Editing Common gate (section)

==Low-frequency characteristics==

[[File:Common gate hybrid pi.PNG|thumbnail|200px| Figure 2: Small-signal low-frequency [[hybrid-pi model]] for amplifier driven by a [[Norton's theorem|Norton signal source]]]]

At low frequencies and under [[small-signal]] conditions, the circuit in Figure 1 can be represented by that in Figure 2, where the [[hybrid-pi model]] for the MOSFET has been employed.

[[File:Common gate output resistance.PNG|thumbnail|200px| Figure 3: Hybrid pi model with test source ''i<sub>x</sub>'' at output to find [[output resistance]]]]
The amplifier characteristics are summarized below in Table 1. The approximate expressions use the assumptions (usually accurate) ''r<sub>O</sub>'' >> ''R<sub>L</sub>'' and ''g<sub>m</sub>r<sub>O</sub>'' >> 1.

{| class="wikitable" style="text-align:center;margin:  1em auto 1em auto 1em auto 1em auto"
!Table 1  !! Definition !! Expression !! Approximate expression
|-valign="center"
! '''Short-circuit current gain'''
|<math> {A_{i}} = {i_\mathrm{out} \over i_\mathrm{S}} \Big|_{R_{L}=0} </math>
|<math>\ 1 </math>
|<math>\ 1</math>

|-valign="center"
! '''Open-circuit voltage gain'''
|<math> {A_{v}} = {v_\mathrm{out} \over v_\mathrm{S}} \Big|_{R_{L}=\infty}</math>
|<math>\begin{matrix} ((g_m+g_{mb}) r_\mathrm{O}+1) \frac {R_L} {r_O + R_L} \end{matrix}</math> 
|<math>\ g_m  R_L </math>

|-valign="center"
! '''[[Input resistance]]'''
|<math> R_\mathrm{in} = \frac{v_{S}}{i_{S}}</math>
|<math> {{R_L+r_O} \over {(g_m+g_{mb}) r_O +1}} </math>
|<math>\begin{matrix} \frac {1} {g_m} \end{matrix}</math>
|-valign="center"
! '''[[Output resistance]]'''
|<math> R_\mathrm{out} =\frac{v_{x}}{i_{x}}</math>
|<math>\ (1+(g_m+g_{mb})r_O)R_S + r_O </math>
| <math>(g_m r_O)R_S</math>
|}

In general, the overall voltage/current [[Gain (electronics)|gain]] may be substantially less than the open/short circuit gains listed above (depending on the source and load resistances) due to the [[loading effect]].

===Closed circuit voltage gain===

Taking input and output loading into consideration, the closed circuit voltage gain (that is, the gain with load ''R<sub>L</sub>'' and source with resistance ''R<sub>S</sub>'' both attached) of the common gate can be written as:
: <math>
{A_\mathrm{v}} \approx \begin{matrix} \frac {g_m R_\mathrm{L}} {1+g_mR_S} \end{matrix}</math> ,
which has the simple limiting forms
: <math>A_\mathrm{v} = \begin{matrix} \frac {R_L}{R_S}\end{matrix} \ \ \mathrm{ or } \ \ A_\mathrm{v} = g_m R_L </math>,
depending upon whether ''g<sub>m</sub>R<sub>S</sub>'' is much larger or much smaller than one.

In the first case the circuit acts as a current follower,  as understood as follows: for ''R<sub>S</sub>'' >> 1/''g<sub>m</sub>'' the voltage source can be replaced by its [[Norton's theorem|Norton equivalent]] with Norton current ''v<sub>Thév</sub> / R<sub>S</sub>'' and parallel Norton resistance ''R<sub>S</sub>''. Because the amplifier input resistance is small, the driver delivers by [[current division]] a current ''v<sub>Thév</sub> / R<sub>S</sub>'' to the amplifier. The current gain is unity, so the same current is delivered to the output load ''R<sub>L</sub>'', producing by Ohm's law an output voltage ''v<sub>out</sub> = v<sub>Thév</sub>R<sub>L</sub> / R<sub>S</sub>'', that is, the first form of the voltage gain above.

In the second case ''R<sub>S</sub>'' << 1/''g<sub>m</sub>'' and the Thévenin representation of the source is useful, producing the second form for the gain, typical of voltage amplifiers.

Because the input impedance of the common-gate amplifier is very low, the [[cascode]] amplifier often is used instead. The cascode places a [[common-source]] amplifier between the voltage driver and the common-gate circuit to permit voltage amplification using a driver with ''R<sub>S</sub> >> 1/g<sub>m</sub>''.