I very frequently, read within these forums, the unit of measurement – Watts being misapplied! So frequently that it is clear to me that many don’t know that our audio circuits promote the development of 3-powers, if you will; with only one of them being usable, towards producing actual work, such as moving a speaker-cone, and only that one, is a Watt!
The 2 other powers, if you will, are also present within the amplifier – speaker circuit.
The three powers, again if you will, are rooted in three different types of opposition, to current flow: Inductances, Capacitances, Resistances – LCR
The first 2, dissipate zero, actual power / real power / power that can perform work. Yet they still produce Voltage and Current measurements, which gives the deceptive impression that they do dissipate power. This is referred to as ‘Reactive Power’, and it is assigned a unit measurement, called Volt-Amps-Reactive (VAR),
not Watts. The mathematical symbol for ‘Reactive Power’ is the capital letter Q
. ‘Real Power’, power that can produce work, is measured in Watts
and is symbolized by the capital letter P
. The combination of ‘Reactive Power’ and ‘Real Power’, is called ‘Apparent Power’, and it is the product of a circuit’s Voltage (E) and Current (I), without reference to phase angle
. Apparent power is measured in Volt-Amps (VA)
and is symbolized by the capital letter S
= Mathematical Symbol: P
= Unit of Measurement: Watts
= Mathematical Symbol: Q
= Unit of Measurement: Volt-Amp-Reactive VAR
= Mathematical Symbol: S
= Unit of Measurement: Volt-Amps VA
Prime PQS Facts:
is a function of a circuit’s dissipative, resistive elements, R
is a function of a circuit’s reactance X
is a function of a circuit’s total impedance Z
All algebraic equations relating these 3 types of power to R/X/Z
, use scalar quantities. As such, complex starting quantities such as Voltage, Current, and Impedance, must be represented by their polar magnitudes, not by real or imaginary rectangular components.
When calculating Real Power from Current and Resistance, I must use the polar magnitude for Current, and not merely the ‘Real’ or ‘Imaginary’ portion of the Current.
When calculating Apparent Power from Voltage and Impedance, both formerly complex quantities must be reduced to their polar magnitudes for the scalar arithmetic.
Breaking It Down Mathematically:
There are two equations presented below, for each respective calculation of a power types P & Q
There is three equations presented below for Apparent Power – S
Making It Practical:
Application of the math with pictorial aid.
Purely Resistive Circuit: True Power, Reactive Power, and Apparent Power
Purely Reactive: True Power, Reactive Power, and Apparent Power
Resistive & Reactive Circuit: True Power, Reactive Power, and Apparent Power
Introduction To The Power Triangle:
PQS are related through a graphing, trigonometric form, commonly referred to as the ‘Power Triangle’.
When graphically depicted the relationship can become more intuitive, and easier to grasp.
P = Adjacent, Q = Opposite, and S = Hypotenuse. The opposite angle is equal to the circuits Impedance-Z, phase angle. Phase angles, must be calculated to determine the Real Power, within all AC Circuits. Overlaying these additional calculations, is commonly referred to a Power Factor calculations.
Introduction to Power Factor:
The Power Triangle graphically indicates the ratio between the amount of dissipated (or consumed) power and the amount of absorbed/returned power. It’s also the same angle, as that of the circuit’s impedance in ‘polar form’. When expressed as a fraction, this ratio between Real Power and Apparent Power is called the Power Factor.
Real Power and Apparent Power form the adjacent and hypotenuse sides of a right triangle, respectively, the Power Factor ratio is also equal to the cosine of that phase angle.
For the purely resistive circuit, the Power Factor is 1 (perfect), because the Reactive Power equals zero. Here, the Power Triangle would look like a horizontal line, because the opposite (Reactive Power) side would have zero length.
For the purely inductive circuit, the Power Factor is zero, because true power equals zero. Here, the Power Triangle would look like a vertical line, because the adjacent (Real Power) side would have zero length.
The same could be said for a purely capacitive circuit. If there are no dissipative (Resistive) components in the circuit, then the Real Power must be equal to zero, making any power in the circuit purely reactive. The Power Triangle for a purely capacitive circuit would again be a vertical line (pointing down instead of up as it was for the purely inductive circuit).
Power Factor is a critically important aspect to consider in an AC circuit. A Power Factor less than 1 means that the circuits must produce, and support more Current than what would otherwise be necessary with zero reactance in the circuit, to deliver the same amount of Real Power to the resistive load.
Amplifier Measurement Standards:
The most commonly used Amplifier Measurement Standards (CEA, IHF, EIA, FTC), only use purely resistive loads. However, the amplifier – speaker network, is comprised of Inductances, Capacitances, and Resistances (LCR) This translates into Wattage Scores, estimated from loads that are purely resistive, being invalid.
Further to this point, the duration of the tests is often less than 1-second, in the range of 13ms, with frequency agitations that are either single tone, dual tone, or various impulses, and short sweeps.
In real world application, the loads are very complex, frequency bandwidth is wide (spanning octaves), and duration of usage is long, often spanning hours.
The Straight Goods:
A Watt is only a Watt, when the Current & Voltage angles are in phase, everything else is quite literally a waste, as in a waste of energy! Incidentally, the Power Factor of any circuit in which the Current & Voltage envelopes are running in phase, have a Power factor of 1, and when its 1, work can be done!
Unfortunately, figuring all of this out, is very complicated, and as such; consumers don’t honestly have any idea, what the Real Power capabilities of their amplifiers are, which has resulted in them calling just about anything a Watt!
If I may, I would like to offer up a fair rule of thumb, if your amplifiers power ratings have been estimated using any of the following standards: CEA, IHF, EIA, FTC. Take the RMS values and simply divide them by two, and you will have a reasonable estimate of their 24/7/365 power capabilities, in to their highest rated nominal load, which is usually 8-Ohms.