Why a mixer to combine two frequencies?

Here's something I don't understand: let's say I want to combine two signals of different frequencies and derive a beat frequency from them. (Just like in a superhet radio receiver where the tuned RF signal is reduced down to IF using a local oscillator.)

I don't see why I need a mixer to do this. It seems to me that a simple circuit like this would do the trick, by simply combining the two signals:


Would this not work? I look at it this way: instead of electronic signals, I think about sounds of two different frequencies that combine, with no mechanism other than the transmission medium, the air. In this case we can clearly hear the two beat frequency tones, one subtractive, the other additive, and extract them if we want to. Why wouldn't this work with electrical signals? Won't they simply cancel and reinforce each other to produce the two beat frequency signals that we can then extract (using the LC filter shown here)?

I'm not sure why we need a "mixer" circuit to do this. From my reading I understand that there's something about mixers operating in a non-linear way that makes them work, but I'm not sure why. (I'm not building anything using this; just curious about general principles.)

Adding (or subtracting) two different frequencies does not give the sum and difference frequencies.
You just end up with the two frequencies together.
Do the math and you will see.
To get the sum and difference frequencies you need to multiply them together and that requires a non-linear (mixer) circuit.
Again, do the math.

If you can hear sum or difference frequencies than that means there is a non-linear process involved.
What are the conditions where you "can clearly hear the two beat frequency tones, one subtractive, the other additive..." and how do you "extract them".

Seconding what crutshchow said, when you have two sinusoidal voltages and simply add them together, you simply don't get any new components at any other frequencies. Only when you combine the two while also distorting them do you create new frequencies. It takes some non-linearity to cause the distortion. Here's another way to think about it...when you hear two different tones of sound with your ears and if those tones are pure and they are not too loud, you only hear those tones, you don't hear anything else, right? Take the alternative to an extreme, if simply adding two tones together (again, if the tones are pure and if the volume is not too loud) were to create a sum and difference, how could you possibly listen to any music and not have it sound completely polluted with many many sums and differences? You don't because at reasonable volumes, your ear is operating linearly and not distorting the sound. If the volume is cranked up very high, the speakers probably begin to operate non-linearly and your ears also start to be non-linear and then you hear harmonics as well as sum and difference products, and the tones (or the music) become a real mess. Its pretty obvious. Why? Well, I was taught this effect by using mathematics to show how when you add one sin(x) plus a second sin(y) all you get is sin(x)+sin(y). Nothing creates sin(x-y) or sin(x+y). Now, if you multiply sin(x) times sin(y) the result is very different, and you have sin(x), sin(y), sin(x+y) and sin(x-y) all in the result. A mixer (or any non-linear circuit) does this multiplication. Again, its easiest to see if you do the math, just as crutschow says.

A lot of confusion arises over the term 'mixer' - as it applies to two completely different techniques.

In this case it's actually a 'multiplier' for RF mixers, but for audio mixers it's an 'adder'.

There's also mixers for cakes, and concrete!

As well as Social ones.

Your extremely simple "circuit" is shorting together the outputs of where they came from which might kill one or both signals and/or damage the outputs. A adder can be made with a current-limiting resistor in series with each signal then connect the free ends of the resistors together to make the resulting added signal.

crutschow said:

Adding (or subtracting) two different frequencies does not give the sum and difference frequencies.
You just end up with the two frequencies together.
Do the math and you will see.
To get the sum and difference frequencies you need to multiply them together and that requires a non-linear (mixer) circuit.
Again, do the math.

Click to expand...

I'd like to be able to (do the math). I'm quite willing to be shown it--you know, sine, cosine, omega, all that. Actually, that would be the best way for me to understand this concept.

I'm confused again by something you said. Why would you need to multiply the frequencies? Wouldn't you be adding and subtracting them?

So far as sound is concerned, I'm pretty sure I've heard beat frequencies resulting from two tones being played together, but I can't point to anything concrete or specific.

carbonzit said:

Why would you need to multiply the frequencies? Wouldn't you be adding and subtracting them?

Click to expand...

Absolute not.
As previously stated, adding and subtracting does not mathematically (or in reality) generate new frequencies.
If you put two different frequencies into an opamp adder circuit you simply end up with the two frequencies combined in the single output.
No new frequencies are formed.
If new frequencies were generated (called IM distortion in amplifiers) it would be impossible to record and store music obtained from different studio microphones as is commonly done.

You have to multiply them to get the sum and difference frequencies.
Perhaps this article will help.

crutschow said:

You have to multiply them to get the sum and difference frequencies.
Perhaps this article will help.

Click to expand...

Thanks. I'll study up on that. (Unless I can find a better explanation somewhere else.)

Just to clarify things, although the signals have frequencies, the addition or multiplication is of the instantaneous voltages of the signals.

Even distortion like clipping the signal(s) will give frequency-mixing effects - an overdriven guitar amp input (or distortion/overdrive/fuzz pedal) does just that.

A practical demonstration.

Connecting two signal sources to a display, using three 100 Ohm resistors:

On the scope, we see this:


A 100kHz sinewave superimposed on a higher frequency signal which the scope shows as a band of signal.

But looking on the spectrum analyser, we see:


Just the 1MHz signal and the 100kHz signal.
(And the image of the 100kHz signal, which is just there because of the way the analyser works and is set-up).

But if we change the resistive network for a diode ring mixer (OK so it is a home made lash-up based on an SBL-1, that I made some time ago in the hope that it would give me some good results at low frequencies. All in all it was not a great success.):


On the scope we see this:


The typical output of a balanced mixer.
If the balance was better, the zero crossing points would be sharp nulls.
Note that this is NOT an conventional Amplitude Modulated signal, it is a Double SideBand Suppressed Carrier signal.

Looking on the spectrum analyser:


We do not see the 100kHz signal, that does not pass through the balance mixer.
The 1MHz carrier is 30dB down on the two side frequencies at 0.90 Mhz and 1.1 Mhz.
If the mixer balance was better, the 1Mhz signal could be supressed down into the noise.


JimB

that's a good demonstration. a "balanced mixer", whether a diode ring or multiplier will give suppression of the two original signals. a single-ended mixer, such as an RF mixer used in many radios and receivers must have the two original signals, and the unwanted sum or difference signal (in the frequency domain) filtered out. a simple analog "mixer" such as the two resistors, or using an op amp with a resistive summing node on the input, really should be called a summing circuit, or an "analog adder". the term "mixer" though is quite common, especially in audio. in audio, the balanced mixer or multiplier is often called a "ring modulator" to keep it from being confused with a "mixer". a common audio use of a multiplier is for speech inversion scrambling, which multiplies the signals from an audio input with an oscillator (usually about 3khz) which creates sum and difference frequencies of the input speech and the oscillator, and the high sideband filtered out. since low frequencies are further from the 3khz frequency of the oscillator, and high frequencies closer to it, the spectrum of the speech is inverted. at the receiving end, passing the inverted speech through an identical modulator and filter restores the original audio. back in the 1980s, some ship-to-shore HF stations used speech inversion scrambling with a twist, the oscillator was frequency modulated by rectifying the envelope of the speech, and using the envelope signal to wobble the oscillator. this provided a bit more privacy on ship-to-shore phone calls, but anybody with an analog music synthesizer (the patch cord type, not the later types that were all pre-programmed) could patch together a working descrambler in less than a minute...

edit: i was going to ask if anybody knew where i could find an EMC-1 synthesizer, but i just realized i could probably put a demo of that scrambler system together using GnuRadio...

OK, I think I'm getting this, at least a little. The "mixing" (actually the creation of sum & difference frequencies) can be explained by the following trig identity:



Multiplying the two waves (represented by their cosines here) results in the sum and difference.

But what I still don't get is this: does the mixer really multiply the frequencies? Or is this just a mathematical explanation of how it works? To me, a multiplier would multiply the two instantaneous voltages (say the signal and LO frequencies), producing a product of ever-changing voltages. Is this what actually happens, electrically, in the mixer? Or is this all just a result of the nonlinearity of the circuit producing harmonics (in this case, just second-order ones).

I'm trying to wrap my head around how this works, both electrically and mathematically. It's not enough to just accept that "that's the way it works". Thanks for bearing with me here.

Yes, basically it is the non-linearity of some active component such as a diode which causes the mixing process.

If you have a look here:
https://ntuemc.tw/upload/file/2012060714010872a74.pdf
From about page 12 onwards it does describe mathematically the process of mixing.

Your trig identity shows the basis of the mixing, resulting in just two output frequencies.
In reality there are many more output frequencies due to the non-linearity of the mixer creating harmonics and higher order intermodulation products.

JimB

In the link that JimB provides, look very closely at pages starting at page 12. You will see that the mathematical description of the non-linearity begins with a power series of terms. In the first example, the diode is a square law device and so the function it imposes on current passing through it includes a component of the current squared. It is these higher powered terms that is applied to the incoming sin(x) and cos(y) signals and provides the multiplication. Since, in theory, any non-linearity will be described by some sort of power series, any non-linearity has the potential to generate the sum and difference terms that we seek.

Thanks JimB. Your link was a good review for me.

carbonzit said:

To me, a multiplier would multiply the two instantaneous voltages (say the signal and LO frequencies), producing a product of ever-changing voltages. Is this what actually happens, electrically, in the mixer?

Click to expand...

yes, it's multiplication of the instantaneous voltages. for a balanced modulator, this is a 4 quadrant function, hence the additional name for a balanced modulator of "4-quadrant multiplier", and the labeling of the inputs "x" and "y". many analog circuits using differential amplifiers and op amps (a 4 quadrant multiplier like the MC1495 is a series of diff amp circuits internally) were not only designed for communication systems, but as analog computational building blocks as well.