What is the 'characteristic frequency' of a certain impedance?

Question

I'm not familiar with that term. Could somebody explain it to me? Is it the frequency at which the phase of the AC current running through the impedance goes to zero?

Edit: Here is the context in which it is used.

Answer 1

The impedance of a parallel connected resistor and capacitor is

$$Z = R\,||\frac{1}{j\omega C} = \frac{1}{\frac{1}{R}+ j\omega C} = R \,\frac{1}{1+j \omega RC} $$

This is of the form of a resistance times a dimensionless, frequency dependent quantity.

There is a particular frequency of interest,

$$\omega_0 = \frac{1}{RC}$$

which is the characteristic frequency. At this frequency, the impedance is

$$Z_0 = R \,\frac{1}{1+j \omega_0 RC} = R \,\frac{1}{1+j} = \frac{R}{\sqrt{2}}e^{-j\frac{\pi}{4}}$$

Answer 2

In the text, the input impedance of the scope is mentioned, which should be written in the following form: "Resistance times a dimensionless quantity". This could only be the following form:

|Zs|=|Rs||Xs|=Rs/SQRT[(1+w²Rs²Cs²)].

Then, they ask for the "characteristic frequency". For my opinion, this can only be the frequency at which we have Rs=1/wCs or w=1/RsCs.

Of course, that is the frequency where |Zs| is Zs,max/SQRT(2)=Rs/SQRT(2).

Update: The question arises WHY we define such a "characteristic frequency" in connection with the scope input. The answer is as follows:

If we set RsCs=Ts we can say that the characteristic frequency of this parallel combination (Rs||Cs) is identical to the inverse time constant Ts. Now - in order to realize a frequency-independent scope input we must use a probe having exactly the same (internal) time constant Tp=Ts. In this case (tuned probe) we have a complex voltage divider with a fixed division ratio (mostly 1:10) which is independent on the frequency. Hence, we require that the scope input has the same characteristic frequency as the probe (1/RsCs=1/RpCp).