elements, the LC is combined in series. In the shunt
elements, the LC is combined in parallel. Screw this up and you
will create a bandstop filter instead of a bandpass filter! Just
refer to our figures above and you can't go wrong.
Below we compare lumpedelement
Chebyshev bandpass filters with passband 4 to 6 GHz, and 0.5 dB
ripple. The first plot is for N=3, the second plot is for N=5. Note
that the skirts are much steeper for the higherorder filter. Again,
the return loss of 10 dB is a function of the specified ripple of
0.5 dB. Which filter you would pick must be determined by how much
rejection you need at specific frequencies.
One other "curiosity"
about lumped element Chebyshev bandpass filters is that the lower
skirt is steeper than the upper skirt. What gives here? The explanation
is that the skirts would look equal, if you plotted the frequency
on a logarithmic scale.
