I have started reading the Self's book on power amp design, and at some point he analyzes a few non linear distortion models, such as the following:
Cubic distorion: Vout = (Vin)^3
Cubic + Linear distortion: Vout = Vin - (Vin)^3
"Stitched" square distortion: Vout = (Vin)^2 * sign(Vin) (to retain negative voltages)
Square root distortion, etc..
For each of those models, he states which harmonics are involved (ie. for the cubic distortion, the 3rd harmonic) and how the amplitude of the generated harmonics changes as a function of the input signal amplitude (what percentage of the output signal is the harmonics). I would like to know how to get these results mathematically. Thanks in advance.
Cubic distorion: Vout = (Vin)^3
Cubic + Linear distortion: Vout = Vin - (Vin)^3
"Stitched" square distortion: Vout = (Vin)^2 * sign(Vin) (to retain negative voltages)
Square root distortion, etc..
For each of those models, he states which harmonics are involved (ie. for the cubic distortion, the 3rd harmonic) and how the amplitude of the generated harmonics changes as a function of the input signal amplitude (what percentage of the output signal is the harmonics). I would like to know how to get these results mathematically. Thanks in advance.
Are the equations you show not the mathematical form you need?
Example: Vout = (Vin)^3 if you have a Vin of 2V, the 3rd harmonic is 2^3 = 8V.
(It is a bad example because normally the equation is Vout = Vin + (x*Vin)^3 where x is a fraction << 1. For Vin = 2V and x = 0.1 the 3rd harmonic would be 0.008V or 0.4% 3rd if my algebra checks out).
Or do I misunderstand you?
Jan
Example: Vout = (Vin)^3 if you have a Vin of 2V, the 3rd harmonic is 2^3 = 8V.
(It is a bad example because normally the equation is Vout = Vin + (x*Vin)^3 where x is a fraction << 1. For Vin = 2V and x = 0.1 the 3rd harmonic would be 0.008V or 0.4% 3rd if my algebra checks out).
Or do I misunderstand you?
Jan
The harmonic distortion is by definition measured with a sine wave input, so assume Vin = A sin(omega t) or Vin = A cos(omega t) and calculate the output. You can use trigoniometric identities or split up the sine or cosine into two complex exponentials, if that makes it easier.
The equations you showed are all static, so you can use any nonzero real value for omega's numerical value, omega = 1 rad/s for example.
The equations you showed are all static, so you can use any nonzero real value for omega's numerical value, omega = 1 rad/s for example.
You have to express f(k sin(x)) in terms of sin(nx) for integer n. Sometimes this its relatively easy algebra (if f() is a polynomial for instance), and sometimes its hard and numeric methods are needed.
In the general case the relative harmonic content depends on the amplitude of the signal (k above).
In the general case the relative harmonic content depends on the amplitude of the signal (k above).
I see what you mean. If an input signal is expressed as Vin = Asin(ωt), then for cubic distortion, Vout = A^3sin^3(ωt).
Using the cubic power reduction formula of sine, we get Vout = A^3(3sin(ωt) - sin(3ωt))/4. This sin(3ωt) term is the 3rd harmonic. With some manipulation we get that Vout = A^3(3/4 sin(ωt) - 1/4sin(3ωt)).
Using the cubic power reduction formula of sine, we get Vout = A^3(3sin(ωt) - sin(3ωt))/4. This sin(3ωt) term is the 3rd harmonic. With some manipulation we get that Vout = A^3(3/4 sin(ωt) - 1/4sin(3ωt)).