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Amplifier Output Impedance: The Amplifier - Loudspeaker Interaction

Amplifier Output Impedance
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One of the most commonly misconstrued topics on audio amplifiers is output impedance and what it means for your loudspeakers. Many articles touch on one aspect of the subject without considering others. Some have misinformation altogether. This article should give a good understanding on how amplifier output impedance affects the interaction between the amplifier and loudspeaker.

The best way to think of the output terminals of an audio amplifier is an AC voltage generator or source. AC meaning alternating current, where the signal current (the waves that make up music in this case) varies sinusoidal with time. When current varies with time, so too does the voltage through a resistive load. AC Voltage, Voltage AC, or VAC are just nomenclatures for the same thing and all can be used interchangeably.

It is also important to keep in mind that the standard of measuring AC voltage is in RMS. In engineering, values of AC voltage and current are assumed to be RMS unless otherwise specified such as peak-to-peak, or peak. A quality digital multimeter on the AC function will measure RMS values as well. The significance of RMS is beyond the scope of this discussion, but in simple terms, AC voltage or current measured in RMS allows the electrical power delivered to an external load from an AC source to be directly compared to the electrical power delivered to an external load from a DC or direct current source.

1. Ohm's Law

For the rest of this discussion we will use the Engineering standards:

  • Current in Amps (AC or DC) = I

  • Voltage (AC or DC) = V

  • Resistance in Ohms = R

Along with the AC voltage model, the output of an amplifier has output impedance, or more simply resistance, in series with the output voltage generator. This is due to the internal construction of the amplifier. This alone is not very exciting, but how this impedance interacts with a loudspeaker load is.

Amplifier Output Model
Figure 1: Amplifier Output Model

Ohm’s Law:

V = IR

Since there is no external load applied across the output of the amplifier, there is no current flow. The circuit is said to be incomplete or “open.” Applying Ohm’s law, we can first find the current and prove that it is zero.

I = V / R = 5V / (2-ohms + infinity-ohms) = 0A

In the denominator due to the open circuit, we have an infinite resistance term. Any number divided by infinity is equal to 0.

Using Ohm’s law again we can find the voltage dropped across the 2-ohm output impedance.

0A x 2-ohms = 0V. The voltage drop across the amplifier’s output impedance is 0V since there is no current. Therefore, at the output terminal, we would measure with a multi-meter the full 5V developed by the amplifier.


Now let’s add a 16-ohm loudspeaker load across the output.

Output with 16ohm Load
Figure 2: Output with 16ohm Load

The output of the amplifier is no longer open as there is now a completed loop for current to flow. Using Ohm’s law, we can determine the current. V/R = I

5/(2 + 16) = 0.278 Amps. Since the loop is closed and we have current flow, Ohm’s law tells us that the output impedance of our amplifier will now consume some voltage due to the current. This means that only a percentage of our 5V generated will reach our speaker! The voltage across the speaker is simply (0.278A x 16) = 4.44V! We have lost 11% of our 5V potential across the amplifier’s own output impedance!


What if the speaker load was 8-ohms?

Output with 8ohm Speaker
Figure 3: Output with 8ohm Speaker

The loop current would be 5/(2 + 8) = 0.5A. The current has increased since the total load seen by the amplifier has decreased. The voltage across the speaker load is now

(0.5A x 8) = 4V. We have now lost 20% of our 5 volts!


How about a 2-ohm load?

Output with 2ohm Load
Figure 4: Output with 2ohm Load

Our speaker would see 2.5V, only 50% of the 5V available! This makes sense intuitively since both the output impedance and the speaker load must consume the total 5V available and their resistances are equal.


2. A Real-World Loudspeaker

So far we have been using the terms resistance and impedance interchangeably, but there is a difference. The term impedance implies that the value, in ohms, varies with frequency. A resistance, such as a resistor, is fixed in value across frequency.

Amplifier output impedance, although called impedance, is usually relatively fixed across frequency and only undergoes minor fluctuations. It is analogous to a resistor in most cases. This terminology is confusing I know, but stick with it.

A loudspeaker on the other hand is almost always highly reactive, meaning its impedance varies wildly with signal frequency. A typical 8-ohm moving cone transducer can have impedance as low as 4-ohms at bass frequencies and an impedance as high as 70-ohms or more at high frequencies! The 8-ohm rating is simply an average for the usable range of the driver, hence why speakers are listed as an 8-ohm impedance, not an 8-ohm resistance.

(Note there are some speaker types with very flat impedance characteristics. These speaker types are considered to be easier to drive by an amplifier.)

From the examples above, it should be clear that the voltage developed across the speaker will vary with the impedance! This will cause the speaker to produce some frequencies louder than others since the load voltage fluctuates and thus power delivered to the load also fluctuates.

Using a loudspeaker that has a 4-ohm minimum and a 70-ohm maximum and the same 5V 2-ohm output impedance amplifier from before: Vmin = 3.33V and Vmax = 4.86V. This is a 46% change in the voltage across the speaker due to its changing impedance!


3. Lowering the Output Impedance

Now let’s do the same example with an amplifier still developing 5V but has an output impedance of 1ohm: Vmin = 4V Vmax = 4.93V. Now the percent change in voltage across the speaker is only 23%.

This change in voltage due to varying loads (in this case the loudspeaker) is known as regulation. The better the regulation, the smaller the change in voltage delivered to a load due to impedance fluctuation of the load itself. Lowering the output impedance of the source improves load regulation. Thus, the apparent voltage of the speaker will remain more constant across all frequencies. An ideal amplifier would have an output impedance of 0-ohms, but we live in a world bound by the laws of physics. However, some amplifiers can get very close.

A. Speaker Resonance

Does this mean that a 0-ohm output impedance amplifier is better?

Not necessarily....

The translation from drive voltage to sound in terms of Sound Pressure Level (SPL) is very complex. This is especially true around the resonance of the loudspeaker where output impedance will have the biggest impact in the 'feel' and reproduction of music.

Resonance of a speaker is determined by the input frequency, where the mass and spring behavior of a moving diaphragm has the highest level of energy storage. At the resonant frequency, speaker excursion (cone moment) is largest for a fixed input power to the speaker. Coincidentally, the electrical impedance of a speaker is largest at resonance.

This actually works in our favor by sympathetically reducing the amplifier input power. Impedance of the speaker can get so large that the input voltage to the speaker will approach the open circuit voltage of the amplifier. For this to be the case, (as seen with ohm's law), the current delivered into the speaker is lower. This has a tendency to reduce the amplifier's output power at the resonant frequency and normalize the SPL to some degree.


4. Speaker Control

Perhaps a less obvious reason for designing an amplifier with low output impedance is speaker control and damping. However, speaker control is possibly the most important factor in determining how a speaker will sound with a given amplifier, although no to the extent often described.

Have you ever tried connecting the two terminals on a small DC electric motor with a piece of wire? What happens if you then try to spin the shaft? From childhood tinkering or physics class in school, you may have noticed that it becomes much more difficult to spin than if you leave the terminals open.

Just as an electric motor will consume power by turning electric energy into rotational energy, rotational energy (as an input on the shaft of the motor) will be converted to electrical energy. Motors not only consume power, but they are generators too. Adding the short between the two terminals of the motor makes it difficult to spin since the energy produced has nowhere to go. Therefore, the motor attempts to cancel out what you just did to it by making it harder to spin the shaft. This is called Back EMF or Backwards Electro-motive Force. The faster you spin the shaft, the more rotational resistance felt. In an ideal world with zero losses, it would be impossible to spin the shaft since every bit of energy produced would be spent trying to undo the rotational movement you just made. However, motors have losses due to the resistance of the armature windings and magnetic field losses.

Well, a loudspeaker is no different than an electric motor. In fact, it is just a linear motor that converts electricity into linear motion rather than rotational. The lower the amplifier output impedance is, the closer we come to the short circuit motor analogy. This short consumes any external forces imposed upon the cone of a speaker and the linear motion is resisted.

If you have a loose woofer lying around you can do a quick test to prove this. Tap on the cone gently, with your ear hovering just above the speaker. You should hear a deep low note. Now short the terminals together with a piece of wire and tap. The low note is gone and the speaker sounds very “restricted.” Even this tiny amount of movement from the tapping has been canceled out. This should also give you an idea on how sensitive a speaker motor is!

This property is known as control or dampening of the loudspeaker. An amplifier that exhibits low output impedance has better control over the motion of the cone since the speaker is better forced to move to the voltage signal from the amplifier and is less affected by external forces.

External forces are mainly composed of the cone’s momentum, the suspension or springiness of the cone that returns it back to its natural resting state, and acoustic loading from air pressure. All of these effects get magnified at higher excursions where the cone travels more distance.

A. Resonance - Yet Again

As discussed above, interesting things happen at resonance where the loudspeaker excursion tends to be highest. This is where damping becomes most important since amplifier output impedance determines the control of the woofer.

For example, if a woofer cone is driven outward, it has some momentum and loading that will make it move nonlinearly to the amplifier signal. If we just consider momentum, the cone would want to continue moving in the same direction even if the amplifier attempts to drive it in the other direction. This is especially true with low bass frequencies including resonance, which require lots of cone movement. An amplifier with lower output impedance is said to have tighter bass and better, faster bass control. The linear overshoot and nonlinearities due to cone mass, suspension, and acoustic loading is minimized.

B. Damping Factor (DF)

In the world of Hi-Fi damping factor is used to describe loudspeaker control and is a ratio of loudspeaker impedance to amplifier (or source) output impedance. The idea that damping factors must be extremely high (i.e. very low amplifier output impedance) for a speaker to sound good and have tight bass is a misnomer and has spread like wild fire.

This is nonsense!

As discussed above, when you short any motor, including a loudspeaker, it has higher resistance to move, but can still move. This is the best damping factor that can be achieved!

In a loudspeaker with high magnetic efficiency, the primary loss mechanism is the DC resistance of the copper winding itself. This loss allows the speaker to still move, since the back EMF can be dissipated in this resistance.

Therefore, what we care about for high DF is not an arbitrarily low ratio, but an amplifier output impedance that is sufficiently low relative to the DC resistance (DCR) of the speaker.

For Example:

Let's say an 8-ohm loudspeaker has a DC resistance of 7-ohms.

  • An amplifier with an output impedance of 0.7-ohms is 10% of the DCR!

  • The DF is: 7 / 0.7 = 10

In reality, this is perfectly acceptable for good loudspeaker control since we are only 10% away from a perfect ideal short or an output impedance of 0-ohms.

However, those who spread the DF "gospel" will tell you this way too high and you need a factor of at least 10000000000.....

Because DF is a ratio, it blows up very quickly as output impedance drops. But what it doesn't tell you is the DCR of the speaker you are trying to control. Think a damping factor of 100 (0.07ohm output impedance / 1% of the DCR) or 1000 (0.007ohm output impedance / 0.1% of the DCR) will sound any different? - It won't!

To drive this home, the room we place our speakers in will have resonances and room modes of their own that accentuate the SPL specific frequencies. The electrical damping factor is just a tiny drop in the bucket in terms of how a speaker will really sound.


Wrapping Up

From this discussion we can conclude that output impedance will have an effect on the sound of on a speaker, especially at low frequency and around speaker resonance. However, the blanket statements we all hear that the best amplifier has 'zero output impedance' or a 'damping factor of one million' is nonsense.

In the end, it is up to the speaker manufacturers to choose damping criteria for their loudspeakers based on drivers and cabinet design. Since there is no universal standard, an amplifier with moderate output impedance may actually sound better with one speaker type or manufacturer than another.

Furthermore, speaker manufacturers may critically damp their acoustic design and SPL response, assuming a specific amplifier output impedance (or range of output impedances) they expect from the amplifier.

The speaker and amplifier are a complex non-linear interaction. There is no one size fits all!

Isn't that what makes Hi-Fi fun?


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