Diode Mixer

The I-V relationship of a diode is,

(1)   \begin{equation*} I_D(V) = I_s (e^{V_D/\eta V_t}-1) \end{equation*}

V_D : voltage across diode
I_D : current through diode
I_s : reverse saturation current of the diode
\eta : diode ideality factor; 1.2 for schottky diodes; 2 for silicon diodes
V_t = kT/q = 26mV @ RT

The I-V characteristic of a diode is shown in Figure 1.
[visualizer id=”1263″]

For small increments(v(t)) around operating point(V_{D0}), diode current is given by Taylor series expansion as,

(2)   \begin{equation*} I_D(V) = I_D(V_{D0}) + G_d v(t) + {1\over 2!} G_d^{'} v(t)^2 + {1\over 3!} G_d^{''} v(t)^3 \end{equation*}

V(t) = V_{D0} + v(t)

    \begin{eqnarray*} G_d &=& \left. {\partial I_D \over \partial V_D} \right|_{V=V_{D0}} =\left. I_s {1 \over \eta V_t}e^{V_D/\eta V_t}\right|_{V=V_{D0}} = {1 \over \eta V_t}[I_D(V_{D0})+I_s] \approx {I_D(V_{D0}) \over \eta V_t} \\ G_d^{'} &=& \left. {\partial^2 I_D \over \partial V_D^2} \right|_{V=V_{D0}} = \left. {\partial G_d \over \partial V_D} \right|_{V=V_{D0}} = {1 \over (\eta V_t)^2}[I_D(V_{D0})+I_s] \approx {G_d \over \eta V_t}\\ G_d^{''} &=& \left. {\partial^3 I_D \over \partial V_D^3} \right|_{V=V_{D0}} = \left. {\partial G_d^{'} \over \partial V_D} \right|_{V=V_{D0}} = {1 \over (\eta V_t)^3}[I_D(V_{D0})+I_s] \approx {G_d^{'} \over \eta V_t}\\ \end{eqnarray*}

In Eq.(2), second order terms contribute to mixing operation, while other terms and higher order terms produce spurious components or spurs.

Single Ended Diode Mixer

The schematic of single ended diode mixer is shown in Figure 2.

single ended diode mixer

Figure 2. Schematic of single ended diode mixer

Diplexer combines the input RF and LO signals by superimposing them to drive the diode.

Diode is biased at DC voltage (V_{D0}), decoupled from RF and LO signal paths through a DC blocking capacitor(C_B). RF choke blocks the RF/LO signals entering into bias source. High frequency components produced due to diode nonlinearity are filtered by IF filter, allowing only IF component to appear at the output.

The equivalent circuit of a single ended diode mixer is shown in Figure 3.

Let v_{RF}(t) = V_{RF}\cos(\omega_{RF}t) and v_{LO}(t) = V_{LO}\cos(\omega_{LO}t). Then v(t)=v_{RF}(t) + v_{LO}(t) = V_{RF}\cos(\omega_{RF}t) + V_{LO}\cos(\omega_{LO}t).

The diode current(i_D(t)) due to voltage v(t) across it at operating point V_{D0} is,

(3)   \begin{equation*} i_D(t) = I_D(V_{D0}) + G_d v(t) + {G_d^{'}\over 2!} v(t)^2 + \ldots \end{equation*}


    \begin{eqnarray*} v(t)^2 &=& \left(V_{RF}\cos(\omega_{RF}t) + V_{LO}\cos(\omega_{LO}t) \right)^2 \\ &=& \left(V_{RF}^2(1+\cos(2\omega_{RF}t))+V_{LO}^2(1+\cos(2\omega_{LO}t))\right.\\ & ~ & + \left.2V_{RF}V_{LO}[\cos(\omega_{RF}-\omega_{LO})t + \cos(\omega_{RF}+\omega_{LO})t ]\right)\\ v(t)^3 &=& \left(V_{RF}\cos(\omega_{RF}t) + V_{LO}\cos(\omega_{LO}t) \right)^3 \\ &=& {1 \over 4}V_{RF}^3(3\cos(\omega_{RF}t)+\cos(3\omega_{RF}t)) \\ &~& + {1 \over 4}V_{LO}^3 (3\cos(\omega_{LO}t)+\cos(3\omega_{LO}t)) \\ &~& + {3\over 2} V_{RF}^2 V_{LO} (2\cos(\omega_{LO}t) + \cos(2\omega_{RF}-\omega_{LO})t + \cos(2\omega_{RF}+\omega_{LO})t) \\ &~& + {3\over 2} V_{RF} V_{LO}^2 (2\cos(\omega_{RF}t) + \cos(\omega_{RF}-2\omega_{LO})t + \cos(\omega_{RF}+2\omega_{LO})t) \\ \end{eqnarray*}

If \omega_{LO} and \omega_{RF} are closely spaced, then from Eq.(3), we can see that except the IF component(\cos(\omega_{RF}-\omega_{LO})t) due to second order term, all other terms are at much high frequency than IF, and can be filtered out by IF filter.

The output current is

    \[i_{IF}(t) = -G_d^{'} V_{RF}V_{LO} \cos(\omega_{RF}-\omega_{LO})t\]

Conversion gain is

    \[G_{CG} = \left|{i_{IF}\over V_{RF}}\right| = G_d^{'} V_{LO}\]

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