Frequency mixer

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This article is about non-linear mixing operating in the frequency domain. For other types of mixers, see electronic mixer.

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Frequency Mixer Symbol.

In electronics a mixer is a nonlinear electrical circuit that creates new frequencies (called heterodynes) from two signals applied to it. In its most common application, two signals at frequencies f₁ and f₂ are applied to a mixer, and it produces new signals at the sum f₁ + f₂ and difference f₁ - f₂ of the original frequencies. Other frequency components may also be produced in a practical frequency mixer.

Mixers are widely used to shift signals from one frequency range to another, for convenience in transmission or further signal processing. For example, a key component of a superheterodyne radio receiver is a mixer used to move received signals to a common intermediate frequency. Frequency mixers are also used to modulate a carrier frequency in radio transmitters.

Nonlinear electronic components that are used as mixers include diodes, transistors biased near cutoff, and at lower frequencies, analog multipliers. Ferromagnetic-core inductors driven into saturation have also been used. In nonlinear optics, crystals with nonlinear characteristics are used to mix two frequencies of laser light to create optical heterodynes.

[edit] Mathematical description

The input signals are, in the simplest case, sinusoidal voltage waves, representable as

$v_i(t) = A_i \sin 2\pi f_i t\,$

where each A is an amplitude, each f is a frequency, and t represents time. (In reality even such simple waves can have various phases, but that is not relevant here.) One common approach for adding and subtracting the frequencies is to multiply the two signals; using the trigonometric identity

$\sin(A) \cdot \sin(B) \equiv \frac{1}{2}\left[\cos(A-B)-\cos(A+B)\right]$

we have

$v_1(t)v_2(t) = \frac{A_1 A_2}{2}\left[\cos 2\pi(f_1-f_2)t-\cos 2\pi(f_1+f_2)t\right]$

where the sum ( $f 1 + f 2$ ) and difference ( $f 1 - f 2$ ) frequencies appear. This is the inverse of the production of acoustic beats.

[edit] Multiplication implementation

There are various ways of multiplying voltages, many of them quite sophisticated. However, as an example, a simple technique involving a diode can be described. The importance of the diode is that it is non-linear (or non-Ohmic), which means its response (current) is not proportional to its input (voltage). The diode therefore does not reproduce the frequencies of its driving voltage in the current through it, which allows the desired frequency manipulation. Certain other non-linear devices could be utilized similarly.

The current I through an ideal diode as a function of the voltage V across it is given by

$I=I_\mathrm{S} \left( e^{qV_\mathrm{D} \over nkT}-1 \right)$

where what is important is that V appears in e's exponent. The exponential can be expanded as

$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$

and can be approximated for small x (that is, small voltages) by the first few terms of that series:

$e^x-1\approx x + \frac{x^2}{2}$

Suppose that the sum of the two input signals $v 1 + v 2$ is applied to a diode, and that an output voltage is generated that is proportional to the current through the diode (perhaps by providing the voltage that is present across a resistor in series with the diode). Then, disregarding the constants in the diode equation, the output voltage will have the form

$v_\mathrm{o} = (v_1+v_2)+\frac12 (v_1+v_2)^2 + \dots$

The first term on the right is the original two signals, as expected, followed by the square of the sum, which can be rewritten as $(v_1+v_2)^2 = v_1^2 + 2 v_1 v_2 + v_2^2$ , where the multiplied signal is obvious. The ellipsis represents all the higher powers of the sum which we assume to be negligible for small signals.

[edit] Output

As every multiplication produces sum and difference frequencies, from the quadratic term of the series we expect to find signals at frequencies $2 f 1$ and $2 f 2$ from $v_1^2$ and $v_2^2$ , and $f 1 + f 2$ and $f 1 - f 2$ from the $v 1 v 2$ term. Often $f_1,f_2\gg|f_1-f_2|$ , so the difference signal has a much lower frequency than the others; extracting this distinct signal is often the principal purpose of using a mixer in such devices as radio receivers.

The other terms of the series give rise to a number of other, weaker signals at various frequencies which act as noise for the desired signal; they may be filtered out downstream to an extent, but sensitive applications will require cleaner output and thus a more complicated design.

[edit] Switching

Another form of mixer operates by switching, with the smaller input signal being passed inverted or uninverted according to the phase of the local oscillator (LO). This would be typical of the normal operating mode of a packaged double balanced mixer module such as an SBL-1, with the local oscillator drive considerably higher than the signal amplitude.

The aim of a switching mixer is to achieve linear operation over the signal level, and hard switching driven by the local oscillator. Mathematically the switching mixer is not much different from a multiplying mixer, just because instead of the LO sine wave term we would use the signum function. In the frequency domain the switching mixer operation leads to the usual sum and difference frequencies, but also to futher terms e.g. +-3*fLO, +-5*fLO, etc. The advantage of a switching mixer is that it can achieve - with the same effort - a lower noise figure (NF) and larger conversion gain. This come because the switching diodes or transistors act either like a low resistor (switch closed) or large resistor (switch open) and in both cases only minimum noise is added. From the circuit perspective many multiplying mixers can be used as switching mixers, just by increasing the LO amplitude. So RF engineers simply talk about mixers, and mean switching mixers.

[edit] Audio mixer

An audio mixer is a device that linearly combines two or more audio signal sources into a common output signal. It is not intended to produce any sum or difference signals, which would cause distortion of the audio result.

[edit] See also

[edit] External links

Wikimedia Commons has media related to: Radio electronic diagrams

This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".

Frequency mixer

Contents

[edit] Mathematical description

[edit] Multiplication implementation

[edit] Output

[edit] Switching

[edit] Audio mixer

[edit] See also

[edit] External links

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