Constant Voltage Biasing of Common Source Amplifier

The NMOS is biased in saturation with $V_{G S} = V_{A}$ and $V_{D S} = V_{B}$ . Now, on applying a small signal input $V_{i n}$ (having a source resistance of $R_{s}$ ) in series with the bias voltage $V_{A}$ , the voltage at the gate of the NMOS will be $V_{A} + V_{i n}$ . Also, a load resistance $R_{L}$ is connected in series with bias voltage $V_{B}$ such that in the small signal equivalent of the circuit, the other end of $R_{L}$ will be effectively grounded. Doing that makes the voltage at drain $V_{D S} = V_{B} - I_{D} R_{L}$ , which is lower than the required bias voltage $V_{B}$ . So, the bias voltage source is adjusted to a value of $V_{B} + I_{D - b i a s} R_{L}$

Not so obvious note: In this configuration of CS amplifier, we generally sense the output $V_{o u t}$ at the drain terminal of the MOSFET. We have to connect a load like $R_{L}$ (or other similar loads) instead of directly connecting the drain of the NMOS to the bias voltage $V_{B}$ (or to ground in small signal model) because, then the drain and source ends of the MOSFET will be essentially fixed at those voltages allowing no swings at all. So, to convert the $I_{D}$ fluctuations due to the input $V_{i n}$ changes into voltage fluctuations at the output, we pass that $I_{D}$ through a resistor which according to $V = I R$ allows us to sense the voltage across it that changes proportionally with the changing $I_{D}$

The problem with the above configuration is we don't generally have biasing voltage sources whose both terminals are available to tap as one of their terminals is often grounded.
In place of the voltage source $V_{A}$ , we can use a capacitor that is charged to the same bias voltage. We would need a capacitor of infinite capacitance to supply current without dropping in voltage, as there exists a parasitic resistor parallel with the capacitor constantly discharging the capacitor. To tackle this, let's place the voltage source $V_{A}$ as shown in the circuit below.

If the voltage source is directly connected to the capacitor, then the gate voltage of the MOSFET will be fixed at $V_{A}$ not allowing it to sense $v_{i n}$ . So let's place a resistor in between and then see the voltage at the gate due to both the sources $V_{A}$ and $v_{i n}$ .
By applying superposition

with $V_{A}$ only: the steady state voltage of the capacitor will be equal to $V_{A}$ (neglecting the minute parasitic resistor).
Not so obvious note: Capacitors don't block DC currents, they can go on charging to infinity if connected to a constant DC current source. But, in this configuration that doesn't happen because if the capacitor charges more than $V_{A}$ volts, then the direction of current will reverse and it will discharge back to $V_{A}$ and vice-versa.
with $v_{i n}$ only: assuming the capacitor allows the input frequency with no impedance, the voltage at the gate will be

v_{g} = \frac{R_{B}}{R_{B} + R_{S}} * v_{i n}

$R_{B} >> R_{S}$ for most of the $v_{i n}$ to appear as $v_{g}$
Now, to know the safe value of capacitance that will allow the input frequency unimpeded let's include the capacitance in calculating $v_{g}$ at a given frequency $ω$

v_{g} (j ω) = \frac{R_{B}}{R_{B} + R_{S} + \frac{1}{j ω C}} * v_{i n} (j ω)

\frac{1}{ω C} ≪ R_{B} + R_{S} $ $

\boxed {C\gg \frac{1}{\omega_{min}(R_{B}+R_{S})}}

S o, t h e c a p a c i t o r o f t h i s c a p a c i t a n c e w i l l a l l o w t h e i n p u t o f a n y f r e q u e n c y $ ω \geq ω_{m i n} $ B u t, l o a d c a n n o t b e c o n n e c t e d i n p l a c e o f $ R_{L} $, b e c a u s e w e g e n e r a l l y d o n o t k n o w t h e e x a c t c h a r a c t e r i s t i c s o f t h e l o a d, a n d t h o s e m i g h t b e c h a n g i n g a s w e l l . I f w e c o n n e c t s u c h a l o a d i n p l a c e o f $ R_{L} $, i t m o s t c e r t a i n l y w i l l c h a n g e o u r b i a s i n g c i r c u i t p o t e n t i a l l y m o v i n g t h e M O S F E T o u t o f s a t u r a t i o n . O n e o t h e r r e a s o n i s t h a t w e d o n o t i n t e n d t o l e t t h e D C b i a s c u r r e n t f l o w t h r o u g h t h e l o a d (w e o n l y w a n t t h e a m p l i f i e d s i g n a l) . T h e s o l u t i o n t o t h i s i s t o p l a c e t h e l o a d $ R_{L} $ i n p a r a l l e l t o $ R_{D} $ * * t h r o u g h a c a p a c i t o r * * $ C_{2} $ a s s h o w n i n t h e m o d i f i e d c i r c u i t . - - c i r c u i t w i t h o u t C 2 a n d w i t h C 2 s h o w i n g t h e p a t h o f D C c u r r e n t - - I f $ R_{L} $ i s c o n n e c t e d d i r e c t l y w i t h o u t t h e c a p a c i t o r $ C_{2} $, t h e l o a d w i l l b e c o m e a p a r t o f t h e D C p a t h a n d w i l l d r a w a p a r t o f D C c u r r e n t a l t e r i n g t h e D C o p e r a t i n g p o i n t o f t h e M O S F E T . A s w e d o n o t w a n t t h e l o a d t o a l t e r t h e D C o p e r a t i n g p o i n t, w e i s o l a t e i t w i t h t h e h e l p o f $ C_{2} $ . W e w a n t t h i s c a p a c i t o r $ C_{2} $ t o a c t a s s h o r t i n s m a l l s i g n a l m o d e l . (b e c a u s e i d e a l l y, w e w o u l d h a v e k e p t a b a t t e r y o f t h e v o l t a g e e q u a l t o t h e D C b i a s a t t h e d r a i n t e r m i n a l i n p l a c e o f $ C_{2} $ t o n o t l e t a n y D C b i a s c u r r e n t t o f l o w t h r o u g h t h a t p a t h, a n d t h i s v o l t a g e s o u r c e w i l l b e s h o r t e d i n s m a l l s i g n a l m o d e l, s o a s $ C_{2} $ i s i t^{'} s e q u i v a l e n t, i t s h o u l d a l s o o f f e r z e r o i m p e d a n c e i d e a l l y t o t h e a m p l i f i e d s i g n a l - - r e q u i r e s c i r c u i t - -) N o w, t o c a l c u l a t e t h e a p p r o p r i a t e c a p a c i t a n c e o f $ C_{2} $, w e s h a l l l o o k a t t h e s m a l l s i g n a l m o d e l . W e f i x e d t h e v a l u e o f $ R_{D} $ d u r i n g b i a s i n g, h e n c e $ r_{0} | | R_{D} $ c a n n o t b e a l t e r e d n o w . F o r u s t o s e e h i g h e r g a i n a c r o s s $ R_{L} $, m u c h o f t h e t h e c u r r e n t $ g_{m} v_{g s} $ s h a l l f l o w t h r o u g h i t s o t h a t i t c a n s e e h i g h e r v o l t a g e s w i n g s . N o w f o r b e t t e r u n d e r s t a n d i n g, I w o u l d c o n v e r t t h i s c u r r e n t s o u r c e i n p a r a l l e l w i t h $ r_{0} | | R_{D} $ i n t o a v o l t a g e s o u r c e i n s e r i e s w i t h t h e s a m e . N o w w e c a n s e e t h a t $ r_{0} | | R_{D} $, $ C_{2} $ a n d $ R_{L} $ a r e i n s e r i e s w i t h t h e t r a n s f o r m e d d e p e n d e n t v o l t a g e s o u r c e . A c c o r d i n g t o t h e v o l t a g e d i v i d e r r u l e, t h e c a p a c i t o r $ C_{2} $ s h o u l d b e s u c h t h a t $ \frac{1}{ω C_{2}} ≪ R_{L} + (R_{0} | | R_{D}) $ . (- - r e q u i r e s s m a l l s i g c i r c u i t w i t h t r a n s f o r m e d s o u r c e s h o w i n g t h e v o l t a g e d i v i s i o n - -) G e n e r a l l y t h e $ {\hat{V}}_{B} > V_{A} $ a s $ V_{D S} > V_{G S} $ a f t e r b i a s i n g p r o p e r l y . S o w e c a n u s e a v o l t a g e d i v i d e r t o a c h i e v e $ V_{A} $ f r o m $ {\hat{V}}_{B} $ . T h e b r a n c h w i t h $ R_{B} $ p l u s t h e $ V_{A} $ s h o u l d b e e f f e c t i v e r e p l a c e d b y a v o l t a g e d i v i d e r s u c h t h a t t h e T h e v e n i n^{'} s e q u i v a l e n t v o l t a g e a n d r e s i s t a n c e s e e n a c r o s s t h e n o d e o f t h a t b r a n c h a r e s i m i l a r . A s p e r t h e c i r c u i t d i a g r a m^{'} s c o n f i g u r a t i o n b e l o w, i f w e c a l c u l a t e t h e s a m e w e g e t $ R_{t h} = R_{1} | | R_{2} $ a n d $ V_{t h} = {\hat{V}}_{B} \frac{R_{2}}{R_{1} + R_{2}} $ . W e s h o u l d p l a c e $ R_{1} $ a n d $ R_{2} $ s u c h t h a t $ V_{t h} = V_{A} $ a n d $ R_{t h} \approx R_{B} $ . W i t h t h i s c o n f i g u r a t i o n j u s t o n e v o l t a g e s o u r c e i s e n o u g h o f t h e v a l u e $ {\hat{V}}_{B} $, a n d w e s h a l l r e f e r t o i t a s $ V_{D D} $ . (- - c i r c u i t o f f u l l s e t u p - -) L e t^{'} s c a l c u l a t e t h e g a i n b y r e d r a w i n g t h e s m a l l s i g n a l a g a i n . A s $ V_{D D} $ i s s h o r t e d, $ C_{1} $ a n d $ C_{2} $ a r e n e g l e c t e d, a s p e r t h e c i r c u i t w e g e t $ v_{g} = v_{i n} * \frac{R_{1} | | R_{2}}{R_{S} + R_{1} | | R_{2}} \approx v_{i n} $ a n d $ v_{o u t} = - i_{d} * (r_{0} | | R_{D} | | R_{L}) $ w h e r e $ i_{d} = g_{m} v_{i n} $ (- - c i r c u i t o f s m a l l s i g n a l - -)