6.5 Power and Efficiency Calculations

Amplifier Efficiency Classes

Class A:

η_max = 25% (resistive load)
η_max = 50% (tuned load)
P_dissipated = P_supply − P_output ≥ P_output × 3

Class AB:

η_typical = 50–65%
P_dissipated ≈ P_output (at max output)
P_dissipated = P_output × (1/η − 1)

Class D:

η_typical = 80–92%
P_dissipated = P_output × (1/0.85 − 1) = P_output × 0.176  (at 85%)

Example comparison — 1000W amplifier:

RMS Power from Waveforms

Sine wave:

P_rms = V_peak² / (2R) = V_rms² / R
V_rms = V_peak / √2 ≈ 0.707 × V_peak

Square wave:

P_rms = V_peak² / R  (100% duty cycle)
V_rms = V_peak

Music signal (approximate):

Crest Factor ≈ 10–15 dB (typical pop/rock)
V_rms ≈ V_peak / (crest factor linear)
P_average ≈ P_peak × (1/crest_factor²)

Practical implication:

An amplifier rated 1000W RMS delivers that power continuously with a full sine wave. With music at 10 dB crest factor:

P_average = 1000 / 10² = 10W average

The amplifier only works hard during peaks — which is why thermal design must handle peaks, not averages.

CEA-2006 Amplifier Rating Standard

Conditions for valid CEA-2006 rating:

• 14.4V supply voltage
• Continuous sine wave test tone (specified frequency)
• Both channels driven simultaneously (multichannel amps)
• At stated THD+N percentage (typically ≤ 1%)
• Temperature stabilized

Why this matters:

Without CEA-2006, amplifier power ratings are essentially marketing: - "1000W MAX" = peak, single channel, at threshold of destruction - CEA-2006 1000W RMS = continuous, all channels, ≤ 1% THD

Converting from MAX claims:

P_rms_honest ≈ P_max / 4  (rough heuristic for uncertified amps)

A "1000W MAX" amplifier typically delivers 200–300W RMS per channel in honest measurement.

Battery Runtime Calculation

How long will battery power the system without alternator?

t (hours) = C_battery (Ah) × V_battery / P_system

Adjusted for Peukert's law (high discharge rate reduces capacity):

C_actual = C_rated × (I_rated / I_actual)^k

Where k = Peukert exponent (1.1–1.3 for lead-acid, ~1.0 for lithium)

Worked example:

System: 2000W RMS, 70% Class D efficiency → 2857W draw Battery: 100Ah AGM at 12V Average music usage: 30% of peak → 857W average → 71.4A average

t = 100 / 71.4 = 1.4 hours (without Peukert)

Peukert correction (k = 1.2, rated at 5A):

C_actual = 100 × (5 / 71.4)^1.2 = 100 × (0.070)^1.2 = 100 × 0.045 = 4.5 Ah?

Wait — this gives an absurd answer because the Peukert formula scales C relative to the rated discharge rate. Let me restate:

At 5A (20-hour rate), battery holds 100 Ah. At 71.4A, actual capacity:

t = C_rated × (I_rated/I_actual)^(k-1) / I_actual
t = 100 × (5/71.4)^(1.2-1) / 71.4
t = 100 × (0.070)^(0.2) / 71.4
t = 100 × 0.617 / 71.4
t = 0.86 hours = 52 minutes

Bottom line: 52 minutes at 30% usage factor from a 100Ah battery. Add a second battery: ~104 minutes. This is why serious competition systems carry large battery banks.