Amplifier power / SPL calculator for home theater THX reference level
A question that frequently comes up when designing a home theater is 'how powerful an amplifier do I need?'. The answer to the question depends on a few things, in particular how sensitive your speakers are and how far you sit away from them. When we design a home theater we want it to meet industry standards, in particular the THX standard for peak sound pressure level (SPL) at the listening position.
Remember THX Reference Level?
Reference level is a calibrated volume setting used for both movie production (in dubbing stages and post production houses) and reproduction (in screening rooms and theaters). Reference level for all channels except low frequency effects is calibrated by adjusting the audio chain such that a pink noise signal recorded at -20dB relative to full scale creates 85dB SPL as measured with a C weighted meter at the primary seating location. Peaks in movie soundtracks can therefore reach 105dB (0dB relative to full scale). For more read THX Reference Level Explained
The 105dB peak SPL requirement
From the discussion of THX reference level the requirement is clear: amplifiers must be capable of cleanly driving speakers to 105dB SPL peaks at the listening location
. Clean means that the amplifier is not clipping and total harmonic distortion (THD) is low - 0.1 or 0.2%.
Calculating SPL from sensitivity, distance and power
The SPL achievable at the listening location can be calculated if the following variables are known:
*Beware that some speaker manufacturers quote sensitivity as 2.83V/1m. A speaker with 4 ohm nominal impedance uses 2W at 2.83V compared to 1W for a 8 ohm speaker.
Speaker sensitivity - the sound pressure level (SPL) measured at 1m (3.3ft) from the speaker for 1W of input power*.
Listening distance - the distance between the speaker and the listener.
Amplifier power as measured in Watts (W).
Most of the online amplifier power calculators work in the same way. You enter the speaker sensitivity, listening distance and amplifier power and get SPL. We, however, are most interested in the amplifier power required to hit a certain SPL and so it makes sense to structure the calculation slightly differently.
Many people also make a simple mistake which effectively doubles the size of amplifier required. Using the online calculators they enter follow a process of trial and error to determine the amplifier size required for 105dB SPL. The issue is that our THX requirement is not
for 105dB continuous output but
105dB peak output. More on this later.
Calculating the amplifier power required for 105dB peaks at the listening position
Our calculator works differently - we enter the sensitivity and listening distance to get the required amplifier power. SPL required is already known - we want 105dB peaks at the listening position to meet specifications. The calculator is shown below and can be downloaded Max SPL Calculator v6
The main two line items in the calculator are Speaker Efficiency and Distance, which we have already defined above. From THX specifications the Average Program can be generically defined as 85dB and the headroom required as 20dB. The calculator then returns the Amplifier Gain required in dB and the Peak Amplifier Watts. You'll see a number of options in the calculator to deal with various scenarios such as baffle wall mounting and sensitivity specifications that are not 1W / 1M.
Finally we see the Equivalent RMS Watts which is roughly half the Peak Amplifier Watts. The reason for this conversion is that all amplifiers spec sheets list RMS Watts, as measured with a sine wave. A sine wave has a 3dB crest factor i.e. the difference between average and peak is 3dB. So we effectively derate the calculated Peak Amplifier Watts by 3dB to get the equivalent RMS specification. This is an important point, and nearly everyone we hear discussing this topic online fails to derate the amplifier power requirements. They make the mistake of equating peak SPL with RMS Watts. In the above example an amplifier with a specified RMS output of 683W would actually give us 108dB peaks at the listening position, which is 3dB more than we need. People using this approach will choose an amplifier roughly twice as powerful as they need.
Down the rabbit hole...
The approach to calculating amplifier power requirements for THX reference level reproduction presented above is of course a simplification. It will likely still give equivalent RMS amplifier power figures that are higher than needed.
The main reason is that an amplifier's power output is higher for typical movie and music content than it is for sine waves. This Dynamic Headroom varies from amplifier to amplifier and can be as little as 1dB to as much as 8dB. Amplifiers with very robust power supplies tend to have low dynamic headroom and those with weaker power supplies such as your typical AV receiver high dynamic headroom.
A second complication for multi-channel is that often more than one amplifier channels share the same power supply. The power supplies in these amplifiers are generally not designed so that the single channel power output is the same as the power output with all channels driven. The question then becomes what amplifier power specification you should look at when choosing an amplifier...single channel, two channel, all channels driven? Audioholics have a good article
on this topic.
Thirdly we have the fact that speakers are not resistive but reactive loads where the real impedance varies over the frequency range rather than being a fixed nominal 4 or 8 ohms. This can result in over-estimation of amplifier power.
Fourthly the SPL in most listening rooms or home theaters does not fall off at 6dB per doubling of distance as the calculator assumes. The 6dB assumes free space. Often we have less than that, maybe 3-4dB. The importance of this is debatable, however, as I believe our 105dB requirement related to impulsive peaks not sound that has a chance to build up and is overlaid with the same sound that is decaying.
From the brief discussion above it is obvious that things get a lot more complicated in the real world. If you have any questions or corrections relating to this article please leave a comment below!