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
Mixing operation is achieved by square law behavior of the MOS transistor operating in saturation region. The bias voltage 
keep the transistor at the right operating point. MOS transistor drain current 
is given by,
(1) 
where 
(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*}](https://analog.intgckts.com/wp-content/ql-cache/quicklatex.com-38d13d569a17a40f38f9ec7b7a78a33d_l3.svg "Rendered by QuickLaTeX.com")
Except the difference term(
), all other terms are of high frequency and can be removed by IF filter.
Voltage conversion gain is
(3) 
Therefore the conversion gain is influenced by LO amplitude. Higher LO amplitude, higher the gain.