Understanding Ohm's Law, Impedance And Electrical Phase 101
Introduction
Have you ever wondered what makes a loudspeaker “difficult to drive”? Do you wonder what’s so special about an amplifier that is stable into a 4 ohm load? If these are the kinds of questions that leave you mystified, this may be the article for you. Fortunately, there is nothing extraordinarily difficult involved in answering these questions: as long as you have rudimentary math skills and knowledge of the right equations, you will be able to look at a few basic measurements of a loudspeaker, namely the impedance curve, electrical phase curve, and voltage sensitivity, and determine what kind of amplification you’ll need to get the job done.
Terminology
Before we get too far, it’s important to define some terms:

Voltage: The difference in electrical potential between two points measured in volts (V); a measurement of the energy contained per unit of charge.

Current/Amperage: the total flow of electric charges through a surface at the rate of coulomb per time unit. The flow of current through a circuit is usually measured in amps (A) or milliamps (mA).

Resistance: Opposition to the flow of current through a conductor measured in ohms (Ω).

Capacitance: The ability to store an electrical charge measured in Farads (F) or micro Farads (uF).

Inductance: The property of a conductor in which a change in current flow within the conductor creates voltage within itself as well as other nearby conductors. Inductance is measured in Henries (H)

Impedance: Similar to resistance, impedance is the opposition of AC current flow through a conductor in a reactive circuit (i.e. one exhibiting elements of capacitance and inductance).

Phase Angle: The degree to which current flow will lead or lag the voltage waveform in a reactive circuit element.
From left to right, a set of resistors, a capacitor, and an air core inductor.
Ohm's Law
To really kick things off, we will first want to examine Ohm’s Law; in plain English, the law says that the current flow/amperage through a conductor is directly proportional to the voltage potential, with the factor between them being the conductor’s resistance, or in the case of a complex circuit, its impedance. Mathematically, this is simply written as:
where V=Voltage, I=Current, and R=Resistance. In a complex load, we substitute Z for resistance, where Z=Impedance.
The Ohm’s Law Triangle demonstrates the interchangeability of the variables. The line dividing the left and right sections of the triangle represents multiplication; the line between the top and bottom sections represents division.
Active Power
One other important equation to keep in mind is:
P = V * I * COS(Ф) (2)
where P=Power and Ф (Phi) = phase angle. Looking at our prior equations, you’ll find you could also rewrite this as:
P = I^2 * Z * COS(Ф) (3)
and
P = (V^2 / Z) * COS(Ф) (4)
It’s worth pointing out here that power will always be positive, given that it is proportional to the square of the current or voltage; that is to say, energy is always dissipated inside of an electrical circuit, though one may note inductance and capacitance in and of themselves store, but do not dissipate energy. With any luck, these equations haven’t scared you off just yet. Suffice it to say, a strong understanding of the relationships they represent are a key component to answering the questions at the beginning of this article.
The Power Triangle visually demonstrates the power lost due to phase shift between voltage and amperage, and is given above as Reactive power.
The Application
So now that we know these rules exist, how do we apply them to audio? Well, let’s start with the magic number of 2.83 volts. You may see that number pop up in speaker specifications a lot with respect to voltage sensitivity. You’ll also find that the majority of loudspeakers are specified as an 8 ohm nominal load. Going back to equation (1), we can plug in 2.83 volts = X amperes * 8 ohms. To solve for X, we simply divide 2.83 by 8, which yields amperage as 0.354, and gives us the statement 2.83 volts = 0.354 amperes x 8 Ohms. Plugging those numbers into equation (2) to solve for power (and ignoring the phase angle for now), you’ll find that 2.83 volts translates into 1 watt with an 8 ohm load.
OK, what about 4 ohms? Well since the impedance is cut in half, you’re going to have to double the amperage to maintain the ratio, and you’ll get the formula 2.83 Volts = 0.708 Amperes x 4 Ohms. Because amperage goes up by a factor of two as a result of impedance being cut, power goes up by a factor of two as well, meaning that 2.83 volts equates to 2 watts into a 4 ohm load. That’s a pretty significant difference, and explains why a 4 ohm speaker is more demanding of an amplifier than an 8 ohm speaker of the same voltage sensitivity.
Now let’s take a look at what happens when you have an 8 ohm speaker with a voltage sensitivity 3dB less than a comparable 4 ohm speaker. In this example, the speakers will consume equal amounts of power to reach any given SPL; however, the voltage and amperage will be different, as the ratios are defined by the impedance. For example, 1 watt into 4 ohms equates to 2 volts and 0.5 amperes; as mentioned above, 1 watt into 8 ohms equates to 2.83 volts and 0.354 amperes. Looking at this, you’ll note that the 8 ohm load requires 41% greater voltage, but correspondingly less amperage relative to a 4 ohm load. Which is preferable depends on the capabilities of the amplification you’re employing; if you’ve got a meaty amplifier that is stable into low impedance loads, a 4 ohm load may not be a big problem. On the other hand, if you’re utilizing a low end surround receiver, you may wish to avoid the 4 ohm option, as the odds are they simply aren’t built with the demands of a 4 ohm load in mind.
High Sensitivity?
As one more example, let’s take the case of two speaker systems, one with a rated voltage sensitivity of 96dB w/ 2.83V at 1 meter, and the other rated at 90dB under the same circumstances. Let’s assume that the 96dB sensitive speaker dips down to the 3.2 ohm mark (the minimum for a 4 ohm nominal speaker per the IEC method, as well as the minimum allowed per THX Ultra 2 spec), whereas the 90dB sensitive speaker stays at 8 ohms or above. This scenario may not be as uncommon as you’d think: it’s easy to have higher sensitivity (and efficiency in terms of watts) at high frequencies; however, at lower frequencies where you must displace more air, you will tend to need more power to reach a given SPL. A few tricks can be used to help even out this mismatch and retain reasonably flat voltage sensitivity over a wide range of frequencies (i.e. a flat frequency response). One of those tricks is wiring a pair of woofers in parallel: this increases voltage sensitivity by 6dB over a single woofer, at the expense of halving impedance.
So what do the numbers tell us? To hit 96dB at 1 meter, our 90dB sensitive loudspeaker will require 4 watts into an 8 ohm load, or ~5.66 volts and ~0.71 amperes. On the other hand, our 96dB speaker will require 2.5 watts into a 3.2 ohm load, or ~2.83 volts and ~0.88 amperes. Surprisingly, in spite of the rather significant advantage in voltage sensitivity, the 96dB sensitive loudspeaker demands more current to reach a given SPL than the 90dB sensitive loudspeaker. Of course, as mentioned previously, which one is a better fit for your amplifier ultimately depends upon its capabilities; if your amplifier is rated to be stable with plenty of output into low impedance loads, then the tradeoff may well be worth it. However, the numbers above should serve as a warning to those who simply assume that a loudspeaker with high rated voltage sensitivity will automatically be a supereasy load for any amplifier. This may be doubly true when you consider that some loudspeaker manufacturers are more conservative in their voltage sensitivity ratings than others.
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Recent Forum Posts:
shadyJ, post: 952555
What I am taking from the article is that phase angles are OK as long as they don't hover around 45 degrees? Is it as simple as that? So 60 degrees is fine and 30 is fine, but 45 is not fine, at least if you don't have a amp with good cooling? What would the most perfect, most ideal phase angle profile a speaker could have?
If you look at the table included in that section, specifically under the Power (Amp) column, you'll note that as the phase angle goes up, power dissipated in the amplifier goes up until you reach 45 degrees. At that point it falls off until you reach 90 degrees, which no real world loudspeaker will present. Nonetheless, at 30 and 60 degrees, while the power dissipated in the amplifier isn't as high as 45 degrees, it's still significantly higher than at 0 degrees, which is a purely resistive load.
Here's a pic that illustrates pure resistance, inductance, and capacitance circuits.
GONAD!, post: 951678
Pictures would help…and sock puppets…and interpretive dance…
There was a chart that was captioned with a link to sound.westhost.com; that article has one picture that illustrates voltage and and current out of phase by 38 degrees and a frequency of 45Hz.