Nonlinear BJT Mixer

Single Ended BJT Mixer(Nonlinear)

A single ended BJT mixer is shown in Figure 1.

Figure 1. Schematic of a single ended BJT mixer

Figure 1. Schematic of a single ended BJT mixer

In active region the BJT is modeled as I_c \approx I_s e^{v_{be}\over V_t}.

From the Figure 1,

(1)   \begin{equation*} I_c &=& I_s e^{V_b + v_{LO} + v_{RF} / V_t} = \underbrace{I_s e^{V_{b}/V_t}}_{I_{CQ}} ~ e^{v_{RF}/V_t} ~ e^{v_{LO}/V_t} \end{equation*}

If {v_{RF}/ V_t} << 1, then

(2)   \begin{equation*} e^{v_{RF}/V_t} \approx 1 + {v_{RF}\over V_t} \end{equation*}

Typically v_{LO} >> v_{t}. So e^{v_{RF}/V_t} is given by

(3)   \begin{equation*} e^{v_{LO}/V_t} = 1 + {v_{LO}\over V_t} + {1\over 2\!}({v_{LO}\over V_t})^2 + \ldots + {1\over n\!}({v_{LO}\over V_t})^n \end{equation*}

Substituting in Eq(1),

(4)   \begin{eqnarray*} I_c &=& I_{CQ} \left(1 + {v_{RF}\over V_t}\right) \left(1 + {v_{LO}\over V_t} + {1\over 2\!}({v_{LO}\over V_t})^2 + \ldots + {1\over n\!}({v_{LO}\over V_t})^n \right) \\ &=& I_{CQ} \left(1 + {v_{LO}\over V_t} + {1\over 2\!}({v_{LO}\over V_t})^2 + \ldots + {1\over n\!}({v_{LO}\over V_t})^n \right) + I_{CQ} {v_{RF}\over V_t} \left(1 + {v_{LO}\over V_t} + {1\over 2\!}({v_{LO}\over V_t})^2 + \ldots + {1\over n\!}({v_{LO}\over V_t})^n \right) \\ &=& I_{CQ} \left(1 + \underbrace{{V_{LO}\over V_t}\cos(\omega_{LO}t) + {1\over 2\!}\left({V_{LO}\over V_t}\right)^2\cos^2(\omega_{LO}t) + \ldots + {1\over n\!}\left({V_{LO}\over V_t}\right)^2\cos^n(\omega_{LO}t)}_{\text{LO and its harmonics}} \right)\\ &~& + ~ I_{CQ} {V_{RF}\over V_t} \left( \cos(\omega_{RF}t)+ \underbrace{{V_{LO}\over V_t}\cos(\omega_{RF}t)\cos(\omega_{LO}t)}_{\text{mixing action}} + \underbrace{{1\over 2\!}\left({V_{LO}\over V_t}\right)^2\cos(\omega_{RF}t)\cos^2(\omega_{LO}t) + \ldots }_{\text{higher order terms}} \right) \end{eqnarray*}

In Eq.(4), except \omega_{LO}-\omega_{RF} term all other terms are near to LO frequency or higher than LO frequency. Therefore they can be filtered out by output filter.

Votlage conversion gain is,

    \[A_{v} = {I_{CQ} V_{LO} \over 2 V_t^2} R_L\]

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