Square law Mixer


The schematic of a simplified single ended square law mixer is shown in Figure 1.

Figure 1. Schematic of a single ended square law mixer

Figure 1. Schematic of a single ended square law mixer

Mixing operation is achieved by square law behavior of the MOS transistor operating in saturation region. The bias voltage V_g keep the transistor at the right operating point. MOS transistor drain current i_{D}(t) is given by,

(1)   \begin{equation*} i_{D}(t) = {1\over 2} \mu_n C_{ox} {W \over L} (V_{GS}-V_T)^2 = \beta_n (V_{GS}-V_T)^2 \end{equation*}

where \beta_n = {1\over 2} \mu_n C_{ox} {W \over L}

(2)   \begin{eqnarray*} i_D(t) &=& \beta(V_g + v_{RF}(t)+v_{LO}(t) -V_T)^2 \\ &=& (v_{RF}(t)+v_{LO}(t))^2 + (V_g-V_T)^2+ 2 (v_{RF}(t)+v_{LO}(t))(V_g-V_T)\\ &=& \beta\left[ {V_{RF}\over 2}(1+\cos(2\omega_{RF}t)) + {V_{LO}\over 2}(1+\cos(2\omega_{LO}t)) \\ &~& \quad + \underbrace{V_{RF}V_{LO}(\cos(\omega_{LO}-\omega_{RF})t + \cos(\omega_{LO}+\omega_{RF})t)}_{\mbox{sum and difference components}} \\ &~& \quad + (V_g - V_T)^2 + 2(V_g - V_T)V_{RF}\cos(\omega_{RF}t) + 2(V_g - V_T)V_{LO}\cos(\omega_{LO}t) \right] \end{eqnarray*}

Except the difference term(\omega_{LO}-\omega_{RF}), all other terms are of high frequency and can be removed by IF filter.

Voltage conversion gain is

(3)   \begin{equation*} A_v =\left| {i_{IF} R_L \over v_{RF}} \right| = \beta V_{LO} R_L \end{equation*}

Therefore the conversion gain is influenced by LO amplitude. Higher LO amplitude, higher the gain.

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