13.1 Decibels, Power Ratios, and Cascaded Stage Calculations

ACMA Amateur Radio Advanced Recognition Certificate - Theory Exam

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1 Chapter 1: DC Theory, Circuit Laws & Passive Components

1.1 DC Circuit Analysis, Ohm's Law, and Kirchhoff's Laws 1.2 AC Fundamentals, Sine Waves, Peak, RMS, and Phase Relationships 1.3 Inductance, Capacitance, Reactance, and Time Constants

2 Chapter 2: Impedance, Resonance & Passive Filters

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3 Chapter 3: Semiconductor Devices & Power Supplies

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4 Chapter 4: Oscillators & Frequency Synthesis

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5 Chapter 5: RF Amplifiers & Linearity

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7 Chapter 7: Transmitters, Mixers & Receivers

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13 Chapter 13: Test Equipment & RF Measurements

13.1 Decibels, Power Ratios, and Cascaded Stage Calculations 13.2 RF Power Meters, Directional Couplers, and Dummy Loads 13.3 Multirange Meters, Waveform Diagnostics, Oscilloscopes, and Frequency Counters

14 Chapter 14: Advanced RF Analysis & Diagnostics

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15 Chapter 15: Electromagnetic Compatibility & Interference

15.1 Sources of Interference, Coupling Mechanisms, and Audio Rectification 15.2 Receiver Susceptibility, Desensitisation, and Front-End Intermodulation 15.3 Practical RFI Suppression, Filter Design, and Bonding Techniques

16 Chapter 16: Electrical, RF & Station Safety

16.1 Biological Effects of RF Energy and ARPANSA Exposure Standards 16.2 Safe Separation Distances, Power Density, and Duty Cycles 16.3 Station Electrical Safety, Battery Hazards, and Safe Headphone Use 16.4 Mast and Tower Construction, Power Line Clearance, and Lightning Protection

13.1 Decibels, Power Ratios, and Cascaded Stage Calculations

Key Takeaways Explain This Topic AI

The decibel ( dB \text{dB} dB ) is a dimensionless logarithmic unit expressing the ratio between two power levels: dB = 10 log ⁡ 10 ( P 1 / P 2 ) \text{dB} = 10 \log_{10}(P_1 / P_2) dB = 10 lo g 10 ​ ( P 1 ​ / P 2 ​ ) ; for voltages and currents across identical impedances, it scales by a factor of 20: dB = 20 log ⁡ 10 ( V 1 / V 2 ) = 20 log ⁡ 10 ( I 1 / I 2 ) \text{dB} = 20 \log_{10}(V_1 / V_2) = 20 \log_{10}(I_1 / I_2) dB = 20 lo g 10 ​ ( V 1 ​ / V 2 ​ ) = 20 lo g 10 ​ ( I 1 ​ / I 2 ​ ) .

Absolute power references anchor the logarithmic scale: dBm \text{dBm} dBm references 1 mW 1\text{ mW} 1 mW ( 0 dBm = 0.2236 V rms 0\text{ dBm} = 0.2236\text{ V}_{\text{rms}} 0 dBm = 0.2236 V rms ​ across 50 Ω 50\ \Omega 50 Ω ), while dBW \text{dBW} dBW references 1 W 1\text{ W} 1 W ( 0 dBW = + 30 dBm 0\text{ dBW} = +30\text{ dBm} 0 dBW = + 30 dBm ). At 14.2 MHz 14.2\text{ MHz} 14.2 MHz , Advanced class-licence limits are 400 W 400\text{ W} 400 W PEP for C3F/J3E/R3E and 120 W 120\text{ W} 120 W mean for other emissions; 400 W 400\text{ W} 400 W is approximately + 56 dBm +56\text{ dBm} + 56 dBm .

The IARU Region 1 technical standard defines receiver signal strength S 9 S9 S 9 as an input power of − 73 dBm -73\text{ dBm} − 73 dBm ( 50 μ V rms 50\ \mu\text{V}_{\text{rms}} 50 μ V rms ​ across 50 Ω 50\ \Omega 50 Ω ) on HF bands below 30 MHz 30\text{ MHz} 30 MHz , and − 93 dBm -93\text{ dBm} − 93 dBm ( 5 μ V rms 5\ \mu\text{V}_{\text{rms}} 5 μ V rms ​ ) on VHF/UHF bands, with each S-unit representing a 6 dB 6\text{ dB} 6 dB step ( 2 × 2\times 2 × voltage, 4 × 4\times 4 × power).

Mental math shortcuts allow rapid field estimation: + 3 dB ≈ 2 × +3\text{ dB} \approx 2\times + 3 dB ≈ 2 × power, + 6 dB ≈ 4 × +6\text{ dB} \approx 4\times + 6 dB ≈ 4 × power ( 2 × 2\times 2 × voltage), + 10 dB = 10 × +10\text{ dB} = 10\times + 10 dB = 10 × power, + 20 dB = 100 × +20\text{ dB} = 100\times + 20 dB = 100 × power ( 10 × 10\times 10 × voltage), and + 30 dB = 1000 × +30\text{ dB} = 1000\times + 30 dB = 1000 × power.

Cascaded RF stages allow complex system gain, feedline losses, and antenna gains to be computed through simple algebraic addition and subtraction of decibels ( G total = G 1 − L 1 + G 2 − L 2 G_{\text{total}} = G_1 - L_1 + G_2 - L_2 G total ​ = G 1 ​ − L 1 ​ + G 2 ​ − L 2 ​ ), forming the foundation of RF link budget calculations.

Last updated: November 2026

13.1 Decibels, Power Ratios, and Cascaded Stage Calculations

In radio frequency (RF) engineering, signals span an enormous dynamic range. A sensitive HF receiver can detect an incoming signal delivering less than 0.0000000000001 Watt 0.0000000000001\text{ Watt} 0.0000000000001 Watt ( 0.1 picowatt 0.1\text{ picowatt} 0.1 picowatt or 10 − 13 W 10^{-13}\text{ W} 1 0 − 13 W ), while an amateur high-power linear amplifier can generate 400 Watts 400\text{ Watts} 400 Watts ( 4 × 10 2 W 4 \times 10^2\text{ W} 4 × 1 0 2 W ) of RF output power. Expressing and calculating such quantities using linear arithmetic requires unwieldy scientific notation and tedious multi-digit multiplications. Furthermore, human sensory perception of sound and signal strength follows a logarithmic rather than linear response (governed by the Weber-Fechner law).

To compress this vast numerical span into manageable numbers and simplify cascaded circuit calculations into basic addition and subtraction, engineers rely on logarithmic units: the Bel and the Decibel ( dB \text{dB} dB ).

• The Physics and Mathematics of Logarithmic Units
• Origin of the Bel and Decibel

The Bel (symbol B \text{B} B ) was formulated in the 1920s by Bell Telephone Laboratories to quantify signal loss along standard telephone cables, named in honour of Alexander Graham Bell. Because the Bel represents a tenfold increase in power ( 1 Bel = log ⁡ 10 ( P 1 / P 2 ) = 1 1\text{ Bel} = \log_{10}(P_1 / P_2) = 1 1 Bel = lo g 10 ​ ( P 1 ​ / P 2 ​ ) = 1 ), it proved too coarse for day-to-day electrical engineering. Engineers adopted the decibel (symbol dB \text{dB} dB ), which is one-tenth of a Bel ( 1 B = 10 dB 1\text{ B} = 10\text{ dB} 1 B = 10 dB ).

The decibel is strictly a dimensionless relative ratio comparing two power levels, P 1 P_1 P 1 ​ (the output or measured power) and P 2 P_2 P 2 ​ (the input or reference power): Ratio (dB) = 10 log ⁡ 10 ( P 1 P 2 ) \text{Ratio (dB)} = 10 \log_{10}\left(\frac{P_1}{P_2}\right) Ratio (dB) = 10 lo g 10 ​ ( P 2 ​ P 1 ​ ​ )

If P 1 > P 2 P_1 > P_2 P 1 ​ > P 2 ​ , the decibel value is positive , indicating power gain (amplification).

If P 1 < P 2 P_1 < P_2 P 1 ​ < P 2 ​ , the decibel value is negative , indicating power loss (attenuation).

If P 1 = P 2 P_1 = P_2 P 1 ​ = P 2 ​ , the ratio is unity, and dB = 10 log ⁡ 10 ( 1 ) = 0 dB \text{dB} = 10 \log_{10}(1) = 0\text{ dB} dB = 10 lo g 10 ​ ( 1 ) = 0 dB (no change).

Voltage and Current Ratios across Equal Impedances

While the decibel is fundamentally defined in terms of power, engineers frequently measure voltage ( V V V ) or current ( I I I ) rather than power directly. According to Joule's and Ohm's laws, the average power dissipated in a purely resistive load R R R is: P = V 2 R = I 2 R P = \frac{V^2}{R} = I^2 R P = R V 2 ​ = I 2 R

Substituting these power expressions into the decibel formula yields: dB = 10 log ⁡ 10 ( P 1 P 2 ) = 10 log ⁡ 10 ( V 1 2 / R 1 V 2 2 / R 2 ) \text{dB} = 10 \log_{10}\left(\frac{P_1}{P_2}\right) = 10 \log_{10}\left(\frac{V_1^2 / R_1}{V_2^2 / R_2}\right) dB = 10 lo g 10 ​ ( P 2 ​ P 1 ​ ​ ) = 10 lo g 10 ​ ( V 2 2 ​ / R 2 ​ V 1 2 ​ / R 1 ​ ​ )

When the input and output measurements are taken across identical impedances ( R 1 = R 2 = R R_1 = R_2 = R R 1 ​ = R 2 ​ = R , such as within a standard 50 Ω 50\ \Omega 50 Ω transmission system): dB = 10 log ⁡ 10 ( V 1 2 / R V 2 2 / R ) = 10 log ⁡ 10 ( V 1 2 V 2 2 ) = 10 log ⁡ 10 [ ( V 1 V 2 ) 2 ] \text{dB} = 10 \log_{10}\left(\frac{V_1^2 / R}{V_2^2 / R}\right) = 10 \log_{10}\left(\frac{V_1^2}{V_2^2}\right) = 10 \log_{10}\left[\left(\frac{V_1}{V_2}\right)^2\right] dB = 10 lo g 10 ​ ( V 2 2 ​ / R V 1 2 ​ / R ​ ) = 10 lo g 10 ​ ( V 2 2 ​ V 1 2 ​ ​ ) = 10 lo g 10 ​ [ ( V 2 ​ V 1 ​ ​ ) 2 ]

Applying the logarithmic power identity log ⁡ 10 ( x k ) = k log ⁡ 10 ( x ) \log_{10}(x^k) = k \log_{10}(x) lo g 10 ​ ( x k ) = k lo g 10 ​ ( x ) , the exponent 2 moves outside the logarithm as a multiplier: dB = 10 × 2 log ⁡ 10 ( V 1 V 2 ) = 20 log ⁡ 10 ( V 1 V 2 ) \text{dB} = 10 \times 2 \log_{10}\left(\frac{V_1}{V_2}\right) = 20 \log_{10}\left(\frac{V_1}{V_2}\right) dB = 10 × 2 lo g 10 ​ ( V 2 ​ V 1 ​ ​ ) = 20 lo g 10 ​ ( V 2 ​ V 1 ​ ​ )

Similarly, for current flowing through identical resistances ( R 1 = R 2 R_1 = R_2 R 1 ​ = R 2 ​ ): dB = 10 log ⁡ 10 ( I 1 2 R I 2 2 R ) = 20 log ⁡ 10 ( I 1 I 2 ) \text{dB} = 10 \log_{10}\left(\frac{I_1^2 R}{I_2^2 R}\right) = 20 \log_{10}\left(\frac{I_1}{I_2}\right) dB = 10 lo g 10 ​ ( I 2 2 ​ R I 1 2 ​ R ​ ) = 20 lo g 10 ​ ( I 2 ​ I 1 ​ ​ )

Important

The multiplier 10 applies strictly to power ratios : dB = 10 log ⁡ 10 ( P 1 P 2 ) \text{dB} = 10 \log_{10}\left(\frac{P_1}{P_2}\right) dB = 10 lo g 10 ​ ( P 2 ​ P 1 ​ ​ ) The multiplier 20 applies to voltage and current ratios across identical impedances : dB = 20 log ⁡ 10 ( V 1 V 2 ) = 20 log ⁡ 10 ( I 1 I 2 ) \text{dB} = 20 \log_{10}\left(\frac{V_1}{V_2}\right) = 20 \log_{10}\left(\frac{I_1}{I_2}\right) dB = 20 lo g 10 ​ ( V 2 ​ V 1 ​ ​ ) = 20 lo g 10 ​ ( I 2 ​ I 1 ​ ​ )

Caution

If the input and output impedances differ ( R 1 ≠ R 2 R_1 \ne R_2 R 1 ​  = R 2 ​ ), such as when measuring across a 600 Ω 600\ \Omega 600 Ω audio line and a 50 Ω 50\ \Omega 50 Ω RF line, the impedance ratio term cannot be ignored: dB = 20 log ⁡ 10 ( V 1 V 2 ) + 10 log ⁡ 10 ( R 2 R 1 ) \text{dB} = 20 \log_{10}\left(\frac{V_1}{V_2}\right) + 10 \log_{10}\left(\frac{R_2}{R_1}\right) dB = 20 lo g 10 ​ ( V 2 ​ V 1 ​ ​ ) + 10 lo g 10 ​ ( R 1 ​ R 2 ​ ​ ) Blindly applying 20 log ⁡ 10 ( V 1 / V 2 ) 20 \log_{10}(V_1 / V_2) 20 lo g 10 ​ ( V 1 ​ / V 2 ​ ) across mismatched impedances yields mathematically invalid power gain figures.

Absolute Power References: dBm \text{dBm} dBm and dBW \text{dBW} dBW

Because the plain decibel ( dB \text{dB} dB ) is merely a relative ratio, it cannot express an absolute power level without a stated baseline. To quantify absolute power, engineers define reference standards by fixing the denominator P 2 P_2 P 2 ​ to a known physical value. Absolute Power Reference Scales

• Watts (W): 0.001 W 0.1 W 1 W 10 W 100 W 400 W
• | | | | | |
• dBm: 0 dBm +20 dBm +30 dBm +40 dBm +50 dBm +56 dBm
• | | | | | |
• dBW: -30 dBW -10 dBW 0 dBW +10 dBW +20 dBW +26 dBW
• ^ ^
• | |
• 1 milliwatt reference 1 Watt reference
• 1. The Decibel Relative to 1 Milliwatt ( dBm \text{dBm} dBm )

The dBm \text{dBm} dBm expresses power relative to 1 milliwatt 1\text{ milliwatt} 1 milliwatt ( 1 mW = 10 − 3 W 1\text{ mW} = 10^{-3}\text{ W} 1 mW = 1 0 − 3 W ): Power (dBm) = 10 log ⁡ 10 ( P 1 mW ) = 10 log ⁡ 10 ( P 10 − 3 W ) \text{Power (dBm)} = 10 \log_{10}\left(\frac{P}{1\text{ mW}}\right) = 10 \log_{10}\left(\frac{P}{10^{-3}\text{ W}}\right) Power (dBm) = 10 lo g 10 ​ ( 1 mW P ​ ) = 10 lo g 10 ​ ( 1 0 − 3 W P ​ )

In standard 50 Ω 50\ \Omega 50 Ω RF systems, 0 dBm 0\text{ dBm} 0 dBm ( 1 mW 1\text{ mW} 1 mW ) corresponds to specific RMS voltage and current values: V rms = P × R = 0.001 W × 50 Ω = 0.05 ≈ 0.2236 V rms = 223.6 mV rms V_{\text{rms}} = \sqrt{P \times R} = \sqrt{0.001\text{ W} \times 50\ \Omega} = \sqrt{0.05} \approx 0.2236\text{ V}_{\text{rms}} = 223.6\text{ mV}_{\text{rms}} V rms ​ = P × R ​ = 0.001 W × 50 Ω ​ = 0.05 ​ ≈ 0.2236 V rms ​ = 223.6 mV rms ​ V pk = V rms × 2 ≈ 0.2236 × 1.4142 ≈ 0.3162 V pk V_{\text{pk}} = V_{\text{rms}} \times \sqrt{2} \approx 0.2236 \times 1.4142 \approx 0.3162\text{ V}_{\text{pk}} V pk ​ = V rms ​ × 2 ​ ≈ 0.2236 × 1.4142 ≈ 0.3162 V pk ​ V p-p = 2 × V pk ≈ 0.6325 V p-p V_{\text{p-p}} = 2 \times V_{\text{pk}} \approx 0.6325\text{ V}_{\text{p-p}} V p-p ​ = 2 × V pk ​ ≈ 0.6325 V p-p ​ I rms = V rms R = 0.2236 V 50 Ω ≈ 4.472 mA rms ( 6.325 mA pk ) I_{\text{rms}} = \frac{V_{\text{rms}}}{R} = \frac{0.2236\text{ V}}{50\ \Omega} \approx 4.472\text{ mA}_{\text{rms}} \quad (6.325\text{ mA}_{\text{pk}}) I rms ​ = R V rms ​ ​ = 50 Ω 0.2236 V ​ ≈ 4.472 mA rms ​ ( 6.325 mA pk ​ )

2. The Decibel Relative to 1 Watt ( dBW \text{dBW} dBW )

The dBW \text{dBW} dBW expresses power relative to 1 Watt 1\text{ Watt} 1 Watt ( 1 W 1\text{ W} 1 W ): Power (dBW) = 10 log ⁡ 10 ( P 1 W ) \text{Power (dBW)} = 10 \log_{10}\left(\frac{P}{1\text{ W}}\right) Power (dBW) = 10 lo g 10 ​ ( 1 W P ​ )

Because 1 Watt = 1000 milliwatts 1\text{ Watt} = 1000\text{ milliwatts} 1 Watt = 1000 milliwatts , and 10 log ⁡ 10 ( 1000 ) = 30 dB 10 \log_{10}(1000) = 30\text{ dB} 10 lo g 10 ​ ( 1000 ) = 30 dB , the relationship between dBm \text{dBm} dBm and dBW \text{dBW} dBW is a constant offset of 30 dB 30\text{ dB} 30 dB : Power (dBm) = Power (dBW) + 30 dB \text{Power (dBm)} = \text{Power (dBW)} + 30\text{ dB} Power (dBm) = Power (dBW) + 30 dB Power (dBW) = Power (dBm) − 30 dB \text{Power (dBW)} = \text{Power (dBm)} - 30\text{ dB} Power (dBW) = Power (dBm) − 30 dB

Standard Conversion Benchmarks

Power (Watts) Power in dBm Power in dBW RMS Voltage ( 50 Ω 50\ \Omega 50 Ω ) Typical Amateur Application

0.1 pW 0.1\text{ pW} 0.1 pW ( 10 − 13 W 10^{-13}\text{ W} 1 0 − 13 W ) − 100 dBm -100\text{ dBm} − 100 dBm − 130 dBW -130\text{ dBW} − 130 dBW 2.24 μ V 2.24\ \mu\text{V} 2.24 μ V Weak CW / digital signal on VHF

0.5 pW 0.5\text{ pW} 0.5 pW ( 5 × 10 − 13 W 5 \times 10^{-13}\text{ W} 5 × 1 0 − 13 W ) − 93 dBm -93\text{ dBm} − 93 dBm − 123 dBW -123\text{ dBW} − 123 dBW 5.0 μ V 5.0\ \mu\text{V} 5.0 μ V S9 reference on VHF/UHF ( > 30 MHz > 30\text{ MHz} > 30 MHz )

50 pW 50\text{ pW} 50 pW ( 5 × 10 − 11 W 5 \times 10^{-11}\text{ W} 5 × 1 0 − 11 W ) − 73 dBm -73\text{ dBm} − 73 dBm − 103 dBW -103\text{ dBW} − 103 dBW 50.0 μ V 50.0\ \mu\text{V} 50.0 μ V S9 reference on HF ( < 30 MHz < 30\text{ MHz} < 30 MHz )

1.0 μ W 1.0\ \mu\text{W} 1.0 μ W ( 10 − 6 W 10^{-6}\text{ W} 1 0 − 6 W ) − 30 dBm -30\text{ dBm} − 30 dBm − 60 dBW -60\text{ dBW} − 60 dBW 7.07 mV 7.07\text{ mV} 7.07 mV Example low-level RF signal

1.0 mW 1.0\text{ mW} 1.0 mW ( 10 − 3 W 10^{-3}\text{ W} 1 0 − 3 W ) 0 dBm 0\text{ dBm} 0 dBm − 30 dBW -30\text{ dBW} − 30 dBW 0.224 V 0.224\text{ V} 0.224 V Signal generator test output baseline

10 mW 10\text{ mW} 10 mW ( 0.01 W 0.01\text{ W} 0.01 W ) + 10 dBm +10\text{ dBm} + 10 dBm − 20 dBW -20\text{ dBW} − 20 dBW 0.707 V 0.707\text{ V} 0.707 V Local oscillator injection level

100 mW 100\text{ mW} 100 mW ( 0.1 W 0.1\text{ W} 0.1 W ) + 20 dBm +20\text{ dBm} + 20 dBm − 10 dBW -10\text{ dBW} − 10 dBW 2.24 V 2.24\text{ V} 2.24 V QRPp transceiver output stage

1.0 W 1.0\text{ W} 1.0 W + 30 dBm +30\text{ dBm} + 30 dBm 0 dBW 0\text{ dBW} 0 dBW 7.07 V 7.07\text{ V} 7.07 V Handheld VHF/UHF low-power mode

5.0 W 5.0\text{ W} 5.0 W + 37 dBm +37\text{ dBm} + 37 dBm + 7 dBW +7\text{ dBW} + 7 dBW 15.81 V 15.81\text{ V} 15.81 V Standard handheld VHF/UHF full power / QRP

10 W 10\text{ W} 10 W + 40 dBm +40\text{ dBm} + 40 dBm + 10 dBW +10\text{ dBW} + 10 dBW 22.36 V 22.36\text{ V} 22.36 V Intermediate driver stage

25 W 25\text{ W} 25 W + 44 dBm +44\text{ dBm} + 44 dBm + 14 dBW +14\text{ dBW} + 14 dBW 35.36 V 35.36\text{ V} 35.36 V Mobile VHF/UHF transceiver medium power

100 W 100\text{ W} 100 W + 50 dBm +50\text{ dBm} + 50 dBm + 20 dBW +20\text{ dBW} + 20 dBW 70.71 V 70.71\text{ V} 70.71 V Standard HF base transceiver full power

400 W 400\text{ W} 400 W + 56 dBm +56\text{ dBm} + 56 dBm + 26 dBW +26\text{ dBW} + 26 dBW 141.42 V 141.42\text{ V} 141.42 V Advanced SSB PEP limit at 14.2 MHz 14.2\text{ MHz} 14.2 MHz

1000 W 1000\text{ W} 1000 W ( 1 kW 1\text{ kW} 1 kW ) + 60 dBm +60\text{ dBm} + 60 dBm + 30 dBW +30\text{ dBW} + 30 dBW 223.61 V 223.61\text{ V} 223.61 V High-power amateur linear amplifier

Receiver Microvolt Levels and Calibrated S-Meters

Receiver signal strength meters (S-meters) are calibrated in logarithmic units standardized by the International Amateur Radio Union (IARU) Region 1 technical recommendation:

High Frequency (HF, below 30 MHz 30\text{ MHz} 30 MHz ):

An S 9 S9 S 9 indication corresponds to an input terminal voltage of 50 μ V rms 50\ \mu\text{V}_{\text{rms}} 50 μ V rms ​ into a matched 50 Ω 50\ \Omega 50 Ω load.

Computing the equivalent power: P = V 2 R = ( 50 × 10 − 6 V ) 2 50 Ω = 2.5 × 10 − 9 50 = 5.0 × 10 − 11 W = 5.0 × 10 − 8 mW P = \frac{V^2}{R} = \frac{(50 \times 10^{-6}\text{ V})^2}{50\ \Omega} = \frac{2.5 \times 10^{-9}}{50} = 5.0 \times 10^{-11}\text{ W} = 5.0 \times 10^{-8}\text{ mW} P = R V 2 ​ = 50 Ω ( 50 × 1 0 − 6 V ) 2 ​ = 50 2.5 × 1 0 − 9 ​ = 5.0 × 1 0 − 11 W = 5.0 × 1 0 − 8 mW Power (dBm) = 10 log ⁡ 10 ( 5.0 × 10 − 8 ) = − 73.01 dBm ≈ − 73 dBm \text{Power (dBm)} = 10 \log_{10}(5.0 \times 10^{-8}) = -73.01\text{ dBm} \approx -73\text{ dBm} Power (dBm) = 10 lo g 10 ​ ( 5.0 × 1 0 − 8 ) = − 73.01 dBm ≈ − 73 dBm

Very High Frequency / Ultra High Frequency (VHF/UHF, above 30 MHz 30\text{ MHz} 30 MHz ):

Galactic and atmospheric noise levels are substantially lower at VHF/UHF than at HF. The IARU standard specifies S 9 S9 S 9 as 5 μ V rms 5\ \mu\text{V}_{\text{rms}} 5 μ V rms ​ into 50 Ω 50\ \Omega 50 Ω .

Because voltage is scaled by 1 / 10 1/10 1/10 , power decreases by a factor of 100 ( 20 dB 20\text{ dB} 20 dB ): Power (dBm) = − 73 dBm − 20 dB = − 93 dBm \text{Power (dBm)} = -73\text{ dBm} - 20\text{ dB} = -93\text{ dBm} Power (dBm) = − 73 dBm − 20 dB = − 93 dBm

The S-Unit Progression:

Each single S-unit increment (from S 8 S8 S 8 to S 9 S9 S 9 , or S 5 S5 S 5 to S 6 S6 S 6 ) represents a 6 dB 6\text{ dB} 6 dB change in signal level.

A 6 dB 6\text{ dB} 6 dB change represents a 2 : 1 2:1 2 : 1 ratio in voltage ( 20 log ⁡ 10 ( 2 ) ≈ 6.02 dB 20 \log_{10}(2) \approx 6.02\text{ dB} 20 lo g 10 ​ ( 2 ) ≈ 6.02 dB ) and a 4 : 1 4:1 4 : 1 ratio in power ( 10 log ⁡ 10 ( 4 ) ≈ 6.02 dB 10 \log_{10}(4) \approx 6.02\text{ dB} 10 lo g 10 ​ ( 4 ) ≈ 6.02 dB ).

Above S 9 S9 S 9 , the scale is calibrated directly in decibels: S 9 + 10 dB , S 9 + 20 dB , S 9 + 40 dB S9+10\text{ dB}, S9+20\text{ dB}, S9+40\text{ dB} S 9 + 10 dB , S 9 + 20 dB , S 9 + 40 dB . IARU HF S-Meter Scale (f < 30 MHz, 50 Ohm)

• S-Unit: S1 S3 S5 S7 S8 S9 S9+10dB S9+20dB
• | | | | | | | |
• Power: -121 -109 -97 -85 -79 -73 -63 -53 (dBm)
• Voltage: 0.2 0.8 3.1 12.5 25 50 158 500 (uV rms)
• <------------------ 6 dB per S-unit ------------------>
• Rules of Thumb and Mental Math Shortcuts

In field operations and technical examinations, rapid decibel estimation without an electronic calculator is an indispensable skill. All power and voltage ratios can be broken down using the Rule of 3s and 10s :

Fundamental Mental Math Building Blocks

+ 3 dB ≈ 2 × +3\text{ dB} \approx 2\times + 3 dB ≈ 2 × power ( 1.414 × 1.414\times 1.414 × voltage); − 3 dB ≈ 0.5 × -3\text{ dB} \approx 0.5\times − 3 dB ≈ 0.5 × power ( 50 % 50\% 50% power, 0.707 × 0.707\times 0.707 × voltage, the half-power point).

+ 6 dB ≈ 4 × +6\text{ dB} \approx 4\times + 6 dB ≈ 4 × power ( 2 × 2\times 2 × voltage); − 6 dB ≈ 0.25 × -6\text{ dB} \approx 0.25\times − 6 dB ≈ 0.25 × power ( 25 % 25\% 25% power, 0.5 × 0.5\times 0.5 × voltage).

+ 10 dB = 10 × +10\text{ dB} = 10\times + 10 dB = 10 × power ( 3.162 × 3.162\times 3.162 × voltage); − 10 dB = 0.1 × -10\text{ dB} = 0.1\times − 10 dB = 0.1 × power ( 0.316 × 0.316\times 0.316 × voltage).

+ 20 dB = 100 × +20\text{ dB} = 100\times + 20 dB = 100 × power ( 10 × 10\times 10 × voltage); − 20 dB = 0.01 × -20\text{ dB} = 0.01\times − 20 dB = 0.01 × power ( 1 % 1\% 1% power, 0.1 × 0.1\times 0.1 × voltage).

+ 30 dB = 1000 × +30\text{ dB} = 1000\times + 30 dB = 1000 × power ( 31.62 × 31.62\times 31.62 × voltage); − 30 dB = 0.001 × -30\text{ dB} = 0.001\times − 30 dB = 0.001 × power ( 0.1 % 0.1\% 0.1% power).

The Additive Decomposition Technique

Because decibels are logarithms, adding decibels corresponds to multiplying linear power factors , while subtracting decibels corresponds to dividing : Total Ratio = 10 dB 1 + dB 2 + … 10 = Ratio 1 × Ratio 2 × … \text{Total Ratio} = 10^{\frac{\text{dB}_1 + \text{dB}_2 + \dots}{10}} = \text{Ratio}_1 \times \text{Ratio}_2 \times \dots Total Ratio = 1 0 10 dB 1 ​ + dB 2 ​ + … ​ = Ratio 1 ​ × Ratio 2 ​ × …

Any arbitrary decibel value can be factored into sums and differences of 10 dB 10\text{ dB} 10 dB and 3 dB 3\text{ dB} 3 dB blocks:

Example A: Determine the power gain of + 16 dB +16\text{ dB} + 16 dB : + 16 dB = + 10 dB + 3 dB + 3 dB +16\text{ dB} = +10\text{ dB} + 3\text{ dB} + 3\text{ dB} + 16 dB = + 10 dB + 3 dB + 3 dB Power Multiplier = 10 × 2 × 2 = 40 × \text{Power Multiplier} = 10 \times 2 \times 2 = 40\times Power Multiplier = 10 × 2 × 2 = 40 ×

Example B: Determine the power gain of + 27 dB +27\text{ dB} + 27 dB : + 27 dB = + 10 dB + 10 dB + 10 dB − 3 dB +27\text{ dB} = +10\text{ dB} + 10\text{ dB} + 10\text{ dB} - 3\text{ dB} + 27 dB = + 10 dB + 10 dB + 10 dB − 3 dB Power Multiplier = 10 × 10 × 10 ÷ 2 = 1000 ÷ 2 = 500 × \text{Power Multiplier} = 10 \times 10 \times 10 \div 2 = 1000 \div 2 = 500\times Power Multiplier = 10 × 10 × 10 ÷ 2 = 1000 ÷ 2 = 500 ×

Example C: Estimate the power corresponding to + 56 dBm +56\text{ dBm} + 56 dBm : + 56 dBm = + 30 dBm ( 1 W ) + 20 dB + 3 dB + 3 dB +56\text{ dBm} = +30\text{ dBm} (1\text{ W}) + 20\text{ dB} + 3\text{ dB} + 3\text{ dB} + 56 dBm = + 30 dBm ( 1 W ) + 20 dB + 3 dB + 3 dB Power = 1 W × 100 × 2 × 2 = 400 Watts \text{Power} = 1\text{ W} \times 100 \times 2 \times 2 = 400\text{ Watts} Power = 1 W × 100 × 2 × 2 = 400 Watts

Cascaded RF Stage Gain and Loss Analysis

An RF system consists of multiple stages connected in series: signal generators, filters, preamplifiers, power amplifiers, attenuators, transmission lines, and antennas. Cascaded RF System Architecture

• +-------------+ +------------+ +------------+ +-----------+
• | Transceiver | | Coax Cable | | Power Amp | | Feedline | |--+ Antenna
• | Output P_in |--->| Attenuation|--->| Gain |--->| Loss |--->| | Gain
• | (+10 dBm) | | (-1.5 dB) | | (+20 dB) | | (-2.5 dB) | |--+ (+11 dBi)
• +-------------+ +------------+ +------------+ +-----------+

In linear units, first calculate the conducted power reaching the antenna, then apply its isotropic gain to obtain EIRP: P ant = P in g amp l coax l feed , EIRP = P ant g ant P_{\text{ant}} = \frac{P_{\text{in}}g_{\text{amp}}}{l_{\text{coax}}l_{\text{feed}}}, \qquad \text{EIRP} = P_{\text{ant}}g_{\text{ant}} P ant ​ = l coax ​ l feed ​ P in ​ g amp ​ ​ , EIRP = P ant ​ g ant ​

Here g g g and l l l are linear gain and loss factors. In decibels, the diagram gives: P ant (dBm) = 10 − 1.5 + 20 − 2.5 = 26 dBm , EIRP (dBm) = 26 + 11 = 37 dBm P_{\text{ant}}\text{ (dBm)} = 10 - 1.5 + 20 - 2.5 = 26\text{ dBm}, \qquad \text{EIRP}\text{ (dBm)} = 26 + 11 = 37\text{ dBm} P ant ​ (dBm) = 10 − 1.5 + 20 − 2.5 = 26 dBm , EIRP (dBm) = 26 + 11 = 37 dBm

The antenna therefore receives about 0.40 W 0.40\text{ W} 0.40 W ; its EIRP in the specified direction is about 5.0 W 5.0\text{ W} 5.0 W .

Complete Link Budget Analysis

A link budget accounts for all gains and losses from the transmitter power amplifier through the propagation medium to the receiver demodulator detector. The received power ( P rx P_{\text{rx}} P rx ​ ) is formulated as: P rx = P tx − L tx_line + G tx_ant − L path + G rx_ant − L rx_line P_{\text{rx}} = P_{\text{tx}} - L_{\text{tx\_line}} + G_{\text{tx\_ant}} - L_{\text{path}} + G_{\text{rx\_ant}} - L_{\text{rx\_line}} P rx ​ = P tx ​ − L tx_line ​ + G tx_ant ​ − L path ​ + G rx_ant ​ − L rx_line ​

where:

P tx P_{\text{tx}} P tx ​ is transmitter output power in dBm \text{dBm} dBm (or dBW \text{dBW} dBW ),

L tx_line L_{\text{tx\_line}} L tx_line ​ is the transmit feedline and connector attenuation in dB \text{dB} dB ,

G tx_ant G_{\text{tx\_ant}} G tx_ant ​ is the transmit antenna forward gain in dBi \text{dBi} dBi (gain relative to an isotropic radiator),

EIRP = P tx − L tx_line + G tx_ant \text{EIRP} = P_{\text{tx}} - L_{\text{tx\_line}} + G_{\text{tx\_ant}} EIRP = P tx ​ − L tx_line ​ + G tx_ant ​ is the Effective Isotropic Radiated Power ,

• L path L_{\text{path}} L path ​ is the path loss through space in dB \text{dB} dB ,
• G rx_ant G_{\text{rx\_ant}} G rx_ant ​ is the receiving antenna gain in dBi \text{dBi} dBi ,

L rx_line L_{\text{rx\_line}} L rx_line ​ is the receive feedline attenuation in dB \text{dB} dB .

The operating fade margin (safety buffer against fading, rain, or multipath) is: Fade Margin (dB) = P rx − P receiver_sensitivity \text{Fade Margin (dB)} = P_{\text{rx}} - P_{\text{receiver\_sensitivity}} Fade Margin (dB) = P rx ​ − P receiver_sensitivity ​

Step-by-Step Worked Example: VHF Point-to-Point Link Budget

An amateur radio club designs a 144.20 MHz 144.20\text{ MHz} 144.20 MHz (2-meter) point-to-point link across a distance of 45 km 45\text{ km} 45 km . The system parameters are:

Transmitter output power: P tx = 50 Watts P_{\text{tx}} = 50\text{ Watts} P tx ​ = 50 Watts .

Transmit station feedline: 30 metres 30\text{ metres} 30 metres of RG-213 coaxial cable, having an attenuation of 2.8 dB 2.8\text{ dB} 2.8 dB per 30 m 30\text{ m} 30 m .

Transmit antenna: 9-element Yagi with forward gain G tx = + 13.1 dBi G_{\text{tx}} = +13.1\text{ dBi} G tx ​ = + 13.1 dBi .

Free-space path loss ( L path L_{\text{path}} L path ​ ) over 45 km 45\text{ km} 45 km at 144.2 MHz 144.2\text{ MHz} 144.2 MHz is calculated as 108.7 dB 108.7\text{ dB} 108.7 dB .

Receiving antenna: 5-element Yagi with forward gain G rx = + 9.0 dBi G_{\text{rx}} = +9.0\text{ dBi} G rx ​ = + 9.0 dBi .

Receive station feedline: 15 metres 15\text{ metres} 15 metres of low-loss foam coax with L rx = 1.1 dB L_{\text{rx}} = 1.1\text{ dB} L rx ​ = 1.1 dB .

Receiver sensitivity threshold for 12 dB 12\text{ dB} 12 dB SINAD: − 124 dBm -124\text{ dBm} − 124 dBm ( 0.14 μ V 0.14\ \mu\text{V} 0.14 μ V ). Calculate:

Transmitter power in dBm \text{dBm} dBm .

Effective Isotropic Radiated Power ( EIRP \text{EIRP} EIRP ).

Total received power ( P rx P_{\text{rx}} P rx ​ ) in dBm \text{dBm} dBm .

The link fade margin in dB \text{dB} dB .

The equivalent S-meter reading on the VHF receiver.

Step 1: Convert transmitter power to dBm \text{dBm} dBm : P tx (dBm) = 10 log ⁡ 10 ( 50 W 0.001 W ) = 10 log ⁡ 10 ( 50000 ) = 10 × 4.6990 ≈ + 47.0 dBm P_{\text{tx}}\text{ (dBm)} = 10 \log_{10}\left(\frac{50\text{ W}}{0.001\text{ W}}\right) = 10 \log_{10}(50000) = 10 \times 4.6990 \approx +47.0\text{ dBm} P tx ​ (dBm) = 10 lo g 10 ​ ( 0.001 W 50 W ​ ) = 10 lo g 10 ​ ( 50000 ) = 10 × 4.6990 ≈ + 47.0 dBm

Step 2: Calculate Effective Isotropic Radiated Power ( EIRP \text{EIRP} EIRP ): EIRP = P tx − L tx + G tx = + 47.0 dBm − 2.8 dB + 13.1 dBi = + 57.3 dBm \text{EIRP} = P_{\text{tx}} - L_{\text{tx}} + G_{\text{tx}} = +47.0\text{ dBm} - 2.8\text{ dB} + 13.1\text{ dBi} = +57.3\text{ dBm} EIRP = P tx ​ − L tx ​ + G tx ​ = + 47.0 dBm − 2.8 dB + 13.1 dBi = + 57.3 dBm

( + 57.3 dBm +57.3\text{ dBm} + 57.3 dBm is equivalent to 537 Watts 537\text{ Watts} 537 Watts of omnidirectional isotropic radiation).

Step 3: Calculate received signal power ( P rx P_{\text{rx}} P rx ​ ): P rx = EIRP − L path + G rx − L rx P_{\text{rx}} = \text{EIRP} - L_{\text{path}} + G_{\text{rx}} - L_{\text{rx}} P rx ​ = EIRP − L path ​ + G rx ​ − L rx ​ P rx = + 57.3 dBm − 108.7 dB + 9.0 dBi − 1.1 dB P_{\text{rx}} = +57.3\text{ dBm} - 108.7\text{ dB} + 9.0\text{ dBi} - 1.1\text{ dB} P rx ​ = + 57.3 dBm − 108.7 dB + 9.0 dBi − 1.1 dB P rx = − 51.4 dBm + 7.9 dB = − 43.5 dBm P_{\text{rx}} = -51.4\text{ dBm} + 7.9\text{ dB} = -43.5\text{ dBm} P rx ​ = − 51.4 dBm + 7.9 dB = − 43.5 dBm

Step 4: Calculate the link fade margin: Margin = P rx − P sens = − 43.5 dBm − ( − 124.0 dBm ) = + 80.5 dB \text{Margin} = P_{\text{rx}} - P_{\text{sens}} = -43.5\text{ dBm} - (-124.0\text{ dBm}) = +80.5\text{ dB} Margin = P rx ​ − P sens ​ = − 43.5 dBm − ( − 124.0 dBm ) = + 80.5 dB

An 80.5 dB 80.5\text{ dB} 80.5 dB margin ensures exceptionally robust, uninterrupted communication.

Step 5: Determine the equivalent S-meter reading:

On VHF ( > 30 MHz > 30\text{ MHz} > 30 MHz ), S 9 = − 93 dBm S9 = -93\text{ dBm} S 9 = − 93 dBm .

Signal level above S 9 S9 S 9 : Δ P = P rx − ( − 93 dBm ) = − 43.5 dBm − ( − 93.0 dBm ) = + 49.5 dB \Delta P = P_{\text{rx}} - (-93\text{ dBm}) = -43.5\text{ dBm} - (-93.0\text{ dBm}) = +49.5\text{ dB} Δ P = P rx ​ − ( − 93 dBm ) = − 43.5 dBm − ( − 93.0 dBm ) = + 49.5 dB

The receiving station will register an overwhelming signal of S 9 + 50 dB S9+50\text{ dB} S 9 + 50 dB .