Characteristic impedance - low vs high frequencies

Jun 07, 2007 22 Replies

For high frequency > 1 MHz - yes, but for audio frequency is different rules

cable parameters (one kind of phone cable) as:

R = 86.2 Ohm/km (unit length set to 1 km now) L = 0.8 mH/km G = 1.6 uS/km C = 37.8 nF/km

w = 2*PI*f j = imaginary constant as j*j = -1 c = speed of light, 3E8 m/s

and one of two telegraph equation:

gamma = sqrt((R+jwL)*(G + jwC))

gamma = alpha + jbeta

alpha = attenuate in Neper per unit length (1 Neper = 8.686 dB) beta = phase constant in radians per unit length.

and 'w/beta' give phase speed

20 Hz = 0.14 dB attenuate/km and phase speed 10350 km/s or 0.034c 795 Hz = 0.77 dB attenuate/km and phase speed 54347 km/s or 0.181c 20 kHz = 2.1 dB attenuate/km and phase speed 149000 km/s or 0.496c

(all loss exclusive ev. skin effect)

This effect is depend of R and C in cable works like lossy RCRCRC-chain as low pass filter and have dominate effect compare to cable inductance and conductance on low frequency - cable have to low serial inductance and to much parallell capacitance for audio frequnecy range to working as good transmission line - one of first person to discover this effect is a Pupin and use inductance in serie to compensate away much of dominate capacitance in cable - seems very strange (and easy to missunderstud) in first look, and people thinking Pupin make lot sharper low pass filters - is true, indeed, but this works with carefully selected values, distance and suffering high frequency range to make better and lower loss in usable low frequency range for (very) long distance transport.

see:

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2500 Hz going from 115 dB attenuate to 22 dB attenuate with Pupin coil on 100 km long line and over all attenuate going from 0.7 dB/km @ 800 Hz to 0.19 dB/km. Chart show also why phone frequency range have upper limit around 3200 - 3400 Hz...

Is possible to make simular for 20 - 20000 Hz range for kilometers distance between ex. studio and big broadcast FM-transmitter etc.

- but to days is more or less obsolet ie. using modern digital communication replace old analog solution.

But telegraph equations still important and work to many GHz, and two famous example to bulid system without know how or/and respect to telegraph equation is a ATA/IDE-bus and USB 1.1 (USB2.0 is patched up with impedance matching to make 450 Mbit speed)

Characteric impedance on cable is also not so funny in low audio frequency Z = sqrt((R+jwL)/(G+jwC)) (part two of telegraph equation) give for same cable parameters as above: 20 Hz = 4147 Ohm |_ -35.66 degree 795 Hz = 675.7 Ohm |_ -43.3 degree 10 kHz = 205 Ohm |_ -29.6 degree negative degree indicate capacitive reactive load. ie. all impedances is more or less complex.

Depends on how one defines "transmission line effect". For reflection issues, then as noted, there aren't any if the line is short compared to the wavelength. For frequency response effects, then any length of line could have these effects, but this is not really a "distributed line" effect in my view. A lumped model will model it ok. e.g.the total line inductance driving a low load, or line resistance driving a capacitor, or the total lumped capacitance of the line on a source.

Kevin Aylward ka@kevinaylward.co.uk

Maybe Phil was referring to tramsient effects ?

Graham

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