Monday, February 29, 2016

Revision of Circuits and Electronics - Amplifiers - small signal model (Lecture 10)


Lecture Notes:-

To make the MOSFET operate in the saturation region,  the signal first needs to be boosted by applying a DC value, a DC voltage source, V_I , then after that apply  a signal of interest, V_A which operates within the saturation region.

The output is an inversion  of the input and there is a fair bit of distortion as the saturation region is not a straight line. There is amplification for sure  but the signal is distorted so the amplifier is non linear.

 How do we get a linear amplifier?  This can be done with the small signal trick, where we focus on a small piece of the non-linear curve. So after boosting with a DC voltage,  the input voltage needs to be shrunk to a small signal that gives a linear response.

 Let's look at the small signal method
 (1)graphically,   (2) mathematically, and (3)  from a circuit viewpoint -  in the next sequence.

(1) Graphically, recall the small Signal Model Notation  where v_I (total variable)= V_I (dc bias) + v_i (small signal)   and v_O=V_O + v_o. From  this method we will notice that a linear output can be obatined.

(2)  Mathematically,  first we need to substitute v_I=V_I+ v_i  into
v_O=V_S -( k/2(v_I - V_T)^2)*R_L.  The Bias Point Equation is  V_O=V_S - (k/2(V_I - V_T)^2)*R_L  and this is marked with an asterisk, to be used later. Then,v_O=V_O+ v_o is substituted as well. From this point we should realize that v_o  is an amplification of v_i , giving  an equation v_o= A*V_i where A is the amplification.

 Removing the terms of V_O  ( the  bias point equation), and neglecting higher  order terms of v_i, we will get v_o=- (k*R_L(V_I - V_T))*v_i.

 v_o=- g_m*R_L*v_i   where   g_m=k*(V_I - V_T)  so  v_o=- A*V_i . A is a constant w.r.t. v_i
so the circuit behaves linearly for small signals.

 Another way is to differentiate v_O=V_S -( k/2(v_I - V_T)^2)*R_L  w.r.t. v_I   taking v_I=V_I . This will be the slope at V_I.  This value will need  to be multiplied by v_i and it will be equal to v_o as v_o/v_i is  equal to the slope. (see lecture 7 for a similar case)

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