Analog multiplier (mixer)

[STOIKOV]

I've read that in a mixer's output there is the sum and subtraction of two input signals, by multiplying both signals due to non-linearity of a device.

There P1dB curve shows a region where Pout is lineal with respect to Pin, so if the device works in this lineal region, why there is a multiplication in its output ?

Hope my question is clear thanks

 

[VVV]

The output POWER is a linear function of the input. That is, if you supply more input power, you get more output power. That does not mean, however, that the device operates in a linear region.

 

[STOIKOV]

so where is the non-linearity of the device. Maybe the equation of a MOS in saturation (vgs-vt)² , but if this term is non-linear why Pout-vs-Pin is linear in some region. It's some confuse for me.

 

[Miguel Gaspar]

Well, it is not so obvious. There are several techniques, none of them are easy.

whenyou multiply two signals:

u1 = A1 Sin (ω1 t + Θ1)

and

u2 = A2 Cos (ω2 t + Θ2)

u1 * u2 = A1 * A2 * Sin (ω1 t + Θ1) * Cos (ω2 t + Θ2)

Sin (A) * Cos (B) = 1/2 * [Sin (A+B) + Sin (A-B)]

so

u1 * u2 = 1/2 * [Sin {ω1 t + Θ1 + (ω2 t + Θ2)} + Sin {(ω1 t + Θ1)- (ω2 t + Θ2)}]

resulting:
u1 * u2 = 1/2 * [Sin {ω1 t +ω2 t + Θ1+Θ2} + Sin {ω1 t -ω2 t + Θ1 - Θ2}

That gives you the add and sustraction of frequiency

regards

 

[VVV]

You are right, the non-linearity is in the characteristic of the device.
Consider a FET mixer. The characteristic is a parabola (square-law curve).
Therefore, a signal will be amplified by the FET (operating medium-signal):
Vo=A*Vin+B*Vin^2

Notice the square term.

If you have two signals applied to the mixer the input will be Vin=Vx+Vy. Therefore the output will be:
Vo=A*Vin+B*Vin^2=A*(Vx+Vy)+B(Vx+Vy)^2=A*Vx+B*Vx^2+A*Vy+B*Vy^2+2*B*Vx*Vy

The last term is the cross product. For sinusoidal signals, you have:
Vx=X*(sinωx*t), Vy=Y*sin(ωy*t)

Thus, 2*B*Vx*Vy=2*B*X*Y*sin(ωx*t)*sin(ωy*t)

With the substitution sinα*sinβ=1/2*cos(α-β)-1/2*cos(α+β),
you get:
2*B*Vx*Vy=B*X*Y*cos((ωx-ωy)*t)-B*X*Y*cos((ωx+ωy)*t)

The first term is the one we are usually interested in, the difference frequency. Note that it is proportional to the amplitude of the input signals. Therefore, the output power is proportional with the square of Vx, for example.
But so is the input power, for the input power is also proportional with the square of the input voltage, say Vx. The input power for signal Vx is: Pxin=0.5*X^2/Zin
Pout=0.5*X^2*Y^2/Zout

As you can see, the ratio of the powers, Pout/Pin is a linear function (of either input signal).