The linked article doesn't give an adequate explanation of the operation of the scheme they are describing.
Figure 1. Typical voltmeter block diagram. Source: Radio Electronics.
The scheme of Figure 1 is much more typical of digital voltmeters.
- A sample and hold reads the applied voltage and holds it (usually on a capacitor).
- The Successive Approximation Register (SAR) outputs a 50% signal on its digital output.
- The DAC converts this to an analog value which is fed back to the comparitor. The comparitor signals to the SAR whether the input is higher or lower than 50%.
- The SAR now switches to 75% or 25% depending on the previous result.
- This process continues successively homing in on the input value.
An alternative method is to generate a "staircase" output from the DAC by feeding it with an increasing binary count. When the DAC output exceeds the input signal the count value is latched and converted to a digital reading.
Figure 2. Pulse counting. Source: Blogspot.
Your referenced article may be describing a system similar to Figure 2. A ramp signal is generated. (This will be a staircase as described in the previous paragraph.) Pulses are counted until the staircase crosses the input value. The pulse count is then converted into a meter reading.
I understand that finally we get our measure by ADC the signal that we know is proportional to the voltage.
That's fine and is explained above.
Question is how is this proportional signal produced and what relations are used.
That is also explained above. We generate a binary pattern, feed that to the ADC and compare the ADC output with the signal to be measured. Then adjust the ADC and try again.
What gets measured really, is this the current measured by the number of charges per time, and how it is measured?
Charges per unit time is current. The definition of the ampere is one coulomb per second. \$ 1\;A = 1\;C/s \$.
Your voltmeter measures potential difference which is not the same. Current is the flow but voltage is the pressure causing the flow. For resistors the two are related by Ohm's law, \$ V = IR \$. If a current is flowing between two points but there is no resistance to flow then there will be no voltage drop measurable between those two points.
Voltage, electric potential difference, electric pressure or electric tension (formally denoted ∆V or ∆U, but more often simply as V or U, for instance in the context of Ohm's or Kirchhoff's circuit laws) is the difference in electric potential energy between two points per unit electric charge. The voltage between two points is equal to the work done per unit of charge against a static electric field to move the test charge between two points. This is measured in units of volts (a joule per coulomb). Source: Wikipedia, Voltage.
One of the beauties of digital voltmeters is that their input impedance is very high - typically 1 or 10 MΩ. They steal tiny amounts of current from the circuit they are measuring.