Hi all, brief theory about the Z0 probe:
Z0, gamma (R, L, C, G)
1k tip Zin -> ================================ Zl (termination, oscilloscope side, 50 ohms)
|<-------------l ------------->|
The input impedance Zin of the lossy transmission line (nonzero R & G line parameters) is given by
Zin = Z0 * [Zl + Z0*tanh(gamma*l)]/[Z0 + Zl*tanh(gamma*l)],
where
characteristic impedance of cable Z0 = sqrt[(R + j*omega*L)/(G + j*omega*C)],
propagation constant of cable gamma = sqrt[(R + j*omega*L)*(G + j*omega*C)],
load (terminating) impedance is Zl,
l is cable length,
j is imaginary unit,
tanh is tangent hyperbolic,
and
L, C, R, G are cable parameters (per one meter),
omega = 2*pi*frequency.
And now:
1. When the cable is lossless (R = G = 0), the Zin formula can be rewritten to
Zin = Z0 * [Zl + j*Z0*tan(beta*l)]/[Z0 + j*Zl*tan(beta*l)],
where
characteristic impedance of cable Z0 = sqrt(L/C) and it is pure real number,
beta = omega*sqrt(LC) and it is also pure real number.
From this Zin formula, it results that when we terminate the lossless line with load Zl = Z0,
the impedance seen at the cable input (Zin) is exactly Z0.
As there are no reflections at the Zl (reflection coefficient is 0 if Zl = Z0), there can't be also
reflections at unmatched cable input returned toward Zl. Due this fact, any voltage measured at
the cable input will be also measured at the cable load Zl, of course delayed due to nonzero cable length.
The situation changes when the Zl is not eq. Z0 (but hopefully close). In this case, there are reflections at Zl (osci side) and there are also
reflections at the cable input (1k tip side), because the reflected signal going back through cable sees 1k + some parasitic C and L.
Note that at Zin side, there are nearly total reflection due to such impedance discontinuity.
How to improve this situation:
A. Of course, try to match Zl to Z0 (better osci etc.;-).
B. Terminate cable at Zin side so that reflections spreading from Zl see impedance close to Z0 and are not reflected back.
The solution is to plug 50 ohms resistor at the cable input. In this case, the Zin changes to 25 ohms approx., so the division ration doubles.
To increase sensitivity back, the 1k can be replaced by 470 ohms resistor (but the load of measured circuit increases).
2. When the cable is lossy (nonzero R and/or G) and loaded (terminated) by pure real impedance Zl (osci 50 ohms)
reflecting only L and C parameters of cable.
There are the reflections at the the Zl and there are also reflections at the cable input if it is also not matched (like in prev. example).
Thus, the signal measured at the Zl is jammed by multiple reflections from Zl mismatch and cable input impedance mismatch.
The situation is even worse as the parameters R and G are significantly frequency dependent and the transmission line becomes dispersive,
i.e., the signal will spread out in time.
There are too limit cases:
a. The cable is not too lossy (up to some desired frequency, e.g. cable with good dielectric, short cable,...).
Then the matching of complex Z0 and pure real Zl is good enough so that the reflections coming to Zl measurement point are small
in amplitude and does not disturb the original signal too much.
b. The cable is too much lossy.
In this case, the reflections are attenuated by cable itself (imagine cable with 10 dB att -> first reflection is attenuated by 20 dB (supposed total refl. at Zin point) -> 0.1 amplitude)
and they are also hopefully marginal at the measurement point Zl.
What can be done to dump reflections coming to Zl measurement point?
A. Match Zl to complex Z0 -> hard to do as Z0 varies with frequency.
B. Use better cable.
C. Use long cable to increase reflections attenuation -> not good idea, we lost the sensitivity.
C. Match cable at the cable input to reduce reflections of reflections from Zl.