Friis Formula

Assuming you connect the power source with power of (Pt) to an transmit antenna with gain of Gt. What will the power be received (Pr) at a receive antenna with a gain of Gr?

This is a very important question when designing wireless systems. Following the concepts of the energy conservation and the gain of antenna, the Friis formula provide good estimation of the receive power. Friis formula is expressed in (1)

(1)

Where r is the distance between the transmit and receive antenna and is the wavelength of the RF signal.
It is obvious that the power is inversely proportional to the square of the distance r. And it is also noticed that the power is directly proportional the the square of the wavelength. Therefore, the receive power lower when the frequency is higher.

Caluculation of S-parameters

S-parameters is defined to express circuits with incident wave and reflect wave instead of voltage (v) and current (i)

Considering a multi-port network, the calculation of S-parameters are classified as transverse coefficient () and reflection coefficient ()

The calculation of the transverse coefficient is shown below:

* Transverse coefficient

Imaging when we performing the experiment to measure the S21 of a 2-port device (for example an attenuator.)

At first step, we get a signal generator (SG) which is constructed of a voltage source (V) with a output impedance of Zo. Then, connect the signal generator to a power meter with input impedance of Zo. Consequently, the power from the source (Pavs) can be measured.

Fig. 1 First step of measuring the transverse parameters (find Pavs).

The second step is to insert the DUT between the signal generator and power meter. The power meter can measure the output power with the insertion of DUT (Pout). Than the transverse coefficient can be calculated as Pout/Pavs.

Fig. 2 Second step of measuring the transverse parameters (find Pout).

These two steps are easily to be understand. But how to calculate?
Let us see Fig. 3. In Fig. 3, it is observed that the voltage at point A (V1) is equal to since the output of SG is terminated to a perfect match load.
Similarly, the voltage at point B (V2) is equal to
Then the S21 can be calculated as V2/V1.

Fig. 3 The calculation of the transverse coefficient.

From the calculation steps, we can find that the calculation is similar to the measurement. It is notice that although S21 is calculated as voltage ratio. But S21 is not equal to the voltage gain since S21 is the voltage ratio between measuring Pout and Pavs.

The useage of dB

Giving a power (P), the definition of dB is:



Why shall we making the definition like this?
Let us exam some example:









From the example above, we can found that dB is very close to the scientific notation.
The coefficient part of the scientific notation is refer to the unit of dB.
The exponent part of the scientific notation is refer to the number of dB above tens.

Therefore, the number covers different scale can be expressed simply in scientific notation as well as dB