Common Data : V_DD = 1.8V , L_min  = 0.18µm, µ_0N = 350cm²/V s, µ_0P  = 100cm²/V s, |V_TH | = 0.55, t_ox = 3.8nm, λ = 0.07V⁻¹, χ = 0.1.

For the curved surface of the cylinder, the unit vector is which gives  Parameterize ,Since we are confined to the first octant The flux through the slant surface is .  The top and the bottom caps are in the  directions, the contribution from these two give zero by symmetry.  There are two more surfaces if we consider the first octant, they are the positive x-z plane and the positive y-z plane., the normal to the former being in the direction of  while that for the latter is along  directions. The flux from the former is , while that from the latter is .  Adding up all the contributions, the total flux from the closed surface is zero. This is consistent with the fact that the divergence of the field is zero.

1. Consider the amplifier in Fig. 27, where of the transistors are and that of are Given

1. Calculate the poles at nodes and , if and Also calculate the phase margine for unity gain feedback.
2. How will you redesign the amplifier to have a phase margine of ?
3. Calculate the slew rate of the opamp for b part only.

Figure 27: Figure for Question 1

a.) The pole at node is given by

Here

Threfore

The pole at node is given by

The net gain of the amplifier is given by

Next we find the gain at the second pole frequency as follows.

Hence we can conclude that the gain touches at a slope of Hence we calculate the unity gain frequency as

Hence

Now we can calculate the phase margin as

Therefore the phase margin is

b.) To redesign the amplifier for a phase margin of we first find the desired position of and the corresponding value of Since , we have We can write

Therefore

will give the required phase margin.

2. An amplifier has forward gain of with two poles at and frequencies.

1. For find the phase margine with unity gain feedback if
   1. 
   2. 
   3. 
2. If the unity gain closed loop amplifier exhibits a frequency peaking of near the gain crossover frequency, then what is the phase margine?

3. Consider the circuit shown in Fig. 28

1. What is the advantage of using transistors and ?
2. Determine the of the opamp .
3. Derive the maximum output swing
4. Derive the value of needed for a phase margine of 60^o  at unity gain feedback when
5. Determine the slew rate of the opamp

Figure 28 : Figure for Question 3

The divergence of the field is 3. The flux, therefore, is  3 times the volume of the cone which is The flux is thus  The  direct calculation of the flux involves two surfaces, the slant surface and the cap, as shown in the figure. The cap is in the xy plane and has an outward normal . The flux (because on the cap z=1 and the cap is a disk of unit radius).  Thus it remains to be shown that the flux from the slanted surface vanishes. At any height z, the section parallel to the cap is a circle of radius z. Since, the height and the radius of the cap are 1 each, the semi angle of the cone is 45⁰.

Thus the normal to the slanted surface has a component  along the z direction and  in the x-y plane. The component in the xy plane can be parameterized by the azimuthal angle  and we can write  . The area element can be written as  , the factor  appears because the length element is along the slant. Thus the contribution from the slanted surface is  . Using  , this integral can be evaluated and shown to be zero.

4. Consider the opamp shown in Fig. 29

Figure 29 : Figure for Question 4

1. Calculate the gain of opamp if and for
   1. 
   2. 
   3. 
2. Calculate the output swing of the opamp
3. Calculate slew rate if and
   1. and
   2. and

This problem is to be attempted similar to the problem  5 of the tutorial, i.e., by closing the cap and subtracting the contribution due to the cap. The divergence being 3, the  flux from the closed cone is 3 times the volume of the cone which gives   The contribution from the top face (which is a disk of  radius 2 ) is . Thus the net flux is zero.  (You can also try to get this result directly as done in problem 4, where we showed that the flux from the curved surface is zero).

5.Design a PMOS input stage two stage opamp. The specifications to be met are , . Assume and stability needed is

6.Consider the amplifier shown in Fig. 30 employing a cascode stage as a amplifier. Assume node is dominant. Sketch the frequency response and explain how the circuit can be compensated .

Figure 30: Figure for Question 6

1. If the circuit is modified as shown in Fig. 31, then sketch the frequency response for

Figure 31: Figure for Question 6

1. 
2. 
3. 
2. If the circuit is modified as shown in Fig. 32, then sketch the frequency response for and

Figure 32: Figure for Question 6

1. 
2. 
3.