# Pi-Network Transfer Function

Discussion in 'Electronic Design' started by Jumbaliah, Nov 2, 2004.

1. ### JumbaliahGuest

I'm trying to come up with some component values for a pi-network low
pass filter for my future DDS project. I started to try and figure
out the basic transfer function for a 3rd order pi-network (two caps
and one inductor). It doesn't look right to me so I wanted a second
opinion.

Vi-+--Z2--+--Vo
| |
Z1 Z3
| |
Gnd Gnd

What I have is Vo/Vi = Z3/(Z2+Z3).

The equations I used to reach this was the following:
I2 is a current loop from Vi -> Z1 -> Gnd -> back to Vi
I1 is a current loop from Gnd -> Z1 -> Z2 -> Z3 -> Gnd

eq1 Vi = I2*(Z1) - I1*(Z1)
eq2 Vo = I1*(Z3)
eq3 0 = -I2*(Z1) + [Z1 + Z2 + Z3]*I1

I arranged eq1 and eq3 to solve for I2. I solved for I1 in eq2 and
sub'd that into eq1 and 3. I factored and cancelled and flipped the
equation till I got Z3/(Z2+Z3). It doesn't seem right that Z1 isn't
involved. I must have missed something here.

Would anyone have the general transfer function for a nth degree
pi-network?

Thanks.
Burnboy

2. ### Jens TingleffGuest

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There are loads of books with schematics and tables of component values for
filters.
What you might have overlooked is that Z1 is loading a perfect voltage
source, and so there is no effect on voltage V1 from Z1. Simple

You could put a current I1 into node 1, that would make Z1 a part of the
trans-resistance transfer function.

Best Regards

Jens

- --
Key ID 0x09723C12,
Analogue filtering / 5GHz RLAN / Mdk Linux / odds and ends
http://www.tingleff.org/jensting/ +44 1223 211 585
"I lost my heel" "Oh, don't worry about him!" 'Bringing up Baby'
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3. ### Allan HerrimanGuest

You seem to be missing something, namely the input and output
impedances.

Vi--Za--+--Z2--+----+--Vo
| | |
Z1 Z3 Zb
| | |
Gnd Gnd Gnd

Za is the source impedance, and Zb is the load impedance. Vi
represents the potential of the Thevenin source, rather than the
actual voltage across the output terminals of the driver.

In a typical RF system, Za and Zb will both be 50 ohm, and (close to)
resistive.

Regards,
Allan

4. ### Kevin AylwardGuest

This is called "filter design". There are no *useful* completely general
formulas for general filters. However, this is a subject extensively
studied such that there are loads of programs out there that will
calculate suitable values for specific filters, e.g. SuperSpice, and
other general filter design programs. I suggest a web search.

Kevin Aylward

http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.

5. ### BanGuest

A pi filter is not usable for your application, if the input impedance is
low impedance. Here it is better to use a 3-pole T-filter consisting of
L-C-L. with the values 120u/27n/22u you will get around 100k bandwidth into
50R load. The coils should have a Q of 20 or higher.
If you put 47R in front to have also a 50R source impedance the bandwidth
goes up to 138k, but stays almost Bessel without over- or undershoot.
Look up the transfer function in google, find the Laplace transform like:
1/(1+1.752S+1.23S^2+0.338S^3) ; S= j*f/f_fil ; f/f_fil is the normalized
frequency operator with f_fil 3dB down.

6. ### Joe McElvenneyGuest

Hi,

You cannot separate the source and load impedances from the
rest of the transfer function. A quick way to design one though,
without the resulting maths, is to use a table of coefficients
for a Butterworth network. They are found in many books and just
need scaling for cut-off frequency and impedance.

Alternatively, split the inductor down the middle to get a
couple of back-to-back L-networks (choosing an intermediate load
impedance). There will be lots of possible values, only limited
if you must have a specific loaded-Q. This latter method is
common when designing tx tank circuits.

If you must have the complete transfer function here is what
my spice program says -

W(s) = RL / [RS + RL + (RS.RL.C2 + L + RS.RL.C1)s + ...

(RL.C2 + RS.C1)L.S^2 + RS.RL.C2.C1.L.s^3]

where C1 is the source-end capacitor.

Cheers - Joe

7. ### JumbaliahGuest

Thanks for the replies everyone.

I've looked in my textbooks and a bit online but so far haven't found
useful tables of component values for pi-networks. I agree with you
all that I should have been using input and output impedances. I've
updated my diagram and formulas to suit.

I originally started using a filter design program from here:

It is simple enough however when I chose my filter parameters it
calculated some capacitance and inductance values. I then wanted to
change them to standard values but it doesn't recalculate the other
components. This is why I wanted the transfer function of an nth
degree pi-network. This way I could get approx values from the filter
software, adjust some parts to standard values, then solve for the
changes in remaining parts.

I figure right now I'll shoot for a filter with passband of 0-30MHz,
stopband at 35MHz+. Maybe a 7th order Chebyshev or Butterworth.

How did you get that transfer function from spice, Joe? Do you happen
to have one for the 7th order?

With a 7th order Pi-network with input and output impedances I've got
6 equations that would require a lot of paper to solve.... I only went
as far as the third substitution .

Thanks,
Burnboy

8. ### John MilesGuest

Pi networks are sort of old-school. They were popularized by Collins
and other manufacturers back in the days when vacuum-tube power
amplifiers ruled the ham-radio roost. They could match the final
amplifier's relatively-high plate impedance to a low-Z antenna,
providing low-pass filtering at the same time. I don't see them
mentioned very often in current literature.

The ARRL Handbook would probably be a good place for further research on
pi networks. If you can find the old "Single Sideband Principles and
Circuits" volume by Sabin, Bruene, and Schoenike, it will probably go
into more detail. Not sure what else to recommend as far as books you
can actually find outside of a well-stocked university library.

-- jm

9. ### BanGuest

Use spice to evaluate the components. Here is a 7th order Tschebycheff(0.2dB
ripple) at 32.5MHz which will be 30dBdown at 50MHz. But beware, the
overshoot is high with a pulsed input and unsuitable for Video or
Oscilloscope applications.

___ ___ ___ ___ ___
+-|___|--UUU-+--UUU-+--UUU-+--UUU-----o
| 50R 0.33u | 0.56u| 0.47u| 68n |
/ \ --- --- --- .-.
( ~ ) --- --- --- | |50R
\_/ |150p |150p |68p | |
| | | | '-'
| | | | |
=== === === === ===
GND GND GND GND GND
(created by AACircuit v1.28 beta 10/06/04 www.tech-chat.de)

10. ### Reg EdwardsGuest

programs -

LPF_HPF

and

BANDPASS1

from website below.
----
............................................................
Regards from Reg, G4FGQ