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Self Bias and Source Bias for JFETs and DE-MOSFETs Self bias and source bias FET circuits do not have a very simple formula for determining the drain current. The development of their equations is a little more involved than a simple KVL/Ohm’s Law solution. Instead, they will focus on the device transconductance equation. Consider a basic self bias configuration: Vgs I d I dss 1 V gs ( off ) 2 Expanding this yields 2 2Vgs Vgs I d I dss 1 2 Vgs( off ) V gs ( off ) I d I dss 2 I dssVgs Vgs ( off ) I dssVgs 2 Vgs( off ) 2 Noting that g m0 2 I dss Vgs( off ) The two can be combined to I d I dss g m 0Vgs I dssVgs 2 Vgs( off ) 2 Due to the fact that gate current is approximately zero Vgs I d Rs Substituting this into the expanded transconductance equation yields 2 I d I dss g m 0 I d Rs I dss I d Rs 2 Vgs( off ) 2 A modest rearrangement yields 2 0 I dss Rs 2 I 1 g m 0 Rs I d I dss 2 d Vgs( off ) It should be noticeable that this is a quadratic equation of the form ax2+bx+c and can be solved using the quadratic formula: y b b 2 4ac 2a The positive option in the numerator may be ignored as this occurs for Vgs less than Vgs(off). The result is 1 g m 0 Rs 1 2 g m 0 Rs I d 2 I dss g m0 Rs 2 It is useful to graph this in terms of the current ratio using the parameter gm0Rs. To use the graphs below simply find gm0Rs along the horizontal axis and trace up to the curve. Then move across to find the ratio of Id to Idss. Multiply this by Idss to find the drain current. Source Bias For source bias, the original circuit is modified to include a negative power supply below the source resistor. The solution for this circuit is very similar to the preceding but with one twist, namely the voltage across the source resistor is no longer equal in magnitude to the gate-source voltage but rather the gate-source voltage plus the source power supply. Therefore, Id Vgs Vss Rs This leads to Vgs I d Rs Vss This is substituted into the expanded transconductance equation as before, simplified and solved with the quadratic formula as before. To make the resulting equation more clear, it us useful to define a “swamping factor”, ks=Vss/Vgs(off). This yields the final equation: 1 g m 0 Rs 1 k s 1 2 g m 0 Rs 1 k s I d 2 I dss 2 g R m0 s Notice that if no source supply is used ks=0 and this equation reverts backs to the self bias equation. Also, note that if ks is very large, meaning that the source supply is much greater than the gate-source off voltage, the current may be approximated as Id The plots were created using FooPlot at fooplot.com Vss Rs