Hmmm.... - you want a network analyzer for > $20000 in one chip
Originally Posted by willywalker
ok... I try
very important formula is:
Z is (unknow) complex impedance
Z0 i reference impedance (mostly case real value)
Γ is reflections cofficient 'gamma', HP call this for s-parameter (here S11)
Z - Z0
Γ = ---------- (Γ also called 's' as S-parameters in old HP-litterature)
Z + Z0
and after normalisation z = Z/Z0
z - 1
Γ = ------
z + 1
And if you via measure bridge (weathstone-bridge or using directive coupler etc.) measure reflected amplitude and phase angle compared to sending wave + know reference impedance (50 Ohm in moste case), ie. make know Γ in amplitude ratio of sending voltage and phase angle compare to sending wave.
For example measured reflected amplitude (voltage) 0.4472 and lagging 63.43 degree compare to sending wave => Γ = 0.4472 |_ 63.43 degree in polar notation or 0.2+i0.4 in rectangular notation.
1 + Γ
z = ---------
1 - Γ
put in value (rectangular notation)
1 + 0.2+i04
z = --------------- = 1+i1
1 - 0.2+i0.4
z*Z0 = Z
1+i1 * 50 = 50+i50 Ohm or 70.71 |_ 45 degree Ohm
ie. 50 Ohm serial with 50 Ohm inductive reactace (you must know measure frequency for make inductance value in Henry)
Above calculation is working only if complex unknow load is coupling directcly to measure bridge without any kind of transmissions line or other error.
If you have transmission line (same impedance as reference impedance) say 10 degree lag travel time from source (bridge) to load and back again, above Γ = 0.4472 |_ 63.43 value turns now to Γ = 0.4472 |_ 73.43 (10 degree extra lag) or 0.1275+i0.4287 in rectangular notation
If you not know this extra delay and calculate directly as above formula, give
ie. wrong measured value depend of unknow distance of transmissions line.
1 + 0.1275+i0.4287
z = ---------------------- = 0.8466+i0.9072
1 - 0.1275+i0.4287
z*Z0 = Z
0.8466+i0.9072 * 50 = 42.33+i45.36 Ohm
Other moment is loss on transmissions cable, If you using exact multiple of half wave lengt transmissions line (no measured extra cable depend delay) and you lose 1 dB in amplitude between source and load (0.5 dB each direction)
1 dB loss is same as 0.8913 of sending voltage, ie reflected value have only 0.8913 of wanted theroretic value from load as Γ = 0.8913 |_ 0 degree * 0.4472 |_ 63.43 = 0.3986 |_ 63.43 or in rectangular notation 0.1783+i0.3565
and put in formula:
Impedances going closer to 50 Ohm same way using PI-attenuator to make better return loss from load with low return loss - and loads real value going more or less hide.
1 + 0.1783+i0.3565
z = ---------------------- = 1.048+i0.8886
1 - 0.1783+i0.3565
z*Z0 = Z
1.048+i0.8886 * 50 = 52.42+i44.43 Ohm or 68.72 |_ 40.28 degree Ohm
If you have both 10 degree extra lag and 1 dB attentuation in used measure cable to load, give shortly result of:
measured: Γ = 0.398.6 |_ 73.43 degree => 45.14+i41.01 Ohm (60.99 |_42.26) formula above compare to rigth value of 50+i50 Ohm (70.71 |_ 45)
If you know this transmissions line depend influence (here 0.8913 |_10.00e0), you can use this 'constant' to take away influence of cable, ie. 0.398.6 |_ 73.43 / 0.8913 |_10.00 = 0.4472 |_63.43 before using above formula and make correct 50+i50 Ohm result
Conclusion: you must know every part of measure bridge and cable character and length to load if you want measure complex load with resonable accurate.
Ie. you must make calibration process (easiest via open, short, load-measure) to make reference plane, error correction constant to hide most of errors in bridge and used cable (length).
mostly RF-book handle matemathic behind complex load and using smithchart etc.
one of many books describe this:
Microwave transistors amplifiers, analysis an design, Guillermo Gonzales and possible to find here in board
If remember rigth hp/Agilents more or less famous appnote 95 describe how S-parameters, network analyzer and math work behind of this.
You can using chip with EXOR input to measure angle between reference and measured signal, and using RSSI-circurits to measure amplitude. If I remember right, Analog Device have type of chip including both RSSI measure (ca 60 dB dynamic) and phase-comparator up to 500 MHz (or 1 GHz??)
Exor-comparator type of phase angle measures can only measure near +90 to -90 degree range and you need reference or measured wave delayed 0 and 90 degree and switch between under measure process (ie. I and Q-measure) to make decision if measured phase angle comming from capacitance or inductive reactance.
You need MCU/computer power to correct measures from error corrections constants and calculate value as above formula. I thinking is not possible to find in one chip solution - yet...
For easy calculate with complex number in above formula I using 'free42' from http://home.planet.nl/~demun000/thomas_projects/free42/
This calculator is perfect to work with complex impedances etc.
Is a RPN-calculator, ie. use '1' <enter> '2' '+' for answer '3'.
to make or split up complex number: '1' <enter> '2' <orange button> 'complex' for answer '1 i2' and after this you can use '+-/*' and all other math function same as normal number without extra moment or special mode for operating on complex numbers.
ie '1 i2' <enter> '3' '*' give answer ' 3.0 i6.0' without fuzz, you _not_ need make '3' as '3 i0' before complex operation as needed on hp32sii or hp33s...
Input and view mode between rectangular and polar notation of complex numbers adjusts via <orange button> 'mode' and select 'rect' or 'polar' to wanted show and input notation.
Yes, you can find function to convert complex numbers to different notation , but this way is absolute fastest.
To make storage register to accept complex numbers, you must make this moment one time:
'RCL' <select REGS> <enter> <orange button> 'complex' 'STO' <select REGS>
after this you can using STO 00-25 to storing real and complex numbers.
(I hopfully thinking rigth now...)