Astable Oscillator Circuit with 555 Timer

As we already analyze one simple Astable circuit with 555 Timer, now we will analyze one improved circuit with the same 555 timer. The oscillator circuit is shown on Picture 1. The main difference in this circuit is that resistor R3 now is connected to the trigger pin of the 555 timer and to the output of the circuit via the capacitor C3.

Picture 1: Astable oscillator circuit with 555 Timer

Time-domain analysis

The results of the transient analysis for this circuit in time domain is shown on Picture 2. These results are for the configuration of the circuit as shown on Picture 1. The power supply voltage is Vcc = 5 V DC. Resistor R3 = 270 Ω, R1 = R2 = 500 Ω, while the capacitors are C1 = 100 nF and C2 = C3 = 1 nF. As measured in the simulation, the frequency of the output signal is about f = 1.826 MHz.

Picture 2: Transient analysis - output voltage Vo wave form (time-domain)


The frequency of the output signal of this circuit can be changed, if we change the values of the components of the circuit. For example, we run simulations with three different values for resistors R1 and R2, and we got these results:

>> For R1 = R2 = 500 Ω -> f = 1.826 MHz;
>> For R1 = R2 = 820 Ω -> f = 1.662 MHz;
>> For R1 = R2 = 1 KΩ -> f = 1.632 MHz;


Just to notice here, the model of the NE555 timer circuit used in these simulations is idealised. In practice, this circuit has some upper limit for frequency of operation. This limit depends on the model and technology of the manufactured circuit. Usually, the older circuits had limit below 1 MHz, but the newer 555 Timers can have upper limit frequency of few MHz-s. Above the upper limit the circuit will be unstable.

Modulators and Feedback

There are numerous modulators, and here it is not objective to give an extensive overview, only the basic topologies are discussed.

PDM modulators

PDM modulators have resulted from the digital signal processing domain. In more and more equipment, the signal is available in digital form. For a switching amplifier it must be converted into a 1 bit signal at a high frequency. Sometimes, as with DSD audio data, this is even the native format. The output stage acts as a 1 bit D/A converter. Because the length of each bit is constant, and only the presence or non-presence of a bit is controlled, this is called Pulse Density Modulation (PDM). To convert a multi-bit signal to a 1-bit signal, oversampled noise shaping is used. Picture 1 shows a general noise shaper.

Picture 1: Noise shaper


The input signal Bin(z) has a larger number of bits than Bout(z). (When the input signal is analogue, a similar structure in the analogue domain constitutes a sigma-delta modulator). The block called "Quantizer" reduces the number of bits by simply passing only the most significant bits to Bout(z). The least significant bits, which are the error, are added to the input after passing through a transfer function J(z). It is easy to calculate Bout:

Bout(z) = Bin(z) - ε(z)(1 - J(z))

Suppose J(z) = z-1, one clock delay. The system is now a first-order noise shaper. Bin(z) is a 16 bit signal at 256fs and Bout is a 1 bit signal at 256fs. In that case, the quantizer transfers 1 bit to the output. The other 15 bits are the error signal. Bout equals:

Bout(z) = Bin(z) - ε(z)(1 - z^-1)

With z = e^( 2πj(f/256fs)), we see that for low frequencies (audio) the error in the output signal approaches zero. The error reaches a maximum for f = 128fs. See Picture 2.

Picture 2: Noise distribution as a function of frequency


Applying Bout to a 1 bit D/A converter and filtering above 20 kHz reconstructs the original signal. In the time domain such a noise shaper is a way to convert resolution in the amplitude domain to resolution in the time domain. It outputs bits at high speed in such a way that the average is the intended output (which has a higher amplitude resolution). This way it is also easy to see that although the D/A converter is only 1 bit, it should have a 16 bit accuracy.
To convert the audio signal to 256fs, an oversampling interpolating filter must proceed the noise shaper. A two times oversampling filter works as follows. Suppose the spectrum of the signal sampled at fs looks like Picture 3. This signal is converted to a sampling frequency of 2fs by inserting a sample of value zero after every original sample. See Picture 4. Because every sample is a Dirac pulse of proportional height, the frequency spectrum stays exactly the same.

Picture 3: Spectrum of the signal

Picture 4: Inserting zero samples


Next, the signal is applied to a digital filter at 2fs that filters out the middle replica, see Picture 5. After that, the frequency spectrum of the signal looks exactly like it has been sampled at 2fs. These techniques, oversampling interpolating filtering and noise shaping are essential for all digital PDM systems, although the exact realisation may vary.

Picture 5: Filtering out the middle replica


Assume, the 256 times oversampling for a CD player D/A converter is done in two stages. A four times oversampling filter is followed by a 64 times linear interpolator. The direct use of a 256 times oversampling filter is also possible, but the filter would be very large. A linearly interpolating filter is easier to build, and at 4fs the distortion that it creates has only little effect in the audio band. Then, at 256fs, a second order noise shaper suffices to get a 1 bit signal with 16 bit resolution in the audio band. Unfortunately 256fs = 11MHz which is too high for power switching.
Another possibility is to use only 32fs with an eighth order noise shaper.
Noise shapers with a higher order than three are prone to instability, and it is necessary to manipulate the system when it becomes potentially unstable. Extensive simulations are necessary for evaluation. Even in this case, the switching frequency is 1.4 MHz. The high switching frequencies are a general problem of PDM modulators. Bit-flipping techniques can reduce the average frequency at which the output changes somewhat.

Digital PWM modulators

Digital PWM modulators offer a lower switching frequency than PDM modulators. The Pulse Amplitude Modulated (PAM) samples are converted to PWM. This could be done by giving each pulse a length that is proportional to the original amplitude. However, for CD quality the internal clock frequency would have
to be 2^16 * 44.1 kHz = 2.9 GHz, which is way too high. Furthermore, the frequency spectrum of the PWM signal would not equal that of the PAM signal. This can be calculated, but for a better understanding it is best to realise that natural sampling yields the best results because it does not introduce harmonic distortion. In natural sampling, the audio signal is compared to a triangle or sawtooth waveform (more details below). When we convert a digital PAM signal directly to PWM, it looks as if, looking in the analogue domain, we compared the sawtooth waveform to a step-like representation of the signal instead of the signal itself. This is called uniform sampling. See Picture 6. It introduces harmonic distortion, which depends on many factors including the signal frequency, the switching frequency and the modulation depth.

Picture 6: Natural sampling versus uniform sampling


To approximate natural sampling, linear or higher order interpolation between two or more samples is used to approach the natural PWM pulse width. When the pulse width has been calculated, the sample instant can be the beginning or the end of the pulse (single sided modulation) or the middle (double sided modulation). There are more aspects that deserve attention, but a full discussion of these would be beyond the scope of this article.

Analogue PWM modulators

In the analogue domain a PWM signal can be generated by comparing the audio signal to a triangle or sawtooth waveform. This technique, called natural sampling, is the basis of almost all analogue modulators. See Picture 7. When the momentary value of the input signal is larger than the triangle, the output of the switch is high. It is easy to see that in this way the pulse width at the output is proportional to the input voltage. The modulator does not introduce harmonic distortion, only (multiples of) the carrier frequency and (multiples of) harmonics of the modulating frequency around the carrier.

Picture 7: Open-loop class D modulator


The main problem is the lack of feedback. Output stage inaccuracies, nonlinearities, timing errors and supply voltage variations all contribute to the distortion. We will discuss feedback here, as it is so closely related to the modulator. Picture 8 shows a modulator with feedback. Both inputs to the comparator have triangular waveforms. Picture 9 shows the waveforms for zero and positive output voltage. At zero output voltage, the feedback signal intercepts the reference triangle in such a way that the duty cycle is 50 %. When the output voltage is not zero, the rising and falling slope of the feedback triangle are different, leading to a larger (or smaller) duty cycle.

Picture 8: Modulator with feedback

Picture 9: Signals at the input of the comparator of the feedback modulator


The slew rate of the feedback signal must always be smaller than the slew rate of the reference triangle. Otherwise, the amplifier starts oscillating at a very high frequency. This constitutes a compromise between switching frequency and loop gain. The slew rate requirement can roughly be translated to the demand that the loop gain of the amplifier at the switching frequency is smaller than 0.5. Thanks to the integrator, the open loop frequency transfer of the amplifier is first order, so that the loop gain at a certain frequency has a maximum that is related to the switching frequency. A way to get more loop gain at low (audio) frequencies is by introducing a range with second order frequency response in the loop. As long as the loop gain is back to first order at 0 dB, stability is ensured. This can be done in the modulator by adding a second integrator before the comparator while bypassing it for high frequencies. In practical realisations of a feedback modulator, the triangle is generated by adding a square wave to the input of the integrator. The feedback properties of this type of modulator can also be used when the input signal is generated by a digital modulator. Because in that case the bitstream is already clocked, the negative input of the comparator can be tied to ground. Other techniques, like the one cycle control technique or pulse edge delay error correction, are similar to this modulator in their attempt to control the integral of the switched output voltage.

The high frequency oscillation that occurs in a feedback modulator when the feedback signal is too large, is exploited in the self-oscillating class D modulator. See Picture 10. The comparator is equipped with some hysteresis to control the switching frequency. Other factors that influence the switching frequency are the integrator time constant and the output voltage. For large output voltages, the frequency approaches zero. This can cause aliasing problems that can be overcome by using a comparator with a variable hysteresis dependent on the input voltage. In that way the oscillator frequency is kept constant over a wide range of output voltages.

Picture 10: Self oscillating class D modulator


In the situations above, feedback is successfully taken before the output filter. The combination with feedback after the filter is more troublesome.

Output Filter in Class D Amplifier

The output filter is a low pass filter that reduces the switching frequency. When designing the filter, the load impedance is part of the equation. Thus, the load can seriously affect the frequency transfer. For different loudspeakers, the impedance over the audio range can vary from 1 Ω to as much as 30 Ω, with a phase from +56º to - 67º. Picture 1 shows the impedance of a 3-way loudspeaker system that was used in the listening tests.

Picture 1: Loudspeaker impedance


The output filter, however, is designed for a real and constant load impedance. The result of connecting the loudspeaker is shown in Picture 2. The flat line is the simulated transfer of an ideal class D filter followed by a fourth order Butterworth filter with a corner frequency of 30 kHz, loaded with the specified load impedance of 4 Ω. The other line shows what happens when the loudspeaker is connected. The transfer deviates several dB’s from the flat line. This will colour the sound impression. Another problem is that any non-linearities in the filter show up in the distortion figures.

Picture 2: Simulated class D frequency transfer with resistor and loudspeaker load


Feedback can reduce these problems considerably, but because of the phase shift, feedback after (part of) the filter is complicated. In general, the filter (or the filter in combination with lead compensation) must have a first order frequency transfer at 0 dB to ensure stable operation. This is extra complicated by the connected load, which is a part of the filter. High feedback factors can not be realised and feedback around a filter with more than 2-nd order behaviour is very rare. Even when these problems are overcome, the filter prevents further integration because it contains elements that can not be integrated on chip. For sufficient suppression of the carrier frequency, typically a fourth order filter is necessary. In this case, the amount of filtering in practical situations is limited by parasitic capacitances and resistances. Furthermore, two coils and two capacitors are already considered to be many external components. Using only a second order filter is a solution, but the amount of switching ripple can cause EMI problems and the application area of the amplifier will be limited.

Switching Amplifiers - Class D

Linear amplifiers are amplifiers with a linear output stage, in which there exists a voltage drop across the output transistors to generate the correct output voltage. Even though most of these amplifiers use some sort of switching, they are not to be confused with switching amplifiers. Switching amplifiers are amplifiers with a switching output stage. This means that the transistors in the output stage have a switch function. Any simultaneous occurrence of voltage across and current through these transistors is undesirable.

The class D principle

A typical class D amplifier consists of a modulator that converts an analogue or digital audio signal into a high frequency Pulse Width Modulated (PWM) or Pulse Density Modulated (PDM) signal followed by the output stage, often a half bridge power switch (Picture 1). The output of the switches is either high or low, and changes at a frequency that is much higher than the highest audio frequency. Typical values are between 200 kHz and 500 kHz. The frequency spectrum of the PWM signal in the audio band is the same as the frequency spectrum of the audio signal. An LC filter filters out the high frequency switching components, so that the audio signal is available at the output of the filter. Ideally, the switches do not dissipate and neither does the filter, so the efficiency can be very high.

Picture 1: Principle of PWM amplifier


For a 10 kHz sinewave, a switching frequency of 350 kHz, and a filter with a 30 kHz Butterworth characteristic, the signals look like Picture 2. In this case, the audio frequency is close to the corner frequency of the filter, so some phase shift can be observed between the PWM signal and the audio signal.

Picture 2: Class D output signal (before and after the filter)

Output stage

Picture 3 shows a typical class D output stage. It is a class AD stage, which is used for most class D amplifiers. It is a simple inverter. When the input signal is positive, M2 conducts. When it is negative, M1 conducts.

Picture 3: A typical class D output stage


The diodes D1 and D2 are needed because the transistors are unidirectional switches. Suppose the output signal is positive, and the output current Io is also positive. When M1 is switched on, this is OK, but when M2 is switched on, the coil in the output filter still tries to keep the current Io, forcing the output voltage below -VS, causing D2 to conduct. With DMOS transistors as switches, the intrinsic diodes can be used. However, the intrinsic diode of a DMOS transistor can have a long recovery time (several hundred ns) or cause latch-up. In that case external (shottky) diodes are a solution, although not a desirable one. It is also possible to build DMOS transistors with a fast-recovery intrinsic diode.

Switching speed

High switching speeds are necessary to keep switching losses small. Typical values of today’s integrated designs are tens of nanoseconds. Because of the large gate-source capacitances of M1 and M2, this leads to large peak currents. Also, the high speed switching in combination with wires and (gate) capacitances can cause ringing, overshoot, and delays. For a low distortion it is important that the switching times of M1 and M2 are equal. Tuneable coils between M1 and M2 can provide a solution. However, both the fact that these coils can not be integrated and that they need to be tuned make this an unattractive solution. With high speed switching, the risk of common conduction of M1 and M2 increases. The introduction of a "dead zone" in which both transistors are turned off is a common solution, although this introduces extra distortion in the audio signal. Another option is a handshake procedure to check if the other transistor is turned off.

Power supply

In pure feed-forward systems a stable power supply is extremely important, because any deviation from the nominal value shows up in the output signal. For an output signal of 16 bit accuracy, the power supply should have a 16 bit stability. Common solutions are feedback from the pulsed output or feed-forward correction by referring the triangle waveform to the supply voltage. Another supply issue arises from the use of NMOS devices that are preferable thanks to the lower R on per area. The gate of M1 needs a voltage that is higher than VS. A bootstrap capacitor or a charge-pump can provide such a voltage.

Cross-over distortion

M1 and M2 have a certain Ron resistance. D1 and D2 have a certain voltage drop when conducting. Suppose the output current is positive. During conduction of M1, the voltage will be a little lower than VDD because of Ron1. During conduction of D2, the voltage will be a little lower than -VS due to the voltage drop. So all the time the voltage is lower than it should be. When the output current is negative, the same reasoning shows that the output voltage is too high. This results in crossover distortion. It can be solved by connecting the transistors to a tap of the output inductor or a separate supply voltage.

Class BD output stage

An alternative to the class AD stage is the class BD stage. In class BD there are three possible output voltages: positive, negative, and zero. There are several ways in which this can be implemented, but the simplest one is shown in Picture 4.

Picture 4: Class BD modulator with output filter


In quiescent, the signal at A and B is the same PWM signal with 50 % duty cycle. The signal A-B across the filter is therefore zero. For a positive output voltage, the duty cycle of A is increased and that of B decreased. The difference signal A-B is now a voltage that varies between 0 and VS. Similarly, for negative output voltages, A-B varies between 0 and -VS. The pulse frequency of A-B is doubled compared to A and B, which is favourable for speed requirements. Balanced current design has the same qualities and the topologies are very similar. The difference between a bridge class BD stage and a bridge class AD stage is subtle. The topologies are exactly the same. In the class AD case, however, A is always the inverse of B, so that A-B alternates between -VS and VS.

Resonant output stage

A way to generate the high frequency pulses for a PDM modulator is to use a quasi-resonant converter. This converter gives 1 bit each time it is switched on. The bit is not a squarewave, but the positive half of a sinewave. This is irrelevant, as long as the area under the signal is the same each time. For the topology in Picture 5 (Lf and Cf are the output filter) this is true, virtually independent of output current and voltage. Switching occurs when the current is zero, giving better efficiency and lower switching noise. The large number of filter components make this topology not very attractive for use with integrated circuits.

Picture 5: Quasi-resonating converter

Summary and Conclusions

The design of a class D output stage is not a trivial matter. In general, an output stage will not be able to preserve the exact frequency content of its input signal. To summarise the limitations that were encountered, it is easy to start with an important audio amplifier specification: low distortion. With feedback directly from the switched output, very good high power PWM signals can be generated. The output filter, however, introduces additional distortion and deviations of the specified frequency transfer when non-resistive loads are connected. Feedback after the filter is difficult, and high feedback factors can not be realised. Even when these problems are overcome, the filter prevents further integration because for sufficient suppression of the carrier frequency, typically a fourth order filter is necessary. It is not possible to eliminate the external two coils and two capacitors without introducing a much larger switching residue.

Linear amplifiers have a low complexity, can have a low distortion, but show limited possibilities for reduction of the dissipation. To reduce the dissipation to very low values, complex switching schemes are necessary, and a large number of external elcos makes such a solution little attractive. Switching amplifiers can have a very low dissipation, but suffer from switching noise at the output, an external filter, a load dependent frequency transfer and difficulties in achieving a low distortion. The idea that a mix of these two systems may be beneficial is not new, and there are a lot of possibilities for such combinations.

Class H Amplifier

It is not necessary to use two power supplies like in class G amplifier. Because the signal peaks generally last only a short time, the energy can be supplied by a capacitor. This technique is referred to as Class H (Picture 1).

Picture 1: Class H amplifier



During low output voltages, the switch is in the position as drawn in Picture 1. During signal peaks the switch lifts the lower side of the elco to the power supply, such that the upper output transistor sees a voltage of approximately 2Vdd. The time that the signal is "high" should not be too long. A large elco is required for high power at low frequencies. Switching according to the envelope of the signal, as is sometimes done with class G amplifiers, is riskier as it is impossible to tell how long an envelope will last.


The advantage of class H is that only one power supply is needed. As such it is ideal for car audio applications. To prevent the need for four lifting elcos, it is then built like a bridge amplifier with a signal dependent common mode level. Picture 2 shows a class H bridge amplifier. The common mode level is normally half the supply voltage. When the load voltage must be higher than Vdd, the common mode level of the bridge is increased such that one half of the bridge remains at a constant voltage close to ground and the other half gets the lifted supply voltage. See Picture 3 for the waveforms of the two bridge halves for a sinusoidal output.

Picture 2: Class H Bridge amplifier

Picture 3: Wave forms of both bridge halves for a sinusoidal output


For normal audio signals, and even for a rail-to-rail sinewave, only one lifting circuit would suffice. In practice this is not implemented, because it’s a bad habit to test audio amplifiers with near rail-to-rail squarewaves, which give the lift elco not enough time to recharge.

Class G Amplifier

A class B amplifier has relatively high efficiency for a rail-to-rail sinusoid. This figure is relatively good because the output signal is close to the supply lines a considerable part of the time, with a limited voltage drop across the output transistors. The output signal for audio, however, is close to zero most of the time, with only few excursions to higher levels. Thus the average voltage drop across the output transistors is large, causing the poor efficiency figures for audio.

An amplifier in class G uses multiple supply voltages. At lower power levels, the lower supply voltage is used. When the signal becomes too large for this supply, the higher power supply takes over, and delivers the output power. In this way the average voltage drop across the output transistors is reduced and the overall efficiency can be improved. There are two basic ways in which class G amplifiers are realised. The difference is the way of switching between the supply voltages. Picture 1 shows the upper half of a possible output stage. The upper transistor is switched on during signal peaks increasing the power supply of the lower transistor that controls the output voltage from Vdd1 to Vdd2. Another way to use this circuit is opening the lower transistor totally during signal peaks, giving the higher MOST the role of output transistor.

Picture 1: Serial Class G amplifier



A disadvantage of this circuit is that there are always two elements in series. At low output voltages, the diode decreases the efficiency. During signal peaks the two transistors are in series, so that the output current has to pass two VDS voltage drops. Picture 2 shows a parallel topology that does not suffer from these problems. It needs special precautions in the driver circuitry, however, to prevent high VGS reverse voltages across the upper left output transistor.

Picture 2: Parallel Class G amplifier


In general, the need for multiple supplies may be a problem. If a transformer is used in the power supply, multiple taps are a good solution, but if a car battery is used, it is more problematic. Another problem with this type of amplifiers is the distortion caused by the switching between the two amplifiers. By using a comparator with hysteresis and delay to decide between the supplies, the number of changes can be reduced, but this is a very inelegant way to reduce the total distortion. Another way to limit switching distortion is by switching between the two amplifiers gradually. However, this cuts down the efficiency a little.

Linear Amplifiers Class A, B and AB

Linear amplifiers, are amplifiers with a linear output stage, in which there exists a voltage drop across the output transistors to generate the correct output voltage. Even though most of these amplifiers use some sort of switching, they are not to be confused with switching amplifiers. The output stage of a power amplifier has perhaps the greatest influence on performance and cost. The output stage must also operate at high power levels, often at elevated temperatures, where difficult loads, high voltages, and high currents may exist. Indeed, there is often a trade-off between heat generation and sound quality.

Every real amplifier has some unavoidable limitations on its performance. Some of the main limitations which should be considered are:

>> Limited bandwidth. In particular, for each amplifier there will be an upper frequency beyond which it finds it difficult or impossible to amplify signals.
>> Noise. All electronic devices tend to add some random noise to the signals passing through them, degrading the SNR (signal-to-noise ratio).
>> Limited output voltage, current and power levels. This means that a given amplifier can't provide output signals above a particular level. In other words, there is always a finite limit to the output signal size.
>> Distortion. The actual signal pattern will be altered due to non-linearities in the amplifier.
>> Finite gain. A given amplifier can have a high gain, but this gain can't be infinite, so may not be large enough for a given purpose. That's why multiple amplifiers or stages are often used to achieve a desired overal gain.

The various limitations demands the various changes in the design of the amplifiers. That lead us to the concept of amplifier classes.

Class A, B and AB

The output transistors in a push-pull class A power amplifier remain in conduction throughout the entire cycle of the audio signal, always contributing transconductance to the output stage signal path. In contrast, the output transistors in a class B design remain on for only one-half of the signal cycle. When the output stage is sourcing current to the load, the top transistor is on. When the output stage is sinking current from the load, the bottom transistor is on. There is thus an abrupt transition from the top transistor to the bottom transistor as the output current goes through zero.
The formal definition of classes A and B is in terms of the so-called conduction angle. The conduction angle for class A is 360 degrees (meaning all of the cycle), while that for class B is 180 degrees. More accurately, the definition should really be the angle over which the transistor contributes transconductance to the output stage and signal current to the output. This precludes many so-called nonswitching amplifiers from being called class A. Such amplifiers include bias arrangements that prevent the power transistor from completely turning off when it otherwise would. Most power amplifiers are designed to have some overlap of conduction between the top and bottom output transistors. This smoothes out the crossover region as the output current goes through zero. For small output signal currents, the output transistors are in the overlap zone and the output stage effectively operates in class A. These amplifiers are called class AB amplifiers because they possess some of the characteristics and advantages of both class A amplifiers and class B amplifiers. Most push-pull vacuum tube amplifiers operate in class AB mode. Class AB output stages have a conduction angle that is greater than 180 degrees, although sometimes only slightly so.

Class A

A simple example of a Class A amplifier stage is a common emitter amplifier circuit. Class A amplifiers have the general property that the output device(s) always carry a significant current level, or they have a large quiescent current. The quiescent current is defined as the current level in the amplifier when it is producing an output of zero. The main disadvantage of Class A amplifiers is that current is flowing through the output transistor and its resistor even when there is no signal. Power is being used but no sound or other form of output activity occurs. Such amplifiers are inefficient because they waste 50% of the energy supplied to them. If an amplifier is to produce enough output power to drive a motor or high-wattage speaker, we must design the output stage of the circuit to avoid such waste. The most inefficient amplifier is single ended. More efficient amplifier can be made by employing a double ended or push-pull arrangement. On Picture 1 is shown an example of output stage in push-pull arrangement which works in Class A. This arrangement employs a pair of transistors, one is an NPN, the other is a PNP bipolar transistor.

Picture 1: Push-Pull output stage in Class A


The transistors in this circuit can be controlled using a pair of input voltages, V1 and V2. Therefore, the currents I1 and I2 can be altered independently, by wish. In practice, the easiest way to use the circuit is to set the quiescent current to half the maximum level we except to require for the load. Then adjust the two transistor currents "in oposition". It is the imbalance between the two transistor currents that will pass through the load, so this means the transistors "share" the burden of driving the output load.

Class B

Simply by changing the quiescent current or bias level in class A amplifier and then operating the system slightly differently, we can make another forms of amplifier. The simplest alternative is the Class B arrangement. To illustrate how this work, consider the circuit shown on Picture 2. This arrangement again employs a pair of transistors. However, their bases (or inputs) are now linked by a pair of diodes. The current in the diodes is mainly set by a couple of constant current stages which run a bias current, ibias, through them. If the forward voltage drop across each diode is Vd, then the voltage of the input to the base of the upper transistor is Vin + Vd, while the voltage of the input to the base of the lower transistor is Vin - Vd. Taking into account that the base-emitter junction of a bipolar transistor is essentially a diode, then the voltage drop between the base and the emitter of the transistors will also be Vd, by absolute value. That leads to the very interesting result where the emitter voltages in the circuit shown on Picture 2 will be V1 = V2 = Vin.

Picture 2: Class B output stage amplifier


This result has two implications. First, when Vin = 0, the output voltages will be zero. Since the voltages above and below the emitter resistors RE will both be zero, it follows that there will be no current at all in the output transistors. The quiescent current level is zero and the power dissipated when there is no output is also zero. So, this circuit has perfect efficiency. The second implication is that as Vin is adjusted to produce the signal, the emitter voltages will both tend to follow it. When the load is connected to the output circuit, it will draw current from one or the other transistor, but not from both. When a positive voltage is produced, the upper transistor conducts and draws the current through the load and the lower transistor is Off. On the ther hand, when a negative voltage is produced, the lower transistor conducts and draws the current through the load and the upper transistor is Off. This again means that the system is highly efficient in power terms.

When power efficiency is the main requirement, then Class B is very useful. However, for this circuit to work as explained, it requires the voltage drops across the diodes and the base-emitter junctions of the transistors to be exactly the same. In practice, this is impossible for many reasons. Firstly, no two physical devices are absolutely identical. The diodes and the transistors will have differently doped and manufactured junctions, designed for different purposes. The currents through the transistors is far higher then through the diodes. The transistors will be hotter than the diodes due to the higher power dissipation. When the applied voltage is changed, it takes a time for a PN junction to react and for the current to change. Also, the transistor can't be turned off right away and stop conducting. As a result, the transistors tend to lag behind any swift changes.

The overall result of the above effect is that the Class B arrangement tends to have difficulty whenever the signal waveform changes its polarity and the transistors turns on and off. The result is what is called crossover distortion and this have a very bad effect on small level or high speed waveforms. This problem is enhanced due to non-linearities in the transistors, meaning that the output current and voltage don't vary linearly with the input level. The effect of the crossover distortion is shown on Picture 3. It is proportionately greater in small signals.

Picture 3: Crossover distortion (as the signal swings between positive and negative)


So, the Class A is very power inefficient, while the Class B is far more efficient, but it can lead to signal distortions. The solution is to find a half-way which will take advantages of both arrangements and will minimize the problems. The most common solution is Class AB amplification.

Class AB

The Class AB arrangement can be seen to be very similar to the Class B circuit. In the example shown on Picture 4, it just has an extra pair of diodes. The change that these makes is, when there is no output, there is a potential difference of about 2 x Vd between the emitters of the transistors. As a consequence, there will be a quiescent current of about Iq = Vd/RE, flowing through both transistors when the output is zero. For small output signals (which requires output currents in the range -2Iq < IL < 2Iq), both transistors will conduct and act as a double ended Class A arrangement. For larger signals, one transistor will be off and the other will supply the current required by the load. Hence for large signals the circuit behaves like a Class B amplifier, this mixed mixed behaviour caused this arrangement to be called Class AB.

 Picture 4: Class AB output stage amplifier

Summary

Class A amplifiers employ a high quiescent or bias current, which causes large transistor currents even when the output signal level is small. Therefore, the power efficiency of class A amplifiers is poor, but they can offer good signal performance due to avoiding problems with effects due to low current level nonlinearities causing distortion. Double ended output design is more efficient than a single ended. Class B has a very low or perhaps zero quiescent (bias) current, and hence low power dissipation and optimum power efficiency. Class B may suffer from problems when handling low level signals. That's why the class AB is often the preferred solution in practice.

Amplifier Noise

Although the noise characteristics of a power amplifier are not as critical as those of a preamp, it is still important to achieve low noise because there is no volume control in the power amplifier to reduce noise from the input stage under normal listening conditions. This is particularly so when the amplifiers are used with high-efficiency loudspeakers. Power amplifier noise is usually specified as being so many dB down from either the maximum output power or with respect to 1 W. The former number will be larger by 20 dB for a 100 W amplifier, so it is often the one that manufacturers like to cite. The noise referenced to 1 W into 8 W (or, equivalently 2.83 V RMS) is the one more often measured by reviewers.
The noise specification may be unweighted or weighted. Unweighted noise for an audio power amplifier will typically be specified over a full 20 kHz bandwidth (or more). Weighted noise specifications take into account the ear’s sensitivity to noise in different parts of the frequency spectrum. The most common one used is A weighting, illustrated in Picture 1. Notice that the weighting curve is up about +1.2 dB at 2 kHz, whereas it is down 3 dB at approximately 500 Hz and 10 kHz.

The noise arising from different sources is usually assumed to be uncorrelated. For this reason, it adds on a power basis. This means that noise voltage adds up on an RMS basis as the square root of the sum of the squares of the various sources. Two noise sources each 10 µV RMS will add to 14.1 µV RMS. Two noise sources, one 10 µV and the other 3 µV will sum to 10.44 = µV. This shows how a larger noise source will tend to dominate over a smaller noise source.

Noise Bandwidth

Most noise sources have a flat noise spectral density, meaning that there is the same amount of noise power in each hertz of frequency spectrum. This means that total noise power in a measurement is proportional to the bandwidth of the measurement being made. This gives rise to the concept of noise bandwidth. A perfect brick-wall filter would have a noise bandwidth equal to its signal bandwidth. Because real filters roll-off gradually, the noise bandwidth is slightly different than the 3 dB bandwidth of a filter (often slightly more).

Noise Voltage Density

White noise has equal noise power in each hertz of bandwidth. If the number of hertz is doubled, the noise power will double, but the noise voltage will increase by only 3 dB or a factor of 2. Thus noise voltage increases as the square root of noise bandwidth, and noise voltage is expressed in nanovolts per root hertz nV/sqrt(Hz). There are 141 sqrt(Hz) in a 20 kHz bandwidth. A 100 nV/sqrt(Hz) noise source will produce 14.1 µV RMS in a 20 kHz measurement bandwidth.
As an aside, so-called pink noise has the same noise power in each octave of bandwidth. Pink noise is usually employed in certain test measurements. Pink noise is created by passing white noise through a low-pass filter having a 3 dB per octave roll-off slope.

A-Weighted Noise Specifications

The frequency response of the A-weighting curve is shown on Picture 1. It weights the noise in accordance with the human ear’s perception of noise loudness. The A-weighted noise specification for an amplifier will usually be quite a bit better than the unweighted noise because the A-weighted measurement tends to attenuate noise contributions at higher frequencies and hum contributions at lower frequencies. A very good amplifier might have an unweighted signal-to-noise ratio of –90 dB with respect to 1 W into 8 W, while that same amplifier might have an A-weighted SNR of 105 dB with respect to 1 W. The A-weighted number will sometimes be 10-20 dB better than the unweighted number.

Picture 1: A-weighting frequency response

Power Supply Noise

The power supply rails in any amplifier are often corrupted by numerous sources of noise. These may include random noise and other noises like power supply ripple and EMI and program-dependent noise from the output stage. The power supply noise can get into the signal path as a result of the signal circuit’s limited power supply rejection ratio (PSRR).
There are two important ways to control power supply noise. The first is to do a better job filtering the power supply rails. This is especially effective for power supply rails that provide power to low-level circuits. The second is to employ circuit topologies that have inherently high PSRR. The ability of a circuit to reject power supply noise usually decreases as the frequency of the noise increases. In other words, PSRR degrades at high frequencies. Fortunately, it is often possible to do a more effective job of filtering the power supply rails at higher frequencies.

Resistor Noise

All resistors have noise. This is referred to as Johnson noise or thermal noise. It is the most basic source of noise in electronic circuits. It is most often modeled as a noise voltage source in series with the resistor. The noise power in a resistor is dependent on temperature. It's determined as:

Pn = 4kTB [W]

k - Boltzman’s constant (k = 1.38 × 10^–23 J/°K);
T - temperature in °K (300°K @ 27°C);
B - bandwidth in hertz [Hz];

So, the resistor noise power density per Hertz, at temperature of 27°C (300°K) is pn = 1.66 × 10^–20 [W/Hz].

The open-circuit RMS noise voltage across a resistor of value R is simply:

en = sqrt(4kTRB)

or, en = 0.129 nV/sqrt(Hz) per sqrt(Ω) Noise voltage for a resistor thus increases as the square root of both bandwidth and resistance. A convenient reference is the noise voltage of a 1 kΩ resistor: 4.1 nV/sqrt(Hz). From this the noise voltage of any resistance in any noise bandwidth can be estimated.

BJT (Shot) Noise

Bipolar transistors generate a different kind of noise. This noise is related to current flow and the discreteness of current. This is called shot noise and is associated with the current flows in the collector and the base of the transistor. The collector shot noise current is usually referred back to the base as an equivalent input noise voltage in series with the base. It is referred back to the base as a voltage by dividing it by the transconductance of the transistor. Once again, the resulting input-referred noise is usually measured in nanovolts per root hertz.
The shot noise current is usually stated in picoamperes per root hertz [pA/sqrt(Hz)] and has the RMS value of:

Ishot = sqrt(2qIdcB)

q - 1.6 × 10^–19 Coulombs per electron;
B - bandwidth in Hertz;

It is easily seen that shot noise current increases as the square root of bandwidth and as the square root of current. An 1 mA collector current flow will have a shot noise component of 18 pA/sqrt(Hz).
The transconductance of a BJT operating at 1 mA is 38.5 mS. Dividing the shot noise current by gm we have input-referred noise en = 0.47 nV/sqrt(Hz). According to the equation for en, we can see that this is the voltage noise of a 13 Ω resistor. At the same time, notice that re’ for this transistor is 26 Ω. The noise voltage for a 26 Ω resistor is 0.66 nV/sqrt(Hz). The input-referred voltage noise of a transistor is equal to the Johnson noise of a resistor of half the value of re’. This is a very handy relationship.

JFET Noise

JFET noise results primarily from thermal channel noise. That noise is modeled as an equivalent input resistor rn whose resistance is equal to approximately 0.6/gm. If we model the effect of gm as rs’ (analogous to re’ for a BJT), we have rn = 0.6rs’. This is remarkably similar to the equivalent voltage noise source for a BJT, which is the voltage noise of a resistor whose value is re’/2. The noise of a BJT goes down as the square root of Ic because gm is proportional to Ic, and re’ goes down linearly as well. However, the gm of a JFET increases as the square root of Id. As a result, JFET input voltage noise goes down as the 1/4 power of Id.

An IEC Variant

The main problem with the IEC signal lies in its need for a noise source. Therefore, a new test signal is proposed that is equivalent to the IEC test signal. The noise source is replaced by 24 square waves of equal amplitude, all a factor 2 in frequency apart. In the frequency range from 10 Hz up to 28 kHz, this simulates pink noise, since the energy per octave is constant. This semi pink noise is filtered to get the IEC frequency characteristics. An additional advantage is that the signal, which had only 24 possible amplitude values, now becomes continuous. Only 100 ms of simulation with this IEC variant suffice, since frequency components below 10 Hz are not present. The square waves are easy to define in a circuit simulator, which will speed up simulations. Also, such a signal can easily be generated in hardware with binary counters or with IC’s that are used as tone generators in electronic organs. To see if this IEC variant is indeed equivalent, the dissipation curves for the three amplifier classes were measured, this time for the IEC signal and its variant. The results in Picture 1 show that the dissipations are almost the same at low output powers. At high output powers, the results differ more. Since heat sink temperature measurements on more than one high power music fragment are not available, it remains unclear if this error is the same for all music fragments at that output power. However, the differences are rather small and only occur when the signal is heavily clipping. At lower, more usual output powers, the IEC variant gives good results.

Picture 1: Measured dissipation of three amplifier classes for the IEC signal and the IEC variant

A Simple Periodic Test Signal

The IEC signal is suitable for measuring the efficiency of audio amplifiers, but for prediction purposes it is less ideal. The signal is difficult to generate when the efficiency of an amplifier has to be simulated in a circuit simulator. In circuit simulators, transient noise sources are rarely available, and usually take a long simulation time. Also, for long term testing (reliability), a noise generator is often not available. A simple, periodic test signal would be welcome. The most important quality is a controlled amplitude probability density function. Suppose the signal is V = f(t), and that it is monotonously rising on t∈ [0,t1]. The distribution function is the chance that f(t) is smaller than a certain value V, is:

Picture 1: The distribution function F(V) and The probability density function f(V)


As it can be seen from Picture 1, the probability density function f(V) is the derivative of the distribution function F(V). So if we want to design a signal with a gaussian amplitude probability density function, we know that f(V) is a gaussian curve. Then, f^-1(V) is the integral of a gaussian curve, which is the normal distribution function. Thus, f(t) must be the inverse of the normal distribution function. Picture 2 shows a possible time function.

Picture 2: Signal with a gaussian amplitude distribution

Picture 3: Frequency distribution of a signal with gaussian amplitude distribution (Picture 2), and of the IEC signal


Unfortunately, the corresponding frequency distribution, shown on Picture 3, is not OK. The higher frequencies are relatively weak. Although the period time of the signal could be chosen a little shorter, the frequency distribution can never match that of the IEC signal. Synthesising a signal that also has a controlled frequency distribution is not straightforward. The signal shown on Picture 2 or a 1/√t signal (which has a 1/f power distribution) can be filtered to produce the signal shown on Picture 4. This signal has a correct frequency distribution, but the amplitude distribution is too wide due to much power in the high amplitudes. Also, the frequency distribution alters when the signal clips, so testing for higher output powers is not possible. When the frequency distribution is of prime importance, the signal of Picture 4 can be useful. However, in practice its application will be very limited.

Picture 4: A signal with an IEC 268 frequency distribution

Conclusion

It seems hardly possible to construct a simple periodic test signal that has all the properties we need to simulate music and speech. Often, however, the amplitude distribution is the most important property. Class G amplifiers, for instance, are not very sensitive to the exact frequency of their output signal. In that case, the signal of Picture 2 is advantageous owing to its very short repetition frequency. In a circuit simulator, a single period will suffice to give a good dissipation prediction.

Measuring and Estimating Amplifier Power Dissipation

It is important that the efficiency of audio amplifiers is measured correctly. Good test signals and adequate measurement procedures are crucial to make fair comparisons between amplifiers and reliably predict the dissipation in practical situations. This is also a vital condition for judging the usefulness of new amplifier topologies.

In literature, the efficiency of amplifiers is usually measured with sinusoidal signals. For amplifiers based on a class D topology, this gives approximately the same results as for audio signals, as long as one bears in mind that the average output power of an audio amplifier while playing normal audio signals is much lower than its maximum sine output power. Some high efficiency audio amplifiers, however, need specific audio characteristics to obtain a high efficiency. Well known topologies in this field are the class G and class H principles. The amplifiers which use knowledge about either the amplitude or the frequency distribution of average audio signals, measurements with sinusoids can give pessimistic results.

The best signal would be a real audio signal, but this has several disadvantages. The question is which audio signal should be taken. Speech? Music? What kind of music? This is not standardised. Furthermore, at least several seconds of audio are necessary to get a good impression, which is not very practical for simulations. Also, a music signal does not give stable readings on meters. In practice, more creative ways were found. Either the efficiency was measured indirectly by measuring heat sink temperatures, or an ad hoc measure is defined. Another possibility is to use the IEC-268 "simulated programme material". The spectral distributions of programme material were measured, and the latter also investigated whether the IEC test signal is useful for evaluating the power rating of loudspeakers. There is, however, no standard test signal intended for measuring or predicting amplifier efficiency. However, we can try to find such a signal. For that reason, please refer to the articles Characteristics of Audio Signals and The IEC-268 Test Signal.

Completeness

Despite the good characteristics of the IEC signal, one can still wonder if these characteristics are complete, do they fully determine amplifier dissipation? To answer this, three amplifiers were built: a standard class AB amplifier, a class H amplifier (an amplifier that lifts the power supply during signal peaks by means of an electrolytic capacitor), and a class D + AB amplifier (an amplifier that has a class AB and a class D amplifier in parallel). The class AB amplifier is only sensitive to the amplitude distribution of its output signal, the frequency is not important. The class H amplifier dissipation is "to some extent" frequency dependent, because charging and discharging the capacitor is not lossless, so the total dissipation of this amplifier depends on both the volume and the frequency of the output signal. Finally, the class D + AB amplifier is also sensitive to both characteristics. The class AB part in this amplifier has to support the output current for high d(Iout)/dt, starting at 1 kHz full scale signal swing. All amplifiers have a maximum output power of 30 W, and have identical heat sinks.

Input to the amplifiers are both the IEC signal and a music fragment that is selected because it has almost identical characteristics (a fragment of "Me and Bobby McGee" by Janis Joplin). Measured are the heat sink temperatures as a function of time of all amplifiers. The results are depicted in Picture 1. The average output power was 2 W, at which the amplifiers were clipping a negligible part of the time. The difference in dissipation between the two signals is insignificant. When the average output power is increased to 10 W, the music and the test signal are clipping a considerable part of the time. Even then, there is hardly any difference between the two, as is shown in Picture 2. The differences that do occur can be explained by measurement inaccuracies or slight differences between the amplitude distributions.

Picture 1: Heat sink temperatures for three amplifier classes and two signals at an average output power of 2 W (no clipping)

Picture 2: Heat sink temperatures for three amplifier classes and two signals at an average output power of 10 W (heavy clipping)


With these results it seems that the amplitude and frequency characteristics fully determine amplifier dissipation, also under clipping conditions. Thus we can trust that the dissipation of audio fragments with the same characteristics as the IEC signal will also cause the same dissipation.

Accuracy

Although the IEC characteristics are a good average, individual fragments can have characteristics that are quite different. The question arises if these fragments produce amplifier dissipations that are also significantly different. To answer this question, it is necessary to measure the dissipation of the three amplifier classes for all audio fragments. Direct measurement of amplifier efficiency for audio signals, however, is difficult. One possibility is measuring the heat sink temperature, as was done in the previous section. This requires a constant ambient temperature and is very time consuming. Another (complicated) possibility is sampling the output voltage and the supply current, and calculate the dissipation. To circumvent these drawbacks, behavioural models of the amplifiers are used, and the dissipation is simulated with C programs, evaluating the dissipated energy per audio sample. With the proper models, it is easy to calculate the dissipations for the various audio fragments. The models were developed with the IEC signal measurement results as reference. To demonstrate the validity for real audio signals, Picture 3 shows the simulated dissipation for both the IEC signal and the fragment of Janis Joplin. The dissipations are practically the same, as they should be. Furthermore, the ratios between amplifier dissipations at 2 W and 10 W deviate less than 15 % from the ratios of the extrapolated increase in heat sink temperatures of Picture 1 and Picture 2.

Picture 3: Simulated dissipation of three amplifier classes (for the IEC test signal and a fragment of Janis Joplin)


After all audio fragments were scaled to equal power, the dissipation they caused was calculated for all amplifier classes. Picture 4 shows the results as a histogram. It has a logarithmic x-axis. The distance between the left border and the right border of each bar is a factor 1.05. The height of the bar indicates how many audio fragments cause a dissipation in that range. The vertical lines indicate the dissipation for the IEC signal. It appears that all fragments have dissipations within +/- 20 % of the dissipation predicted by the IEC test signal. One fragment stands out because it causes a high dissipation in both the class D + AB and the class H amplifier. The large high frequency contents decreases the efficiency of the two amplifiers. Although this is an exceptional case, it is important to realise that the good predictive qualities of the IEC signal might not be valid for an amplifier which is more sensitive to the frequency contents of its input signal. In general, however, the IEC signal is representative for a wide range of audio signals.

Picture 4: Histogram of the simulated dissipation of all audio fragments in 3 amplifier classes (Vertical lines indicate the dissipation for the IEC signal)

Conclusions

For the tested types of high-efficiency amplifiers: a class AB, a class H, and a class D + AB amplifier. The power, the amplitude distribution and the frequency distribution of the output signal fully determine the amplifier’s dissipation. The Peak-to-Average ratio of the signal is not very significant.
The dissipation for a variety of real-life audio signals of constant volume deviates only 20 % from the dissipation caused by the IEC 268 test signal at the same output power. Therefore, this signal is very suitable for measuring audio amplifier efficiency. This must be verified for new amplifiers types, that may be more sensitive to amplitude or frequency distribution deviations.
Two alternative test signals are proposed. For simulation and test purposes, a simple test signal can be used for amplifiers with near frequency independent dissipation (A Simple Periodic Test Signal). When the frequency contents is also important, an IEC look-alike test signal can be used which has the same characteristics as the IEC signal (An IEC Variant), but is easier to generate in simulation and hardware.

The IEC-268 Test Signal

The International Electrotechnical Commission (IEC) has defined a noise input signal representative for normal programme material. It is generated by a pink or white noise source followed by a filter. We will refer to this signal as the "IEC signal", and investigate if it is useful for efficiency measurements (more of this in next articles).

Characteristics

Picture 1 shows that the amplitude distribution of the IEC signal is gaussian.

Picture 1: Amplitude distribution of the IEC-268 test signal and a gaussian curve as reference


Picture 2 shows the IEC signal frequency distribution, together with the distribution of the fragments (discussed in Characteristics of Audio Signals). The IEC signal serves well as a typical audio fragment.

Picture 2: Frequency distribution of the fragments and of the IEC test signal (fat line)

Characteristics of Audio Signals

The test set

In order to compare test signals to realistic audio signals, it is necessary to define a test set of audio fragments. Due to the variation in volume in audio signals, the statistical parameters depend on the length of the time interval that is being analysed. Picture 1 shows the amplitude distribution of complete CD tracks. Compared to shorter fragments with constant volume (see Picture 2), we notice a somewhat larger spread and a clearly different shape which peaks around zero amplitude. This is a result of the sections with a lower volume. Now suppose we would use the distributions of Picture 1 to predict amplifier dissipation. Such a signal has a certain average power that has to be delivered by the amplifier, leading to a certain (predicted) average dissipation. During the loud passages, however, the amplifier has to deliver considerably more power, and when they last longer than the heat sink’s thermal time constant, the amplifier will overheat. Therefore, we have chosen audio fragments with constant volume. Of course it should be noted that "constant" is a relative measure, since the audio waveform itself is not constant. It is assumed that variations in less than seconds will not give rise to the problems described above.
There are chosen 80 fragments from various CD’s, including classical music, pop music, jazz, hard rock, house, heavily compressed music, and speech signals. The length of each fragment is between 3 and 12 s. The volume during each fragment is constant. All fragments were converted to mono and normalised to full scale, with the highest sample just clipping. The number of bits per sample was reduced to 8 to get smoother amplitude distributions. Because the fragments are normalised to full scale, this barely affects the sound impression.

Picture 1: Amplitude distributions (of 36 CD tracks, normalised to 1 at zero amplitude and then scaled to equal power)

Amplitude distribution

The amplitude distribution is determined by counting how many samples with a certain amplitude (28 = 256 levels) occur in one fragment. Picture 2 shows the amplitude distribution of all 80 fragments.

Picture 2: Amplitude distributions (of all fragments, normalised to 1 at zero amplitude and then scaled to equal power)


It confirms that the shape of the amplitude distribution is gaussian. There are a few exceptions, though. Firstly, one curve has two peaks symmetrically around zero amplitude. This is the distribution of a fragment hard-core house music, that contains purely synthesised sounds. Although this is an exceptional case, it shows the importance of realising that certain audio characteristics can differ significantly from the average case. Secondly, we see some very narrow curves. These are the distributions of speech signals. Due to the pauses inherent to spoken word, the distributions peak around zero amplitude.
When discussing amplitude distributions, it is useful to critically examine the Peak-to-Average Ratio (PAR). It is widely acknowledged as a signal property, and identical to the traditional crest factor. Expressed in dB’s, the PAR is defined as:

PAR = 20*log(U(t)max/URMS)


Picture 3 shows the PAR-s of all fragments. Roughly, it is between 10 dB and 20 dB, with an average of 15 dB. This means that "in order to be undistorted" the average audio fragment must have a power at least 12 dB below a full power sinewave.

Picture 3: Peak-to-Average ratios (of all fragments)


Often, the PAR is also used for calculating amplifier efficiencies, resulting in a certain efficiency for a certain PAR of the signal. In that case it is assumed that every fragment is amplified to a level just below clipping. The result is that the amplifier dissipation strongly depends on the PAR. The reason for this is, that the average power (or URMS) also varies considerably, since U(t)max is the clipping point of the amplifier and therefore constant. In Picture 2, however, it can be seen that, when scaled to equal power, the amplitude distributions are almost the same. U(t)max varies, but since the high amplitudes near U(t)max are unlikely to occur, they hardly effect the total dissipation of the amplifier. When a fragment with a large PAR is amplified to equal power as a fragment with a low PAR, there will be some clipping, but this is barely perceptible in normal listening conditions. Only when we increase the volume a lot, the sound quality degrades. Subjective listening tests show that the PAR can be made as small as 6dB before most fragments sound really bad through clipping. A PAR of 6dB means that the output power is half the maximum sine power. From the above we conclude the following: Audio fragments of constant volume generally have a gaussian amplitude distribution with an average PAR of 15dB. Concerning amplifier dissipation, average power is the most important variable, while the PAR does not play a significant role. Amplifier dissipation for gaussian signals must be tested up to half the full sine power.

Frequency distribution

On the same audio fragments, a Fast Fourier Transform (FFT) was performed over the full length. A normal log-log bode plot of the frequency content (Picture 4) does not provide very useful information.

Picture 4: Traditional graph of a Fourier transform of a music fragment (Vertical scale dB’s are relative to full scale for measurement bandwidth 2/Tfragment)


Firstly, there is no need for a high accuracy, so it seems more logical to choose the vertical scale of the plot linear instead of logarithmic. Secondly, efficiency is a matter of power. When an amplifier has a better efficiency for certain frequencies, it is important to know how much power is present in those frequencies, not how much amplitude. So it’s more useful to square the amplitudes. Finally, the squared FFT gives the power of the frequencies in the signal. The frequencies are linearly spaced. With a logarithmic frequency axis, a temptation exists to overemphasise the lower frequencies because they are relatively enlarged. A linear frequency axis might seem a logical choice, but since pitch perception is logarithmic in nature (every octave higher equals a factor two), it is preferable to use a logarithmic axis, and plot the sum of the squared Fourier coefficients. An extra advantage is that the summation smoothens the curve.
Presented in this way, the frequency distribution is a line that starts at (almost) power = 0 at 20 Hz, climbing to power = 1 at 20 kHz. The frequency distributions of all fragments are shown on Picture 5. The average fragment is S-shaped, with a mid-frequency part corresponding to a straight line between (50 Hz, 0) and (3 kHz, 1). This does not come as a surprise when we realise that the notes in a musical scale are fixed factors in frequency apart, in which case a linear frequency distribution requires all notes to be equally loud. In Picture 5, the fragments with much power in the lower frequencies have a house beat or a contrabass. The fragments with much power in the higher frequencies mostly have electric guitars or synthesisers. One fragment in particular stands out because it contains much more high frequencies than the others. It is the intro of Melissa Etheridge’s "Like the way I do", containing a guitar and a tambourine.

Picture 5: Frequency distribution (of all audio fragments)