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  1. #1
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    complex impedance calculator

    Hi,
    I found out how I can calculate the complex impedance of a unkown load by replacing it with two other loads and doing some maths, but what Im really looking for is to calculate the complex impedance of a load by measuring forward and reflected power right after the power amplifier. I remember these formulas were taught at university; how to calculate the complex impedance from 3 different measurements of forward and reflected power, but I cant find the maths in my books, neither on the internet. Or is there an IC available that does all the complex impedance calculation by itself?
    Can anyone help me out on this one?
    Thanks a lot on forehand.

    •   Alt22nd July 2006, 17:24

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  2. #2
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    power calculation complex impedance

    Quote Originally Posted by willywalker
    Hi,
    I found out how I can calculate the complex impedance of a unkown load by replacing it with two other loads and doing some maths, but what Im really looking for is to calculate the complex impedance of a load by measuring forward and reflected power right after the power amplifier. I remember these formulas were taught at university; how to calculate the complex impedance from 3 different measurements of forward and reflected power, but I cant find the maths in my books, neither on the internet. Or is there an IC available that does all the complex impedance calculation by itself?
    Can anyone help me out on this one?
    Thanks a lot on forehand.
    Hmmm.... - you want a network analyzer for > $20000 in one chip

    ok... I try

    very important formula is:

    Code:
    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.


    and using:

    Code:
           1 + Γ     
    z = ---------
           1 - Γ
    
    put in value (rectangular notation)
    
    
           1 + 0.2+i04
    z = ---------------   = 1+i1
           1 - 0.2+i0.4
    
    and denormalisation 
    
    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

    Code:
           1 + 0.1275+i0.4287
    z = ----------------------   = 0.8466+i0.9072
           1 - 0.1275+i0.4287
    
    and denormalisation 
    
    z*Z0 = Z 
    
    0.8466+i0.9072 * 50 = 42.33+i45.36 Ohm
    ie. wrong measured value depend of unknow distance of transmissions line.

    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:

    Code:
           1 + 0.1783+i0.3565
    z = ----------------------   = 1.048+i0.8886
           1 - 0.1783+i0.3565
    
    and denormalisation 
    
    z*Z0 = Z 
    
    1.048+i0.8886 * 50 = 52.42+i44.43 Ohm or 68.72 |_ 40.28 degree Ohm
    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.


    ---

    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...)


    /xxargs



    •   Alt23rd July 2006, 05:59

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  3. #3
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    degree-ohm

    If remember rigth hp/Agilents more or less famous appnote 95 describe how S-parameters, network analyzer and math work behind of this.
    You're right, the good old HP AN95, I've got to dig that one up...
    I'll break my brains on these maths again and then program a mcu to do the thinking for me :D

    Thanks, but still...all additional comments are welcome!



    •   Alt24th July 2006, 21:26

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  4. #4
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    find the complex impedance

    Quote Originally Posted by willywalker
    If remember rigth hp/Agilents more or less famous appnote 95 describe how S-parameters, network analyzer and math work behind of this.
    You're right, the good old HP AN95, I've got to dig that one up...
    I'll break my brains on these maths again and then program a mcu to do the thinking for me :D

    Thanks, but still...all additional comments are welcome!
    curiosity:

    What for range of frequency you thinking measure.
    how you think build measure bridge ???
    power range?

    For low power I thinking weatstone bridge is easiest to build, but needs balun or balanced input to measure and (very) low source impedance or make source impedance part of measure bridge.



  5. #5
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    phase angle calculation from complex impedance

    What for range of frequency you thinking measure.
    3-50MHz

    how you think build measure bridge ???
    directional coupler + log detector

    power range?
    depends on coupler...can go from 1mW...1,5kW

    Possibilities are infinite...




    Willy



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