Frequency mixer

From Academic Kids

In telecommunication, a mixer is a nonlinear circuit or device that accepts as its input two different frequencies and presents at its output (a) a signal equal in frequency to the sum of the frequencies of the input signals, (b) a signal equal in frequency to the difference between the frequencies of the input signals, and, if they are not filtered out, (c) the original input frequencies.

Source: from Federal Standard 1037C

Mixers have 3 ports. The LO port, RF port and IF port. To achieve the sum of two frequencies, input one signal into the LO port and the other signal into the IF port. The RF port will be the output. To achieve the difference of two frequencies, input one signal into the LO port and the other signal into the RF port. The IF port will be the output.

Mathematical mechanism

The two frequencies that are to be mixed are, in reality, sinusoidal voltage waves. They can be represented as:

<math>v_1 = A_1\sin (2\pi f_1 t)\,<math>

<math>v_2 = A_2\sin (2\pi f_2 t)\,<math>

where

  • <math>v_1, v_2\,<math> represent the two varying voltages
  • <math>A_1, A_2\,<math> represent the respective maximum voltages (amplitudes)
  • <math>f_1, f_2\,<math> represent the two frequencies in Hertz
  • <math>t\,<math> represents time

If we can find a way to multiply these two signals by each other at each instant in time, we could apply the following trigonometric identity:

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

We get:

<math>v_1 \cdot v_2 = \frac{A_1 A_2}{2}\left[\cos(2\pi[f_1-f_2]t)-\cos(2\pi[f_1+f_2]t)\right]\,<math>

So, you can see the sum (<math>f_1 + f_2\,<math>) and difference (<math>f_1 - f_2\,<math>)frequencies as required.

Mathematics of the practicalities

The next question is, how are we going to achieve this multiplication? There are complex circuits that tackle this question with increasing accuracy, but the simplest answer is so simple that it is also worth some analysis. It is to use a forward-biased semiconductor diode.

From the diode page we find that the I-V equation for an ideal diode is:

<math>I=I_S \left( {e^{qV_D \over nkT}-1} \right)\,<math>

From the Taylor series page, we see that we can expand the exponential function as below:

<math>e^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!}\,<math> or

<math>e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \dots\,<math>

Now, we are going to start simplifying things (without forgetting that we have done so!)

First we apply a small voltage to a diode that represents our two sine waves added together: <math>v_1 + v_2\,<math>, then we generate a second voltage proportional to the current that flows through the diode (a simple resistor will do this, according to ohm's law).

According to the Taylor series expansion, the second, output voltage from our diode mixer will be related to the following:

<math>v_o = 1+(v_1+v_2)+\frac{(v_1+v_2)^2}{2!}+\frac{(v_1+v_2)^3}{3!} + \dots\,<math>

The terms represent

  • 1, a DC shift, which we shall ignore
  • The original two signals, which we expected and shall ignore
  • a square-law signal: the square of the sum
  • signals equivalent to the cube and higher powers.

We said this was going to be a small signal, compared to the other voltages around like the 0.6 V forward bias that the diode expects, etc. With that in mind, we are going to ignore all cube and higher power terms too for now.

Also ignoring the constant divisor, the square of the sum term expands out to:

<math>(v_1+v_2)^2 = v_1^2 + 2 v_1 v_2 + v_2^2\,<math>

So, among other things, we have achieved our goal to multiply the two signals: we have <math>2 v_1 v_2\,<math> in there.

Spurious signals

(See Spurious emission)

Now, recalling what we found in the previous section, every multiplication produces sum and difference frequencies. From the first two terms alone we can expect signal at the following frequencies: <math>f_1, f_2, 2f_1, 2f_2, f_1+f_2\,<math> and <math>f_1-f_2\,<math>.

If <math>f_1\,<math> and <math>f_2\,<math> are both large and relatively close in value, then by far the smallest of these will be the last, the frequency difference signal. This is the one that is almost exclusively selected in modern, low cost radio receivers that use this simple mixer technology.

Don't forget also that we ignored the cube and all higher order terms earlier. These will produce a plethora of other high frequencies, and a few not so high. Any of these could slip into or break into the passband of the low-cost filters that would follow this diode mixer and it is these that set the main performance limitations of this approach.da:Elektronisk mikser de:Mischer sv:Mixer (elektronik)

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