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A Mathematical Analysis of Frequency Mixing |
It has been said that the equations describing the mixing of two frequencies in a non-linear circuit are second only to Ohm's Law in importance in electronics.
| Assume the voltage-current characteristic of some non-linear circuit can be described by: | |||
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[1] | i = a + be + ce2 | |
| and that we apply as the input voltage "e" the sum of two voltages, e1 and e2. Then i = a + b(e1 + e2) + c(e1 + e2)2, or expanding: | |||
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[2] | i = a + b(e1 + e2) + c(e12 + e22 + 2e1e2) | |
| Assume the two input voltages to be sinusoidal, as | |||
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[3] | e1 = E1sin(xt) and e2 = E2sin(yt) | |
| where x and y are the two angular velocities. Substituting equation [3] into [2], we have | |||
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[4] | i = a + b[E1sin(xt) + E2sin(yt)] + c[E12sin2(xt) + E22sin2(yt) + 2E1E2sin(xt)sin(yt)] | |
| Equation [4] may be simplified with the use of the trigonometric identities for "sin(x)sin(y)" and "sin2(x)" to give: | |||
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i = a + bE1sin(xt) + bE2sin(yt) + ½cE12[1 − cos(2xt)] + ½cE22[1 − cos(2yt)] + cE1E2[cos(x−y)t − cos(x+y)t] | |
| Now gathering like terms: | |||
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[5] | i = (a + ½cE12 + ½cE22) | {part I} |
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+ bE1sin(xt) | {part II} |
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+ bE2sin(yt) | {part III} |
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− ½cE12cos(2xt) − ½cE22cos(2yt) | {part IV} |
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+ cE1E2cos(x−y)t | {part V} |
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− cE1E2cos(x+y)t | {part VI} |
Note that {part I} is a DC component, {part II} is the original frequency "x", and {part III} is the original frequency "y". {Part IV} consists of the second harmonics of x and y. {Part V} and {part VI} are new frequencies at the sum and difference of x and y. Addition of third-power terms to the original non-linearity expression (Equation [1]) would introduce third-order terms throughout the analysis, such as 3x, 3y, 2x+y, x+2y, 2x−y, and x−2y. Higher-order non-linearities would similarly introduce even higher-order products.
This derivation demonstrates that non-linear circuits can produce harmonic and intermodulation distortion, and can be used as modulators and demodulators. It is proof of the operation of the beat frequency oscillator (BFO) and the product detector. The frequency multipliers often used in transmitter designs depend on the harmonics generated. Superheterodyne receivers and transmitters, frequency synthesizers, and the intercarrier method of television reception depend on the sum and difference frequencies. The separate intelligence sidebands which lead to single sideband and independent sideband transmission are predicted by this analysis.
Derivation and expansion of these equations depend only on noticing that non-linear characteristics can be described as a power series, and on trigonometric identities.
