Electromagnetics & Fundamentals

The Decibel Scale (dB)

P1: Core

The Decibel Scale (dB) in RF

RF engineering deals with the largest scales in physics—from Megawatt radar pulses ( $10^6$ W) down to GPS signals below the thermal noise floor ( $10^{-16}$ W). Linear arithmetic cannot handle this 22-order-of-magnitude range. We use Logarithms.

The Decibel Workshop

Interactive Logarithmic Scale

Slide to see how massive changes in Power (Watts) compress into manageable changes in Decibels (dBm).

Linear Power1.00 mWExponential Growth

Logarithmic Level0.00dBmLinear Growth

-140 dBm (fW)0 dBm (mW)+60 dBm (kW)

Apply Rule of Thumb:

Equivalent Voltage (at 50Ω): 223.61 mV

1. Why Logarithms?

Decibels transform multiplication into addition.

Linear: $P_{out} = P_{in} \cdot G_1 \cdot L_1 \cdot G_2 ...$ (Hard math)
Log: $P_{out(dB)} = P_{in(dB)} + G_{1(dB)} - L_{1(dB)} + ...$ (Easy math)

The Core Definition

$dB = 10 \log_{10} \left( \frac{P_1}{P_0} \right)$

Note: For Voltage, the formula is $20 \log_{10}(V/V_0)$ because Power is proportional to $V^2$ .

2. Absolute vs Relative Units

This is the most common confusion for beginners.

Unit	Type	Definition	Example
dB	Relative	A ratio of two powers (Gain/Loss).	"This amplifier has 10 dB gain."
dBm	Absolute	Power relative to 1 milliWatt.	"The Wi-Fi Tx power is 20 dBm (100mW)."
dBW	Absolute	Power relative to 1 Watt.	"The Satellite EIRP is 60 dBW."
dBi	Relative	Antenna gain relative to an isotropic sphere.	"The High-Gain antenna is 6 dBi."
dBc	Relative	Power relative to the Carrier signal.	"The harmonic is at -40 dBc."

Warning: Math Rule:

dBm + dB = dBm (Power + Gain = New Power)

dBm - dBm = dB (Power / Power = Ratio)

dBm + dBm = NONSENSE. To add two powers (e.g., two signals combining), you must convert to Watts, add, and convert back.

3. Rules of Thumb (Mental Math)

Memorize these to verify simulations instantly.

+3 dB = 2x Power
-3 dB = 1/2 Power
+10 dB = 10x Power
-10 dB = 1/10 Power
+30 dBm = 1 Watt
0 dBm = 1 milliWatt
-174 dBm/Hz = Thermal Noise Floor (at room temp).

4. System Application: Dynamic Range

Every RF system lives between two hard limits:

Noise Floor (Sensitivity): The bottom. Determined by Boltzmann's constant, Bandwidth, and Component Noise Figure. Signals below this are lost in static.
Compression Point (P1dB): The top. Where the amplifier saturates and distorts.

Dynamic Range (SFDR) is the "Headroom" between the Noise Floor and the Compression Point.

Example: Simple Link Budget

Let's track a signal from a Wi-Fi router to a phone.

Source Power: 20 dBm (100 mW)
Cable Loss: -2 dB
Tx Antenna Gain: +5 dBi
Path Loss (Air): -80 dB (Distance dependent)
Rx Antenna Gain: +0 dBi

$P_{rx} = 20 - 2 + 5 - 80 + 0 = -57 \text{ dBm}$

Is -57 dBm good?

If Receiver Sensitivity is -90 dBm, we have 33 dB of Link Margin. Excellent connection.