Types of SignalsRoot Mean SquareCurrent TransformersCT ConversionVoltage MeasurementOpAmpsPower CalculationPower GameAnalogue Quiz

Bringing it Together

Basic Power Calculation

Power Calculation Waveforms

VoltageCurrentPowerAvg Power

๐Ÿ’ก Understanding Power Calculation

  • Instantaneous power = voltage ร— current at each point
  • Results in oscillating power curve:
    • Always positive for resistive loads
    • Can go negative with reactive loads
    • Oscillates at twice the line frequency
  • Average (real) power:
    • What meters and inverters measure
    • Useful for billing and power flow
    • Calculated over complete cycles

โšก Practical Implementation

  • Microprocessor sampling:
    • Sample V and I multiple times per cycle
    • Multiply samples to get instantaneous power
    • Average over complete cycles
    • Typical sampling rate: 2-4kHz
  • Signal chain:
    • Voltage divider โ†’ op-amp โ†’ ADC for voltage:
      • Example: 330V โ†’ 3.3V (รท100 divider) โ†’ ADC
      • Firmware multiplies by 100 to get actual voltage
      • This ratio is fixed in firmware for each meter type
    • Calibration in firmware:
      • Fine-tune gain corrections
      • Adjust for component tolerances
      • Store calibration factors in EEPROM
      • Apply phase corrections if needed

โšก CT Signal Chain Example

  • CT and burden:
    • 100A primary current โ†’ 0.1A secondary (1000:1 CT ratio)
    • 0.1A through 10mฮฉ burden โ†’ 1mV (V = I ร— R)
  • Op-amp and ADC:
    • 1mV ร— 1000 gain โ†’ 1V for ADC
    • ADC reads 1V as digital value
  • Firmware processing:
    • Step 1: Get back to secondary current
      • 1V รท (1000 gain ร— 0.01ฮฉ burden) = 0.1A secondary
      • This value is used for calibration during production
    • Step 2: Apply CT ratio to get primary current
      • 0.1A ร— (100A รท 0.1A) = 100A primary
      • This ratio is the "CT Primary" setting on meters
      • User can change this to match installed CT

๐Ÿ“Š Power Quality Considerations

  • Power factor effects:
    • Phase shift between V and I changes power curve
    • Average power reduced by cos(ฯ†)
    • Important for billing and efficiency
  • Harmonics impact:
    • Distorts power waveform
    • Requires higher sampling rates
    • May need additional filtering

Power Factor and Phase Angle

Adjust the phase angle to see how it affects power transfer and power factor:

degrees
VoltageCurrentTotal PowerReal PowerReactive PowerVoltageCurrent

๐Ÿ’ก Reactive Power Balance

  • Apparent Power (V x I): 3600.0 VA
  • Real Power (V x I x cos(ฯ†)): 3600.0 W
  • Reactive Power (V x I x sin(ฯ†)): 0.0 VAR
  • Power Factor: 1.000
  • Represents energy oscillating between source and load

Harmonic Distortion

Adjust the harmonic amplitudes to see how they affect the waveform:

Aliasing Effects with 800Hz Sampling

50Hz FundamentalTrue WaveformSampled PointsADC View0ms5ms10ms15ms20ms

๐Ÿ’ก Understanding Aliasing and Nyquist Filtering

  • Sampling rate: 800Hz (fixed)
  • RMS Values:
    • Ideal Fundamental RMS: 0.707
    • True RMS (all frequencies): 0.707
    • Raw Sampled RMS (with aliasing): 0.707
    • Nyquist-filtered RMS (50Hz only): 0.354(-50.0% error)
  • Key observations:
    • When fundamental = 0, raw sampling still shows power!
    • Nyquist filtering correctly shows near-zero 50Hz component
    • High frequencies alias to look like real power
    • This is why power meters need proper filtering
  • Nyquist Filtering Benefits:
    • Extracts only the 50Hz component magnitude
    • Immune to aliasing of higher frequencies
    • Works because 800Hz sampling is greater than 2x fundamental
    • Perfect for power measurement (we only want 50Hz)
    • Higher frequencies are automatically rejected
  • Practical Implementation:
    • Sample at โ‰ฅ100Hz for 50Hz measurement
    • Use digital filters to extract fundamental
    • Ignore or monitor harmonics separately
    • Results in accurate power readings

๐ŸŽฏ Summary

Power measurement is not as simple as it seems:

  • โšก Requires accurate voltage and current measurement
  • ๐Ÿ”„ Instantaneous power oscillates, often at twice the line frequency
  • ๐Ÿ“Š Power is not as simple as Vrms x Irms (true only for purely resistive loads)
  • ๐Ÿ”ฌ We analyze complete cycles to compute: real power, reactive power, apparent power
  • ๐Ÿ”‹ Reactive power is energy oscillating between source and load (no net transfer)
  • ๐Ÿ’ช Real power represents actual work being done and energy transferred
  • ๐Ÿ”ข Power factor (cos ฯ†) indicates how efficiently power is being used
  • ๐ŸŽญ Harmonics can significantly impact power quality and measurement accuracy
  • โฑ๏ธ High-frequency components require faster sampling for accurate measurement
  • ๐Ÿงฎ Digital processing enables advanced power quality analysis in modern meters