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1.3 Power, Energy, and Resistance Calculations

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ACMA Amateur Radio Standard Recognition Certificate - Theory Exam

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1 Chapter 1: Electrical Principles and Components

1.1 Mathematics, Units and Metric Prefixes 1.2 DC Circuits, Voltage, Current, and Ohm's Law 1.3 Power, Energy, and Resistance Calculations 1.4 Resistor Colour Code and Component Symbol Identification 1.5 Capacitors: Construction, Ratings and Safety 1.6 Inductors: Construction, Factors Affecting Inductance and Combinations 1.7 Capacitor and Inductor Combination Calculations 1.8 AC Circuits, Frequency, Wavelength, and RMS Values 1.9 Reactance and Impedance in AC Circuits 1.10 Resonance, Q Factor and Tuned Circuits 1.11 Transformers: Turns Ratio, Impedance Matching and Eddy Currents 1.12 Rectifiers, PIV and Power Supply Smoothing 1.13 Active Components: Semiconductors, Diodes, and Transistors 1.14 Transistor and FET Symbols, Configurations and External Behaviour

2 Chapter 2: Transmitters and Receivers

2.1 Superheterodyne Receiver Architecture and Block Diagrams 2.2 Frequency Converters, IF Amplifiers and Transceiver Sharing 2.3 Receiver Performance: Sensitivity, Selectivity, and AGC 2.4 Signal-to-Noise Ratio and Receiver Noise Performance 2.5 Transmitter Architecture, Oscillators and Frequency Synthesisers 2.6 Transmission Quality: Frequency Drift, Parasitic Oscillation and Over-Modulation 2.7 Modulation Techniques: AM, FM, and SSB 2.8 Amplifier Classes and Power Amplifiers (Class A, B, C, and AB) 2.9 Modulation, Duty Cycle and Rated Transmitter Power 2.10 Transceiver Control: ALC, Squelch, and Digital Modes 2.11 Digital Transmission Modes and Bandwidth

3 Chapter 3: Transmission Lines and Antennas

3.1 Transmission Lines: Coaxial, Twin-Lead, and Characteristic Impedance 3.2 Standing Wave Ratio (SWR) and Reflection Loss 3.3 Impedance Matching, Baluns, and Antenna Tuning Units (ATUs) 3.4 Resonant Antennas: Dipoles, Verticals, and End-Fed Antennas 3.5 Angle of Radiation, Antenna Height and Current Distribution 3.6 Directional Antennas: Yagis, Gain and Radiation Patterns 3.7 Effective Radiated Power: System Gains and Losses

4 Chapter 4: Radio Wave Propagation

4.1 Electromagnetic Waves: Velocity, Wavelength and Polarisation 4.2 Ground Wave, Direct Wave and Line-of-Sight Propagation 4.3 Ionospheric Layers (D, E, F1, F2) and Skywave Propagation 4.4 Critical Frequency, Maximum Usable Frequency (MUF), and Fading 4.5 Solar Activity, Sunspot Cycles, and VHF Anomalous Propagation 4.6 Seasonal and Diurnal Effects on Band Selection

5 Chapter 5: Electrical and RF Measurements

5.1 Multimeter Operation: Voltage, Current, and Resistance Measurements 5.2 RF Power and SWR Measurement (PEP vs. Average Power) 5.3 Frequency Counters, Dummy Loads, and Test Equipment 5.4 Frequency Standards, Crystal Calibrators and Standard-Frequency Transmissions

6 Chapter 6: Interference and Electromagnetic Compatibility (EMC)

6.1 Harmonic Emissions, Spurious Signals, and Transmitter Filtering 6.2 How Interference Enters Television and Broadcast Receivers 6.3 Masthead Amplifier Overload and Audio-Stage Demodulation 6.4 Electromagnetic Interference (EMI), Breakthrough, Ferrites, and EMC 6.5 Filter Types, Response Curves and EMC Filter Selection 6.6 EMC Good Practice: Minimum Power, Balanced Antennas and Feedline Routing 6.7 Vehicle EMC and Resolving Community Interference Complaints

7 Chapter 7: Safety and Environmental Hazards

7.1 Mains Voltage Safety, Earthing, and High-Voltage Capacitors 7.2 Equipment Approval, Mains Wiring and Fuse Practice 7.3 Station Layout, Cabling and Emergency Shut-Off 7.4 Lightning Protection and Station Security 7.5 Electric Shock: Prevention, Rescue and Emergency Response 7.6 Battery, Chemical and Hearing Safety in the Amateur Station 7.7 RF Exposure: EMR Hazards and Safe Distance 7.8 Antenna Erection, Siting and Power Line Safety

1.3 Power, Energy, and Resistance Calculations

Key Takeaways Explain This Topic AI

Electrical power (P) is the rate of energy transfer, measured in Watts (W): P = V × I = I² × R = V² / R.

Electrical energy (E) is power multiplied by time (E = P × t), expressed in Watt-hours (Wh) or Joules (1 Wh = 3600 J).

Power source internal resistance (r) causes terminal voltage under load to drop according to V_load = V_open_circuit - (I × r).

The Maximum Power Transfer Theorem proves that maximum power is delivered to a load when load resistance equals source internal resistance (R_L = r), yielding 50% electrical efficiency.

Decibels express power and voltage ratios logarithmically: dB = 10 log10(P2 / P1) for power, and dB = 20 log10(V2 / V1) for voltage in equal-impedance circuits.

Last updated: July 2026

1.3 Power, Energy, and Resistance Calculations

In amateur radio engineering, calculating electrical power and signal level changes is essential. Radio transmitters convert DC electrical power into high-frequency RF power, while receivers amplify microvolt-level signals over many orders of magnitude. Master candidate proficiency in power dissipation, internal battery resistance, decibel ratios, and absolute power units (dBm, dBW) is directly assessed on the ACMA Standard Theory examination.

  • 1. Electrical Power and Energy Fundamentals
  • Electrical Power ( P P P )

Electrical Power ( P P P ) is defined as the rate at which electrical energy is transferred, converted, or dissipated per unit of time. The SI unit of power is the Watt (W) , named after James Watt. One Watt represents an energy transfer rate of one Joule per second ( 1 W = 1 J/s 1\text{ W} = 1\text{ J/s} 1 W = 1 J/s ).

By combining the basic definition of power ( P = V × I P = V \times I P = V × I ) with Ohm's Law ( V = I × R V = I \times R V = I × R and I = V / R I = V / R I = V / R ), three equivalent power formulas are derived: Power Formulas: P = V × I ∣ P = I 2 × R ∣ P = V 2 R \text{Power Formulas: } \quad P = V \times I \quad \vert \quad P = I^2 \times R \quad \vert \quad P = \frac{V^2}{R} Power Formulas: P = V × I ∣ P = I 2 × R ∣ P = R V 2 ​

Where:

  • P P P is power in Watts (W)
  • V V V is potential difference in Volts (V)
  • I I I is current in Amperes (A)
  • R R R is resistance in Ohms ( Ω \Omega Ω )
  • Electrical Energy ( E E E )
  • Energy ( E E E ) measures total work performed over a given duration: E = P × t E = P \times t E = P × t

While the SI unit of energy is the Joule (J) , electrical energy is routinely measured in Watt-hours (Wh) or Kilowatt-hours (kWh) :

1 Watt-hour (Wh) = 1 W × 3600 s = 3 , 600 Joules (J) 1\text{ Watt-hour (Wh)} = 1\text{ W} \times 3600\text{ s} = 3,600\text{ Joules (J)} 1 Watt-hour (Wh) = 1 W × 3600 s = 3 , 600 Joules (J)

1 Kilowatt-hour (kWh) = 1 , 000 Wh = 3.6 × 10 6 Joules 1\text{ Kilowatt-hour (kWh)} = 1,000\text{ Wh} = 3.6 \times 10^6\text{ Joules} 1 Kilowatt-hour (kWh) = 1 , 000 Wh = 3.6 × 1 0 6 Joules

Worked Example 1.3: Battery Operating Capacity

A 12 V 12\text{ V} 12 V , 20 Ampere-hour (Ah) 20\text{ Ampere-hour (Ah)} 20 Ampere-hour (Ah) lithium iron phosphate ( LiFePO 4 \text{LiFePO}_4 LiFePO 4 ​ ) battery powers a portable 50W VHF transmitter. On transmit, the radio draws 9.0 A 9.0\text{ A} 9.0 A from the battery. Calculate:

Total energy stored in the battery in Watt-hours.

Theoretical continuous transmit operating time in hours.

Solution: E = V × Q = 12 V × 20 Ah = 240 Wh E = V \times Q = 12\text{ V} \times 20\text{ Ah} = 240\text{ Wh} E = V × Q = 12 V × 20 Ah = 240 Wh Transmit Power Drawn P = 12 V × 9.0 A = 108 W \text{Transmit Power Drawn } P = 12\text{ V} \times 9.0\text{ A} = 108\text{ W} Transmit Power Drawn P = 12 V × 9.0 A = 108 W Operating Time t = E P = 240 Wh 108 W ≈ 2.22 hours (2 hours 13 minutes) \text{Operating Time } t = \frac{E}{P} = \frac{240\text{ Wh}}{108\text{ W}} \approx 2.22\text{ hours (2 hours 13 minutes)} Operating Time t = P E ​ = 108 W 240 Wh ​ ≈ 2.22 hours (2 hours 13 minutes)

2. Resistor Power Ratings and Thermal Dissipation

Every physical resistor converts electrical energy into heat energy. Resistors carry a maximum power rating (e.g., 0.25 W 0.25\text{ W} 0.25 W , 0.5 W 0.5\text{ W} 0.5 W , 2 W 2\text{ W} 2 W , 50 W 50\text{ W} 50 W ) indicating the maximum power the device can continuously dissipate without structural degradation, resistance value shift, or fire risk.

When designing or repairing amateur radio equipment, designers must select resistors with a power rating significantly higher than the calculated actual dissipation—typically applying a safety factor of 2 × 2\times 2 × (derating).

If a 100 Ω 100\ \Omega 100 Ω resistor carries a current of 0.1 A 0.1\text{ A} 0.1 A , the power dissipated is: P = I 2 × R = ( 0.1 ) 2 × 100 = 0.01 × 100 = 1.0 Watt P = I^2 \times R = (0.1)^2 \times 100 = 0.01 \times 100 = 1.0\text{ Watt} P = I 2 × R = ( 0.1 ) 2 × 100 = 0.01 × 100 = 1.0 Watt

In this case, a 2 W 2\text{ W} 2 W or 5 W 5\text{ W} 5 W rated power resistor must be used.

3. Internal Resistance of Power Sources & Maximum Power Transfer

Real-world voltage sources (batteries, power supplies, solar panels) are not ideal. They possess an internal resistance ( r r r ) in series with their ideal electromotive force ( E emf E_{\text{emf}} E emf ​ ). +---[ Ideal Source E ]---+---[ Internal Resistance r ]---+--- ( Positive Terminal )

  • |
  • [ Load R_L ]
  • |
  • --------------------------------------------------------+--- ( Negative Terminal )

When current I I I is drawn by an external load R L R_L R L ​ , a voltage drop occurs across the internal resistance: V load = E emf − ( I × r ) V_{\text{load}} = E_{\text{emf}} - (I \times r) V load ​ = E emf ​ − ( I × r )

  • Where: I = E emf r + R L I = \frac{E_{\text{emf}}}{r + R_L} I = r + R L ​ E emf ​ ​
  • Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem states that maximum power is transferred from a power source to an external load when the load resistance equals the internal resistance of the source ( R L = r R_L = r R L ​ = r ).

Important implications for ACMA exams:

When R L = r R_L = r R L ​ = r , power transferred to the load is at its maximum absolute value.

At maximum power transfer, electrical efficiency is exactly 50% , as equal power is dissipated inside the source ( I 2 r I^2 r I 2 r ) and inside the load ( I 2 R L I^2 R_L I 2 R L ​ ).

In RF antenna systems, matching generator output impedance to transmission line and load impedance (e.g., 50 Ω 50\ \Omega 50 Ω ) maximises RF energy transfer.

4. Decibel (dB) Calculations

The Decibel (dB) is a logarithmic unit used to express ratios of power, voltage, or current. Logarithmic scales are essential in radio communications because signal power ranges from fractions of a picowatt at receiver antennas to kilowatts at transmitter output stages.

Power Ratio Formula Gain/Loss in dB = 10 log ⁡ 10 ( P 2 P 1 ) \text{Gain/Loss in dB} = 10 \log_{10} \left( \frac{P_2}{P_1} \right) Gain/Loss in dB = 10 lo g 10 ​ ( P 1 ​ P 2 ​ ​ )

Where P 1 P_1 P 1 ​ is reference power and P 2 P_2 P 2 ​ is measured power.

Voltage Ratio Formula (Equal Impedance Circuits) Gain/Loss in dB = 20 log ⁡ 10 ( V 2 V 1 ) \text{Gain/Loss in dB} = 20 \log_{10} \left( \frac{V_2}{V_1} \right) Gain/Loss in dB = 20 lo g 10 ​ ( V 1 ​ V 2 ​ ​ )

Common Decibel Rules of Thumb (Must Memorize):

Power Ratio ( P 2 / P 1 P_2 / P_1 P 2 ​ / P 1 ​ ) Voltage Ratio ( V 2 / V 1 V_2 / V_1 V 2 ​ / V 1 ​ ) Decibel Value (dB)

  • 2 : 1 2 : 1 2 : 1 (Doubling) 1.414 : 1 1.414 : 1 1.414 : 1 + 3 dB +3\text{ dB} + 3 dB
  • 4 : 1 4 : 1 4 : 1 2 : 1 2 : 1 2 : 1 (Doubling) + 6 dB +6\text{ dB} + 6 dB
  • 10 : 1 10 : 1 10 : 1 (Tenfold) 3.16 : 1 3.16 : 1 3.16 : 1 + 10 dB +10\text{ dB} + 10 dB
  • 100 : 1 100 : 1 100 : 1 10 : 1 10 : 1 10 : 1 + 20 dB +20\text{ dB} + 20 dB
  • 1000 : 1 1000 : 1 1000 : 1 31.62 : 1 31.62 : 1 31.62 : 1 + 30 dB +30\text{ dB} + 30 dB
  • 0.5 : 1 0.5 : 1 0.5 : 1 (Halving) 0.707 : 1 0.707 : 1 0.707 : 1 − 3 dB -3\text{ dB} − 3 dB
  • 0.1 : 1 0.1 : 1 0.1 : 1 (Tenth) 0.1 : 1 0.1 : 1 0.1 : 1 − 10 dB -10\text{ dB} − 10 dB
  • Absolute Power Units: dBm and dBW

dBm \text{dBm} dBm : Power relative to 1 Milliwatt ( 1 mW = 10 − 3 W ) 1\text{ Milliwatt } (1\text{ mW} = 10^{-3}\text{ W}) 1 Milliwatt ( 1 mW = 1 0 − 3 W ) . P dBm = 10 log ⁡ 10 ( P watts 0.001 W ) P_{\text{dBm}} = 10 \log_{10} \left( \frac{P_{\text{watts}}}{0.001\text{ W}} \right) P dBm ​ = 10 lo g 10 ​ ( 0.001 W P watts ​ ​ )

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

Key Conversions :

0 dBm = 1 mW 0\text{ dBm} = 1\text{ mW} 0 dBm = 1 mW

30 dBm = 1000 mW = 1 W = 0 dBW 30\text{ dBm} = 1000\text{ mW} = 1\text{ W} = 0\text{ dBW} 30 dBm = 1000 mW = 1 W = 0 dBW

50 dBm = 100 W = + 20 dBW 50\text{ dBm} = 100\text{ W} = +20\text{ dBW} 50 dBm = 100 W = + 20 dBW (the HF limit for an amateur standard station using emission mode J3E or R3E; 30 W pY mean power applies in any other mode).

Worked Example 1.4: Multi-Stage RF System Decibel Calculation

An RF signal generator outputs 2 mW 2\text{ mW} 2 mW ( + 3 dBm +3\text{ dBm} + 3 dBm ). The signal passes through a + 13 dB +13\text{ dB} + 13 dB preamplifier, a − 4 dB -4\text{ dB} − 4 dB bandpass filter attenuator, and a + 20 dB +20\text{ dB} + 20 dB power amplifier. Calculate total system gain and final output power in Watts.

Solution: Total System Gain = + 13 dB − 4 dB + 20 dB = + 29 dB \text{Total System Gain} = +13\text{ dB} - 4\text{ dB} + 20\text{ dB} = +29\text{ dB} Total System Gain = + 13 dB − 4 dB + 20 dB = + 29 dB Output Power in dBm = + 3 dBm + 29 dB = + 32 dBm \text{Output Power in dBm} = +3\text{ dBm} + 29\text{ dB} = +32\text{ dBm} Output Power in dBm = + 3 dBm + 29 dB = + 32 dBm Converting +32 dBm to Watts : 32 = 10 log ⁡ 10 ( P 0.001 ) ⟹ 10 3.2 × 0.001 = 1585 × 0.001 ≈ 1.585 Watts \text{Converting +32 dBm to Watts}: \quad 32 = 10 \log_{10}\left(\frac{P}{0.001}\right) \implies 10^{3.2} \times 0.001 = 1585 \times 0.001 \approx 1.585\text{ Watts} Converting +32 dBm to Watts : 32 = 10 lo g 10 ​ ( 0.001 P ​ ) ⟹ 1 0 3.2 × 0.001 = 1585 × 0.001 ≈ 1.585 Watts

Power Distribution at Maximum Power Transfer (50% Load vs 50% Internal Loss)