# Handling a complex load by power amplifiers

#### pma

##### Major Contributor
Handling a complex load by power amplifiers

A power amplifier’s job is to supply a voltage to a loudspeaker and then to deliver whatever current the loudspeaker needs to move its voice coil, producing airwaves with a sound pressure level up to what is specified for the speaker and/or application. The output voltage multiplied with the resulting current constitutes the power that is delivered to the loudspeaker.

If we connect a resistor load to the amplifier’s output then things are simple; a resistor is a pure resistive and constant impedance. Between current through and voltage across the resistor there is zero phase shift, current and voltage have the same wave shape and resistor current can be directly calculated as I = V/R. Resistor power is then V*I or V^2/R. However, a loudspeaker is not a simple resistor, instead it can be seen as a coil with a resistance plus a resonant RLC circuit. This causes the loudspeaker’s ‘voltage to current’ transfer function to become complex, with the resulting impedance varying from the lowest ‘DC’ impedance to the impedance at the speaker’s resonance frequency, which can be much higher. The rated impedance listed in a loudspeaker’s data sheet is just an indication of the average.

More important is that a loudspeaker’s impedance is inductive below and capacitive above resonance, and inductive well above resonance, which means that voltage and current can be and are out of phase for most frequencies. Assuming we use a class A, B or AB amplifier, this poses a peak power issue. With a pure resistive load, at peak voltage the amplifier has to deliver also the peak current to the load. The voltage inside the amplifier is the difference between the amplifier’s power supply voltage and the output voltage. So, when delivering a peak voltage to the resistive load, even at a very high current the power dissipated inside the amplifier is limited, as it’s the product of a high current and a low internal voltage difference.

With a complex load, the phase between voltage and current changes with frequency. So there are frequencies where the current is out of phase with the voltage, asking the power amplifier to deliver the maximum current to the loudspeaker, while the voltage delivered to the loudspeaker is small. This means that the voltage difference inside the amplifier is much larger, compared to a pure resistive load. A high internal voltage difference multiplied by maximum current equals high power, so the power amplifier has to work much harder. This requires a lot from the output stage (the final power transistors) and the power supply, especially with high power loudspeakers with low impedances.

Testing power amplifiers with real loudspeakers instead of resistive dummy loads is difficult; needing to use loudspeaker arrays in isolated rooms or the use of large and expensive R-L-C dummy loads.

Apart from distortion - occurring when the power amplifier’s capabilities fall short of providing clean peak power - the main bottleneck is heat dissipation and even more the peak power in the output transistors. As we shall see later, transistor peak power can be much higher when the amplifier is feeding a complex load than when it is feeding a resistive load.

Characteristics of load used for tests

Two kinds of load were used for further testing, pure resistive load – power resistor 4.7ohm/200W, and a R-L-C dummy load that simulates impedance of the 7” SEAS woofer, the dummy load circuit schematics is shown below:

R1 and L1 are voice coil parameters, C1 and L2 and R6 represent cone parameters. The mechanical resistance is usually drawn in parallel with mass L2, but from a circuit view this has equivalent Q as my circuit and I have taken an advantage of the L2 intrinsic dc resistance to act as R6.

Though impedance Z of the pure resistive load equals to R and is a simple real number, impedance of the complex load Z is a complex number and must be represented in a 2-dimensional complex plane:

in Cartesian form:
Z = R + jX, where
R = Re(Z) … real part of Z, resistance
X = Im(Z) … imaginary part of Z, reactance

or in Polar form:
|Z| = sqrt(R^2 + X^2) … magnitude of impedance Z
Ø = arctan(X/R) … phase angle, represents phase angle between voltage and current

for more basic info on complex notation and complex impedance definition, please visit:

Simulated dummy load impedance is in the further image:

The plots show all the discussed impedance components, Re(Z), Im(Z), |Z| and Ø.

As a comparison there is another image with values of impedance magnitude and phase measured on the built dummy load.

We can see very good conformance of measured values with simulated values. In this image we can also see EPDR plot, which means Equivalent Peak Dissipation Resistance, and is extremely important for evaluation of complex impedance impact to amplifier output stage, as we shall see later. EPDR plot shows an equivalent resistor value that would represent the same peak power loading as the complex load at the individual frequency. We can see that at 1460Hz, the EPDR = 2.66 ohm, though magnitude |Z| is seemingly benign 7.546 ohm with 45° phase angle.

Class AB amplifier driving resistive and complex load – comparison of peak power, simulation

First, let's assume a class B/AB amplifier driving a 4.7 ohm resistive load

As an example, a pair of MJL3281/1302 “200W” transistors is driven from 34Vrms sine source and is supplied from 2 x 55Vdc power supply. The crucial parameter to be investigated is transistor's peak power. The allowed values of peak power are drawn in datasheets as a so called “Active Region Safe Operating Area” (SOA) and the plots show peak value of collector Ic current as a function of collector-emitter Vce voltage and time. The longer the time exposition to peak power, the lower allowed SOA.

So, we need to examine SOA in our circuit example for 4.7 ohm load. We need to see Ic as a function of Vce and compare it to SOA boundary from the datasheet. We shall use log scale for both Vce and Ic to make a comparison with the datasheet SOA plot feasible.

We can see that peak power is 160W and the Ic(Vce) red plot always remains below the 1s SOA blue boundary plot. So, with 1 pair of MJL3281/1302 200W/15A transistors we should be save with 4.7 ohm resistive load, when supply voltage is 2 x 55Vdc and output voltage is 34Vrms (load power 246W). Frequency of the sine source is unimportant in case we use the pure resistive load and power transistors are fast enough to handle the 20kHz bandwidth.

Now, let's move to the complex dummy load that simulates a single speaker, 7” woofer. This is the circuit for simulation

Please note that dc resistance of the dummy load is 6.6 ohm and thus is higher than the 4.7 ohm resistor resistance used in the previous SOA examination. The impedance plots of this load were shown here above.
The impedance plots of this complex load were shown above and we shall concentrate our examination efforts to the frequency where we have lowest EPDR at considerably high phase angle. Both our intuition, when reading the impedance and EPDR plots, and intensive simulation over wide range of frequencies confirm that the frequencies of concerns are near to 1460 Hz. At 1460 Hz, we have impedance magnitude |Z| = 7.546 ohm, phase angle Ø = +45°, but EPDR = 2.66 ohm! So the peak dissipation factor for output transistors equals to 2.66 ohm resistor, with the load that has 7.546 ohm impedance seen in a conventional way.
Let's start a simulation:

We can see that transistor peak stress values have changed significantly, compared to the 4.7 ohm resistor load. Vce(peak) = 102.2 V (same), Ic(peak) = 6.2A (much lower), but P(peak) = 231.3W – much higher than in case of 4.7 ohm load, when it was 160W. The key to this behaviour is a phase angle between load current I(load) and load voltage V(load), that results in different P = Vce*Ic instant power values than in case of a resistor load, where resistor current and voltage are in phase. The simulated circuit SOA is now exceeded, the Ic(Vce) red plot jumps above the SOA boundary blue plot. We would need to add one more pair of output transistors, in parallel with the existing pair, to keep the amplifier on a safe side of SOA, with this complex load.

Impact of complex load to distortion

The impact of complex load impedance to amplifier distortion is not quite straightforward when we look at impedance magnitude and phase plots. Measurements of THD+N vs. frequency and THD+N vs. output voltage at 1kHz were done to illustrate it. The A250W4R class AB amplifier was used for these measurements.

THD+N vs. frequency at 7Vrms output voltage

We can see that distortion with the complex load is lower except of the area between approx. 200Hz – 1700Hz. At 278Hz, the complex load impedance is purely resistive and equals to 5 ohm. Below 278Hz we can also see the capacitive part of impedance and above 278Hz starts inductive part of impedance. At 1700Hz, magnitude |Z| is 8.3 ohm, EPDR is 3.03 ohm and phase +49°. Above 1700Hz, distortion with complex load is lower than with the 4.7 ohm resistor, though phase is getting high up to +83°. At 1700Hz and above it, the complex impedance might be modelled by a series connection of 5.37 ohm resistor and 0.586 mH inductor. Quite predictable from the dummy load circuit schematics.

THD+N vs. output voltage at 1kHz

At 1kHz we have impedance magnitude |Z| = 6.336 ohm, phase = +34° and EPDR = 2.74 ohm. Compared to distortion with 4.7 ohm resistor, the distortion with complex impedance at 1kHz is higher.

The impact of this complex load to frequency response of the amplifier used was negligible.

Conclusion

The simulations show high impact of complex loads to class AB amplifier peak power stress of the output transistors. This is in general valid for all kinds of linear amplifiers. The impact of complex impedances to amplifier distortion is not quite straightforward and may be only guessed from the impedance plots. What might be the issue from the point of view of peak power dissipation does not necessarily reflect in high distortion, in case that output devices are not destroyed.
In case of class D amplifiers, when the output transistors dissipate the power mostly during the switching transitions, it is impossible to draw any conclusions from the data I have posted here. Further investigation will be necessary.

__________________________________________________________________________________________________________________________________________

Several hours later - I have just started to test class D amplifiers with the complex load shown in this thread. First on the bench is AIYIMA A07 and the results are worse than I have expected. The amplifier is unable to handle frequency sweep from 20Hz to 30kHz even at low output voltage 3.5V. It is able to manage the stepped sine frequency test, however. Below are the results, frequency response and distortion vs. frequency. To drive a full range speaker with 1 driver, the amplifier would not be usable.

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Wish someone could show me the FR deviations on class d that are load dependent. With a real inductive load.

I only know V = i x R, from high school. Less R (ohm), less V, easier for the amplifier.

It’s good to know that:
“A high internal voltage difference multiplied by maximum current equals high power, so the power amplifier has to work much harder.’

I only know V = i x R, from high school. Less R (ohm), less V, easier for the amplifier.

It’s good to know that:
“A high internal voltage difference multiplied by maximum current equals high power, so the power amplifier has to work much harder.’
That's perfectly fine for resistance. But any real world circuit has some capacitance and inductance. In fact even an apparently pure resistor can have inductance from its leads and sometimes capacitance from its construction! So surface mount resistors with virtually no leads will be less inductive than a resistor standing up on legs.

If you study high power electricity (10,000 volts and above) and motors and generators (part of my degree), you have to account for reactance all the time. What it means in practice is for a sine wave (e.g. mains) into an inductive load, the voltage leads the current (or the current lags the voltage); for a sine wave (e.g. mains) into an capacitive load the current leads the voltage (or the voltage lags the current. So both waves are not happening at the same time! This is not something that is intuitive!

I only know V = i x R, from high school. Less R (ohm), less V, easier for the amplifier.
Actually it is the other way around. An amplifier is suppossed to be an ideal voltage source, meaning that is should be able to produce the same voltage at its terminals regardless of R. So if R is small that means amp needs to supply more current to generate the same voltage, which means it needs to work harder. That is actually half of the premise of the original post if I understood it correctly, the other half being due to reactance, current and voltage are not always in sync, which means amp needs to supply a lot of current when the voltage is low, which is even more of a problem for the output transistors.

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Thanks a lot for this PMA. I felt like in school again, which is great.

Allow me to carry on one open point from the last discussion please; if we know the output impedance of the amp, can we not calculate how its going to behave for any given load or do you think we need to test amps with various loads to get a realistic understanding of their behavior?

Output resistance is 'complex' and can vary per frequency.
Measuring it can shed some light on expected variances but all speakers are different so there is not 1 single correct 'load' to test with only some standard loads and load examples as used here by PMA.
It should be done under actual load which is not as easy as it looks at higher power levels. Then there is the wiring as well...
But it is always a good thing to measure with more than just 2 or 3 resistive loads.
Maybe there are some 'standard complex loads' around... dunno. My speakers will differ from those anyway.

Output resistance is 'complex' and can vary per frequency.
Measuring it can shed some light on expected variances but all speakers are different so there is not 1 single correct 'load' to test with only some standard loads and load examples as used here by PMA.
It should be done under actual load which is not as easy as it looks at higher power levels. Then there is the wiring as well...
But it is always a good thing to measure with more than just 2 or 3 resistive loads.
Maybe there are some 'standard complex loads' around... dunno. My speakers will differ from those anyway.
I believe the point was (or the piont I took away at least) that if we do a sweep and plot the output of the amp at 4ohm and 8ohm resistive loads, from the graphs generated we can calculate the impedance of the amp, and can know what we need to know about it. Then the issues becomes knowing the speaker better i.e, what kind of a load does it represent. Once those two are known, the behaviour of these components working together can be predicted as long as those behaviour are within the working specs of the components. Is this understanding correct?

You would also need a no-load (or 100ohm load) plot and perhaps even 2ohm load as well.
The output R of an amp is not purely resistive and can react differently with inductive and capacitive loads which real speakers have.
This is what @pma basically showed.
The currents in an amp may not be in phase with the voltage in real life situations which they will be with resistive loads. So while resistive loads are better than no loads they cannot fully 'characterize' an amp.
There is no single 'complex' load that represents the majority of speakers. Which is in essence the crux. But better to have some complex loads than just resistive.

I believe the point was (or the piont I took away at least) that if we do a sweep and plot the output of the amp at 4ohm and 8ohm resistive loads, from the graphs generated we can calculate the impedance of the amp, and can know what we need to know about it. Then the issues becomes knowing the speaker better i.e, what kind of a load does it represent. Once those two are known, the behaviour of these components working together can be predicted as long as those behaviour are within the working specs of the components. Is this understanding correct?

Which I tried to re-summarise here: https://audiosciencereview.com/foru...ments-into-4ohm-2-2uf-load.43162/post-1545663.

If you study high power electricity (10,000 volts and above) and motors and generators (part of my degree), you have to account for reactance all the time. What it means in practice is for a sine wave (e.g. mains) into an inductive load, the voltage leads the current (or the current lags the voltage); for a sine wave (e.g. mains) into an capacitive load the current leads the voltage (or the voltage lags the current. So both waves are not happening at the same time! This is not something that is intuitive!

I'm a chemical engineer, so my coursework re those issues consisted of a few lectures in a heat, electricity, & optics physics course. Would have been nice to have learned more about that before entering industry and wondering why the corporate engineers insisted we undertake an expensive project to install power factor correction systems for 4000 hp motors...and learning that kVA and kW were not the same (definitely not intuitive!!!) It's obviously not just an audio concern.

Good friend of mine got his MSEE in power. Funny story about trying to "roll his own" big caps for an experiment that ended rather explosively.

Thank you @pma! This is the best explanations I have read on this rather complex subject and you cleared up some questions I have had for a long time. My biggest take away is that I thought I didn't have to worry about this stuff because I have an DIY active system with the amps directly driving the speaker drivers and I incorrectly assumed that was an "easy" load.

I believe the point was (or the piont I took away at least) that if we do a sweep and plot the output of the amp at 4ohm and 8ohm resistive loads, from the graphs generated we can calculate the impedance of the amp, and can know what we need to know about it.

The point is that measurements into resistive load only, usually two values here, 4ohm and 8ohm, can show you neither amplifier stress nor distortion with a complex load. The second point is we tend to look at impedance magnitude only and try to predict the results from measurements with resistor loads. This is a big misunderstanding and big mistake.
Measuring it can shed some light on expected variances but all speakers are different so there is not 1 single correct 'load' to test with only some standard loads and load examples as used here by PMA.
Again - you may measure impedance plot of the speaker, which is very simple using LIMP or REW. You get |Z| and Ø plots and for any single frequency point you may substitute the complex impedance just by R + C or R + L, depending on Ø sign. You may choose several important points of the impedance curve and easily prepare a simple circuit that would be a substitute to the complex load at one single frequency. This is important - just at one frequency.
You only need to fulfil 2 conditions:
1) You are able to measure the impedance and read the necessary values from your measurement,
2) You want to do it and you have a will to do it. It is not difficult.
The option is that you may say that it is difficult, that there are so many speaker impedances to be modelled, etc. etc., and you stick with resistor load measurements, trying to persuade everyone that it is enough. Everyone has a choice, hopefully.

The described issue with peak power stress of output transistors of the class B/AB amplifier is universal and applies to any such amplifier. It also shows why it is difficult to design an effective output protection. It clearly shows that the protection based only on output current is insufficient. The current into 4.7 ohm resistor in my example is much higher than that into the complex load (10A peak vs. 6A peak). Nevertheless, peak power into complex load was 230W (exceeding SOA), though into resistor it was only 160W (inside SOA). Complex impedance magnitude at the test frequency was 7.546 ohm compared to 4.7 ohm of the resistor (in fact 4.96 ohm). My point - never believe oversimplified explanations, never stick with primary/secondary school basics. The results would be misleading in the best case. Or completely wrong.

Impedance of the resistor load from post#1 measured with connecting wires

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Wish someone could show me the FR deviations on class d that are load dependent. With a real inductive load.
Like this? With a "real speaker", voltage measured at A07 output. With every speaker you get different plot. Logical result of output LC filters outside FB. But it is quite OT here, now. You may use a search engine and will find more examples.

Like this? With a "real speaker", voltage measured at A07 output. With every speaker you get different plot. Logical result of output LC filters outside FB. But it is quite OT here, now. You may use a search engine and will find more examples.

View attachment 279839

No, i like to see a FR plot of the speaker with a not load dependent amp vs. a load dependent amp. So your AB vs. the A07 would be interesting.

In your plot i see peaking at 8khz. With amirs plots i see peaking at > 20khz. 2db + at 20khz i dont care a shiit. The same at 8khz is at least some difference.

I have no feeling for your plot. Or in other words i dont get it. Thats why it would be much more easy to see the two different FR measured with a mic.

Or gets that peak always shifted down some khz by the inductive load?
Than resistive load measurements are very problematic, couse they give a complete wrong picture about what you hear?

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It has been posited that some amplifiers that rely heavily on negative feedback have rising distortion with frequency and these mechanisms become less effective with reactive loads (at these frequencies).

Is it possible to perform this analysis of a reactive load at higher frequencies where human hearing is more sensitive, for example, 1 or 2 kHz?

- Rich

The point is that measurements into resistive load only, usually two values here, 4ohm and 8ohm, can show you neither amplifier stress nor distortion with a complex load.
True. I was thinking maybe slightly differently. In my understanding there are three factors in play here, characteristics of the amp, the load and their interaction. The point was made that If your 4 ohm and 8 ohm sweeps shows no frequency dependency, meaning that amp output impedance is resistive, then you know you have the best case scenario for the amp wrt varying load impedance. . It does not mean amp will not distort, or behave fine under stress, but it is at least one potential source of non-linearity eliminated.

The second point is we tend to look at impedance magnitude only and try to predict the results from measurements with resistor loads. This is a big misunderstanding and big mistake.
Yes, I agree. I had arguments in this very forum about the impact of reactive loads on amp output characteristics, and am of the opinion that X db best case SINAD does not necessarily mean amp will be transparent under any circumstances. Maybe I am drifting towards the idea that we know, or we can calculate, amps contribution to non-linearities with varying load impedance; what we need to measure and understand better is speakers and their electrical characteristics, which might or might not be part of your point as well.

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Have you run this with a typical 1st or 2nd order LP filter for the case of passive speakers?

Thanks for the post, it's a nice writeup.

Just added measurements of AIYIMA A07 into the complex load to the post #1. It is the answer to the question of @tomtoo how it would work into load with higher inductance and the result is not nice. The amplifier is unable to handle frequency sweep from 20Hz to 30kHz even at low output voltage 3.5V - it gets unstable and turns off. It is able to manage the stepped sine frequency test at low voltage output, however.

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Again - you may measure impedance plot of the speaker, which is very simple using LIMP or REW. You get |Z| and Ø plots and for any single frequency point you may substitute the complex impedance just by R + C or R + L, depending on Ø sign.

Of course it is. The point I was trying to get across is that if you want to test an amp with a complex load (real speaker or equivalent) you basically only test with that particular load. It is A load not a representative load for the majority of speakers. We can't expect to test amps with a bunch of difficult loads just to see how it handles that.

My question is whether or not there is an 'official' complex load that should/could be used to test amps. I am totally cool with your test and simulations with the loads you create for those tests. Those are your simulations tests with your loads.

I mentioned that your simulations and measurements show that simple resistive loads do not tell the whole story. However, it would be handy to have some form of 'standard' difficult load. Unless there is one (such a standard) you are not going to get Amir and others on board.
It looks like you think I am opposing your views and believe that resistive testing is all that is required. I don't. I like to see a standard developed for this. Perhaps even a few 'difficult' loads. I don't think it will happen though despite me thinking it should. The fact is there is no single 'complex' load that would be valid for the majority of speakers out there. One could make some loads that will be very hard for amps but are these realistic ?

No one with any technical background believes all that is needed is a few resistive loads to test with. I like it that you at least make an effort and post that here.

There are very likely amp owners who have had amps die 'inexplicably' thinking their speaker was not such a terrible load while it might well have been. What you showed (and the linked article) is that such things can happen.

My remark about wires is that some people use extremely high capacitive cables that can destabilize amps and do so without the obligatory series inductor. Not about 'normal' cables.
In high-end lala-land there are plenty of idiots around who do crazy stuff like that. These are things that doesn't have to be tested for IMO.

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