Diode

A diode can be used to create a simple unbalanced mixer. This type of mixer produces the original frequencies as well as their sum and their difference. The important property of the diode is its non-linearity (or non-Ohmic behavior), which means its response (current) is not proportional to its input (voltage). The diode does not reproduce the frequencies of its driving voltage in the current through it, which allows the desired frequency manipulation. The current ${\displaystyle I}$ through an ideal diode as a function of the voltage ${\displaystyle V_{D}}$ across it is given by

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

where the important property of non-linearity results from ${\displaystyle V_{D}}$ being in ${\displaystyle e}$'s exponent. The exponential can be expanded as

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$

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

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

Suppose that the sum of the two input signals ${\displaystyle v_{1}+v_{2}}$ 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

${\displaystyle v_{\mathrm {o} }=(v_{1}+v_{2})+{\frac {1}{2}}(v_{1}+v_{2})^{2}+\dots }$

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 ${\displaystyle (v_{1}+v_{2})^{2}=v_{1}^{2}+2v_{1}v_{2}+v_{2}^{2}}$, 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.

Suppose that two input sinusoids of different frequencies are fed into the diode, such that ${\displaystyle v_{1}=\sin at}$ and ${\displaystyle v_{2}=\sin bt}$. The signal ${\displaystyle V_{0}}$ becomes:

${\displaystyle v_{\mathrm {o} }=(\sin at+\sin bt)+{\frac {1}{2}}(\sin at+\sin bt)^{2}+\dots }$

Expanding the square term yields:

${\displaystyle v_{\mathrm {o} }=(\sin at+\sin bt)+{\frac {1}{2}}(\sin ^{2}at+2\sin at\cdot \sin bt+\sin ^{2}bt)+\dots }$

Ignoring all terms except for the ${\displaystyle \sin at\sin bt}$ term and utilizing the prosthaphaeresis (product to sum) identity,

${\displaystyle \sin a\sin b={\frac {\cos(a-b)-\cos(a+b)}{2}}}$

yields,

${\displaystyle v_{\mathrm {o} }=\cos((a-b)t)-\cos((a+b)t)+\dots }$

demonstrating how new frequencies are created from the mixer.

Switching

Another form of mixer operates by switching, which is equivalent to multiplication of an input signal by a square wave. In a double-balanced mixer, the (smaller) input signal is alternately inverted or non inverted according to the phase of the local oscillator (LO). That is, the input signal is effectively multiplied by a square wave that alternates between +1 and -1 at the LO rate.

In a single-balanced switching mixer, the input signal is alternately passed or blocked. The input signal is thus effectively multiplied by a square wave that alternates between 0 and +1. This results in frequency components of the input signal being present in the output together with the product,^[4] since the multiplying signal can be viewed as a square wave with a DC offset (i.e. a zero frequency component).

The aim of a switching mixer is to achieve the linear operation by means of hard switching, driven by the local oscillator. In the frequency domain, the switching mixer operation leads to the usual sum and difference frequencies, but also to further terms e.g. ±3f_LO, ±5f_LO, 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 is because the switching diodes or transistors act either like a small resistor (switch closed) or large resistor (switch open), and in both cases only a minimal 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, while they mean switching mixers.

The mixer circuit can be used not only to shift the frequency of an input signal as in a receiver, but also as a product detector, modulator, phase detector or frequency multiplier.^[5] For example, a communications receiver might contain two mixer stages for conversion of the input signal to an intermediate frequency and another mixer employed as a detector for demodulation of the signal.