- Jan 1, 1970
John said:I'd vote for 15 km.
Figure twisted pair at roughly 0.7, depending.
For high frequency > 1 MHz - yes, but for audio frequency is different
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
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.