Common source

At low frequencies and using a simplified hybrid-pi model (where the output resistance due to channel length modulation is not considered), the following closed-loop small-signal characteristics can be derived.

	Definition	Expression
Current gain	${\displaystyle A_{\text{i}}\triangleq {\frac {i_{\text{out}}}{i_{\text{in}}}}\,}$	${\displaystyle \infty \,}$
Voltage gain	${\displaystyle A_{\text{v}}\triangleq {\frac {v_{\text{out}}}{v_{\text{in}}}}\,}$	${\displaystyle {\begin{matrix}-{\frac {g_{\mathrm {m} }R_{\text{D}}}{1+g_{\mathrm {m} }R_{\text{S}}}}\end{matrix}}\,}$
Input impedance	${\displaystyle r_{\text{in}}\triangleq {\frac {v_{\text{in}}}{i_{\text{in}}}}\,}$	${\displaystyle \infty \,}$
Output impedance	${\displaystyle r_{\text{out}}\triangleq {\frac {v_{\text{out}}}{i_{\text{out}}}}}$	${\displaystyle R_{\text{D}}\,}$

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 ${\displaystyle 1+|A_{\text{v}}|\,}$, 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_L is added at the output node and a Thévenin driver of applied voltage V_A and series resistance R_A is added at the input node. The limitation on bandwidth in this circuit stems from the coupling of parasitic transistor capacitance C_gd between gate and drain and the series resistance of the source R_A. (There are other parasitic capacitances, but they are neglected here as they have only a secondary effect on bandwidth.)

Using Miller's theorem, the circuit of Figure 4 is transformed to that of Figure 5, which shows the Miller capacitance C_M on the input side of the circuit. The size of C_M is decided by equating the current in the input circuit of Figure 5 through the Miller capacitance, say i_M, which is:

${\displaystyle \ i_{\mathrm {M} }=j\omega C_{\mathrm {M} }v_{\mathrm {GS} }=j\omega C_{\mathrm {M} }v_{\mathrm {G} }}$ ,

to the current drawn from the input by capacitor C_gd in Figure 4, namely jωC_gd v_GD. These two currents are the same, making the two circuits have the same input behavior, provided the Miller capacitance is given by:

${\displaystyle C_{\mathrm {M} }=C_{\mathrm {gd} }{\frac {v_{\mathrm {GD} }}{v_{\mathrm {GS} }}}=C_{\mathrm {gd} }\left(1-{\frac {v_{\mathrm {D} }}{v_{\mathrm {G} }}}\right)}$ .

Usually the frequency dependence of the gain v_D / v_G 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_D / v_G. This evaluation is Miller's approximation^[1] and provides the estimate (just set the capacitances to zero in Figure 5):

${\displaystyle {\frac {v_{\mathrm {D} }}{v_{\mathrm {G} }}}\approx -g_{\mathrm {m} }(r_{\mathrm {O} }\parallel R_{\mathrm {L} })}$ ,

so the Miller capacitance is

${\displaystyle C_{\mathrm {M} }=C_{\mathrm {gd} }\left(1+g_{\mathrm {m} }(r_{\mathrm {O} }\parallel R_{\mathrm {L} })\right)}$ .

The gain g_m (r_O || R_L) is large for large R_L, so even a small parasitic capacitance C_gd 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_L ≈ 1 / g_m ≈ V_ov / (2I_D) ; see common gate). Small R_L reduces C_M.^[2] The article on the 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:

${\displaystyle v_{\mathrm {G} }=V_{\mathrm {A} }{\frac {1/(j\omega C_{\mathrm {M} })}{1/(j\omega C_{\mathrm {M} })+R_{\mathrm {A} }}}=V_{\mathrm {A} }{\frac {1}{1+j\omega C_{\mathrm {M} }R_{\mathrm {A} }}}}$ .

The bandwidth (also called the 3 dB frequency) is the frequency where the signal drops to 1/ √2 of its low-frequency value. (In decibels, dB(√2) = 3.01 dB). A reduction to 1/ √2 occurs when ωC_M R_A = 1, making the input signal at this value of ω (call this value ω_3 dB, say) v_G = V_A / (1+j). The magnitude of (1+j) = √2. As a result, the 3 dB frequency f_3 dB = ω_3 dB / (2π) is:

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

If the parasitic gate-to-source capacitance C_gs is included in the analysis, it simply is parallel with C_M, so

${\displaystyle f_{\mathrm {3dB} }={\frac {1}{2\pi R_{\mathrm {A} }(C_{\mathrm {M} }+C_{\mathrm {gs} })}}={\frac {1}{2\pi R_{\mathrm {A} }[C_{\mathrm {gs} }+C_{\mathrm {gd} }(1+g_{\mathrm {m} }(r_{\mathrm {O} }\parallel R_{\mathrm {L} }))]}}}$ .

Notice that f_3 dB becomes large if the source resistance R_A is small, so the Miller amplification of the capacitance has little effect upon the bandwidth for small R_A. 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_A of the original driver.^[3]

Examination of the output side of the circuit in Figure 2 enables the frequency dependence of the gain v_D / v_G to be found, providing a check that the low-frequency evaluation of the Miller capacitance is adequate for frequencies f even larger than f_3 dB. (See article on pole splitting to see how the output side of the circuit is handled.)

• Miller effect
• Pole splitting
• Common gate
• Common drain
• Common base
• Common emitter
• Common collector