Nonlinearity based Mixers


We can get the mixing action by using the nonlinear characteristics of an active device.

The input-output characteristic of a 2-port network with weak nonlinearity is represented by a polynomial as

(1)   \begin{equation*} v_o(t) = f(v_i(t)) = \alpha_o + \alpha_1 v_i(t) + \alpha_2 v_i^2(t) + \ldots + \alpha_n v_i^n(t)  \end{equation*}

If v_i(t) = A \cos(\omega_{RF}t) + B \cos(\omega_{LO}t), then the nonlinearity of the device introduce inter-modulation products in the output at frequencies \underline{p\omega_{RF} \pm q\omega_{LO}}, where p,q are integers and (p+q) is the order of non-linearity.

Due to mixing action the desired components should be at \omega_{RF} \pm \omega_{LO}, where p=q=1. So second order nonlinear terms are required for mixing action. Higher order terms introduce distortion.

A nonlinear device with square law behavior is well suited for mixing operation.
Consider a second order polynomial,

(2)   \begin{eqnarray*} v_o(t) &=& f(v_i(t)) = \alpha_o + \alpha_2 v_i^2(t) \\ &=& \alpha_o + \alpha_2 (A \cos(\omega_{RF}t) + B \cos(\omega_{LO}t)^2 \\ &=& \underbrace{\alpha_o + {1 \over 2}\alpha_2 (A^2+B^2)}_{\text{DC}}      + \underbrace{{1 \over 2}\alpha_2 (A^2 \cos(2\omega_{RF}t) + B^2 \cos(2\omega_{LO}t))}_{\text{RF and LO harmonics}}\\    &~&+ \underbrace{\alpha_2 AB \cos(\omega_{RF}\pm\omega_{LO})t}_{\text{mixing operation}} \end{eqnarray*}

Higher frequency components (RF, LO and their harmonics) are removed by filtering at the output. A device with square law behavior is well suited for mixing operation.

A long channel MOS transistor have nearly square law behavior is a good candidate for mixing operation. Even devices with other nonlinear characteristics does the same job, but introduce higher amount spurs (for example diode, BJT, etc.,).

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