Editing Common source (section)

===Bandwidth===

[[Image:Common source with active load.PNG|thumbnail|200px|Figure 3: Basic N-channel MOSFET common-source amplifier with [[active load]] ''I''<sub>D</sub>.]]
[[Image:Small-signal common source with C gd.PNG|thumbnail|250px|Figure 4: Small-signal circuit for N-channel MOSFET common-source amplifier.]]
[[Image:Small-signal common source with Miller cap.PNG|thumbnail|300px|Figure 5: Small-signal circuit for N-channel MOSFET common-source amplifier using Miller's theorem to introduce Miller capacitance ''C''<sub>M</sub>.]]
Bandwidth of common-source amplifier tends to be low, due to high capacitance resulting from the [[Miller effect]]. The gate-drain capacitance is effectively multiplied by the factor <math>1+|A_\text{v}|\,</math>, thus increasing the total input capacitance and lowering the overall bandwidth.

Figure 3 shows a MOSFET common-source amplifier with an [[active load]]. Figure 4 shows the corresponding small-signal circuit when a load resistor ''R''<sub>L</sub> is added at the output node and a [[Thévenin's theorem|Thévenin driver]] of applied voltage ''V''<sub>A</sub> and series resistance ''R''<sub>A</sub> is added at the input node. The limitation on bandwidth in this circuit stems from the coupling of [[parasitic capacitance|parasitic transistor capacitance]] ''C''<sub>gd</sub> between gate and drain and the series resistance of the source ''R''<sub>A</sub>. (There are other parasitic capacitances, but they are neglected here as they have only a secondary effect on bandwidth.)

Using [[Miller effect|Miller's theorem]], the circuit of Figure 4 is transformed to that of Figure 5, which shows the ''Miller capacitance'' ''C''<sub>M</sub> on the input side of the circuit. The size of ''C''<sub>M</sub> is decided by equating the current in the input circuit of Figure 5 through the Miller capacitance, say ''i''<sub>M</sub>, which is:

::<math>\  i_\mathrm{M} = j \omega C_\mathrm{M} v_\mathrm{GS} = j \omega C_\mathrm{M} v_\mathrm{G}</math> ,

to the current drawn from the input by capacitor ''C''<sub>gd</sub> in Figure 4, namely ''j&omega;C''<sub>gd</sub> ''v''<sub>GD</sub>. These two currents are the same, making the two circuits have the same input behavior, provided the Miller capacitance is given by:

::<math> C_\mathrm{M} = C_\mathrm{gd} \frac {v_\mathrm{GD}} {v_\mathrm{GS}} = C_\mathrm{gd} \left( 1 - \frac {v_\mathrm{D}} {v_\mathrm{G}} \right)</math> .

Usually the frequency dependence of the gain ''v''<sub>D</sub> / ''v''<sub>G</sub> is unimportant for frequencies even somewhat above the corner frequency of the amplifier, which means a low-frequency [[hybrid-pi model]] is accurate for determining ''v''<sub>D</sub> / ''v''<sub>G</sub>. This evaluation is ''Miller's approximation''<ref name=Spencer>
{{cite book 
|author1=R.R. Spencer |author2=M.S. Ghausi |title=Introduction to electronic circuit design
|year=2003
|page=533 
|publisher=Prentice Hall/Pearson Education, Inc. 
|location=Upper Saddle River NJ 
|isbn=0-201-36183-3
|url=http://worldcat.org/isbn/0-201-36183-3}}
</ref> and provides the estimate (just set the capacitances to zero in Figure 5):

::<math> \frac {v_\mathrm{D}} {v_\mathrm{G}} \approx -g_\mathrm{m} (r_\mathrm{O} \parallel R_\mathrm{L})</math> ,

so the Miller capacitance is

::<math> C_\mathrm{M} = C_\mathrm{gd} \left( 1+g_\mathrm{m} (r_\mathrm{O} \parallel R_\mathrm{L})\right) </math> .

The gain ''g''<sub>m</sub> (''r''<sub>O</sub> || ''R''<sub>L</sub>) is large for large ''R''<sub>L</sub>, so even a small parasitic capacitance ''C''<sub>gd</sub> can become a large influence in the frequency response of the amplifier, and many circuit tricks are used to counteract this effect. One trick is to add a [[common-gate]] (current-follower) stage to make a [[cascode]] circuit. The current-follower stage presents a load to the common-source stage that is very small, namely the input resistance of the current follower (''R''<sub>L</sub> ≈ 1 / ''g''<sub>m</sub> ≈ ''V''<sub>ov</sub> / (2''I''<sub>D</sub>) ; see [[common gate]]). Small ''R''<sub>L</sub> reduces ''C''<sub>M</sub>.<ref name=Lee>
{{cite book 
|author=Thomas H Lee
|title=The design of CMOS radio-frequency integrated circuits
|year= 2004
|edition=Second 
|publisher=Cambridge University Press 
|location=Cambridge UK 
|isbn=0-521-83539-9
|url=http://worldcat.org/isbn/0-521-83539-9
|pages=246&ndash;248}}
</ref> The article on the [[common emitter|common-emitter amplifier]] discusses other solutions to this problem.

Returning to Figure 5, the gate voltage is related to the input signal by [[voltage division]] as:

::<math> v_\mathrm{G} = V_\mathrm{A}\frac {1/(j \omega C_\mathrm{M}) } {1/(j \omega C_\mathrm{M}) +R_\mathrm{A}} = V_\mathrm{A}\frac {1} {1+j \omega C_\mathrm{M} R_\mathrm{A}} </math> .

The [[Bandwidth (signal processing)|bandwidth]] (also called the 3&nbsp;dB frequency) is the frequency where the signal drops to 1/ {{radic|2}} of its low-frequency value.  (In [[decibel]]s, dB({{radic|2}}) = 3.01&nbsp;dB).  A reduction to 1/ {{radic|2}}  occurs when ''ωC''<sub>M</sub> ''R''<sub>A</sub> = 1, making the input signal at this value of ''ω'' (call this value ''ω''<sub>3&nbsp;dB</sub>, say) ''v''<sub>G</sub> = ''V''<sub>A</sub> / (1+j). The [[Complex number#Operations|magnitude]] of (1+j) = {{radic|2}}. As a result, the 3&nbsp;dB frequency ''f''<sub>3&nbsp;dB</sub> = ''&omega;''<sub>3&nbsp;dB</sub> / (2π) is:

::<math> f_\mathrm{3dB}=\frac {1}{2\pi R_\mathrm{A} C_\mathrm{M}}= \frac {1}{2\pi R_\mathrm{A} [ C_\mathrm{gd}(1+g_\mathrm{m} (r_\mathrm{O} \parallel R_\mathrm{L})]}</math> .

If the parasitic gate-to-source capacitance ''C''<sub>gs</sub> is included in the analysis, it simply is parallel with ''C''<sub>M</sub>, so 
::<math> f_\mathrm{3dB}=\frac {1}{2\pi R_\mathrm{A} (C_\mathrm{M}+C_\mathrm{gs})} =\frac {1}{2\pi R_\mathrm{A} [C_\mathrm{gs} + C_\mathrm{gd}(1+g_\mathrm{m} (r_\mathrm{O} \parallel R_\mathrm{L}))]}</math> .

Notice that ''f''<sub>3&nbsp;dB</sub> becomes large if the source resistance ''R''<sub>A</sub> is small, so the Miller amplification of the capacitance has little effect upon the bandwidth for small ''R''<sub>A</sub>. This observation suggests another circuit trick to increase bandwidth: add a [[common-drain]] (voltage-follower) stage between the driver and the common-source stage so the Thévenin resistance of the combined driver plus voltage follower is less than the ''R''<sub>A</sub> of the original driver.<ref name=Lee2>
{{cite book 
|author=Thomas H Lee
|title=pp.&nbsp;251&ndash;252
|year=2004
|isbn=0-521-83539-9
|url=http://worldcat.org/isbn/0-521-83539-9}}
</ref>

Examination of the output side of the circuit in Figure 2 enables the frequency dependence of the gain ''v''<sub>D</sub> / ''v''<sub>G</sub> to be found, providing a check that the low-frequency evaluation of the Miller capacitance is adequate for frequencies ''f'' even larger than ''f''<sub>3&nbsp;dB</sub>. (See article on [[pole splitting]] to see how the output side of the circuit is handled.)