# Linearity in Audio, Part Two

### Written by J.I. Agnew

Another common usage of the term “linearity” in electronics refers to linearity in amplification at different levels; this can be called “dynamic linearity.” As an example, let us take an imaginary amplifier, which is meant to have an amplification factor of 10. This means that if we feed a 1 kHz sine wave having an amplitude of 1 volt RMS at its input, we shall expect to see a 1 kHz sine wave having an amplitude of 10 volts RMS at its output, without any additional elements. Before going further, let’s review frequency and phase linearity, which we covered in Part One. Frequency linearity in the 20 Hz – 20 kHz bandwidth would mean that if we feed any frequency from 20 Hz to 20 kHz at 1 volt RMS to the input of our amplifier, we shall see that same frequency at 10 volts RMS at the output. Phase linearity in the 20 Hz – 20 kHz range would require that the phase of a sine wave at any frequency from 20 Hz to 20 kHz would be exactly the same at the output, as it was at the input. (This would imply a linear frequency response over a much wider range than the specified 20 Hz – 20 kHz.) So, frequency linearity looks at the relative amplitude of different frequencies. Phase linearity looks at the relative phase between different frequencies. Dynamic linearity investigates the linearity of the amplification process itself at different amplitudes (signal levels). So, our theoretical amplifier will amplify 1 Vrms into 10 Vrms, 2 Vrms into 20 Vrms, 3 Vrms into 30 Vrms and so on. But can it be expected to continue this relationship between input and output, in a linear fashion, regardless of how high or low we go? If we feed 1,000 volts RMS to the input, will we get 10,000 volts RMS at the output? The answer of course is no. Practical electronic circuits cannot remain linear indefinitely. There are limits imposed, both at high levels and at low levels. Not only will we not get 10,000 Vrms for our 1,000 Vrms input, we also won’t be getting 0.000001 Vrms for our 0.0000001 Vrms! All real-life electronic circuits have noise. This results in the ultimate limit for dynamic linearity capability at low levels. If the signal is lower than the noise floor of the amplifier, then it will remain buried in noise and there will not be any meaningful amplification. On top of this, various non-linear mechanisms in amplifier circuits may prevent linear amplification below a certain point. This would mean that if 1 Vrms becomes 10 Vrms, 0.1 Vrms may only become 0.9 Vrms instead of the expected 1 Vrms that the amplification factor of 10 would produce. At high signal levels, there are power supply voltage, current and bias limitations. The amplifier can only supply so much voltage or current. So, while 1 Vrms becomes 10 Vrms and 3 Vrms becomes 30 Vrms, 4 Vrms may only become 32 Vrms, ruining the linear relationship. A deviation from linearity is any change in the usual amplification factor of the circuit, at different amplitudes (signal levels). This usually results in distortion, which produces additional frequencies within the amplifier that are not present in the input. The type of distortion and the linear range of operation depend on the circuit design and amplification devices used. We shall now leave our theoretical amplifier aside and see how dynamic linearity works in a real triode tube circuit. We will connect the cathode to ground and bias the grid at – 20 VDC to set the operating point. The audio signal fed to the grid will move upwards of – 20 VDC during the positive peaks of the wave and will move lower than – 20 VDC at the negative peaks. As we approach 0 VDC during the positive peaks, the grid starts to attract electrons and draws current. If the circuit supplying the grid with the audio signal cannot deliver this current, the positive peaks of the wave will be compressed and eventually chopped off! This reduces our amplification factor and results in second harmonic distortion (distortion that occurs at twice the fundamental frequency).
Average plate characteristics of an 845 triode from a 1933 RCA data sheet.

“I see music as fluid architecture.” – Joni Mitchell In this issue: Paul McGowan begins a series on the 50th anniversary of PS Audio. Octave Records releases one of its...