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

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.

• Where did you see it used? Can you show us the context where you saw it used? Commented Feb 9, 2015 at 23:24
• The natural frequency of a resonant circuit could be the frequency where the impedance goes to a minimum or maximum. Commented Feb 9, 2015 at 23:25
• More than likely it's the frequency which excites a circuit to exhibit the "certain impedance" specified. For example, at what frequency would 1000 nanofarads in series with 1000 ohms look like an impedance of 10000 ohms? Commented Feb 9, 2015 at 23:40

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

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

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).