A final year project $($FYP$)$ student, Howard, needs to design and construct an RF mixer to downconvert an RF signal $(\backslash ($ V$\_\{\backslash$mathbf$\{$R F$\}\}\backslash ))$ of $27$ MHz from the output of a low$-$noise amplifier $($LNA$).$ The local oscillator $($LO$)$ signal $\backslash (\backslash$left$($V$\_\{\backslash$text $\{$LO $\}\}\backslash$right$)\backslash )$ has a square waveform oscillating between $0$ V and $1\mathrm{.}0$ V $.$

$($a$).$ Now, Howard is given one expensive high$-$performance MOS transistor. Draw a schematic diagram of an RF mixer circuit that he can construct using this MOS transistor $($and if needed, other cheap passive components$).$ Assume the MOS transistor has a threshold voltage of $0$ V $.$

$(4$ marks$)$

$($b$).$ Determine the linear magnitude of the maximum possible voltage conversion gain of the RF mixer in $($a$).$ Then calculate this maximum voltage conversion gain in dB $.$ Assume that the expensive MOS transistor behaves as an ideal switch.

$(5$ marks$)$

$($c$).$ If Howard needs to construct a single$-$balanced mixer as an improvement of that in $($a$),$ what is the minimum number of MOS transistors required? Draw a schematic circuit diagram of such a single$-$balanced RF mixer. Assume the MOS transistor has a threshold voltage of $0$ V $.$

$(6$ marks$)$

$($d$).$ Determine the linear magnitude of the maximum voltage conversion gain of the singlebalanced RF mixer in $($c$).$ Then calculate this maximum voltage conversion gain in dB $.$

$(4$ marks$)$

$($e$).$ Now, Howard finds from his own electrical measurements that the expensive MOS transistor has a threshold voltage of $1\mathrm{.}5$ V $.$ Draw a new schematic diagram of the RF mixer in $($a$)$ with proper biasing network $($and if needed, additional part of the circuit$)$ using only passive components.

Note: In drawing $($a sketch of$)$ the schematic circuit diagrams above, please use standard circuit symbols and all circuit elements as well as other important features $($e$.$g$.$ input and output ports$)$ must be labeled properly.