EE5320: Analog Integrated Circuit Design; Assignment 1

Nagendra Krishnapura (nagendra@iitm.ac.in)due on 2nd February 2015

f()

Vi Vo

+-

Ve f(Ve)

Vi

+

(a)

(b)Vnl+

f()Vi Vo

VoA

(c)

ke

Figure 1: Problem 2

1. Fig. 1(a) shows a nonlinearity f enclosed in a nega-tive feedback loop with a feedback fraction .

If the overall nonlinear transfer characteristic fromVi to Vo is given by g(), determine the coefficientsof Taylor series of g(Vi) around Vi = 0 in terms ofthe Taylor series coefficients of f(). For simplicity,assume f(0) = 0. Determine these up to the thirdorder.

In Fig. 1(a), deterine the small signal linear gaink+e = Ve/Vi. Fig. 1(b) shows f() preceded by again ke. In other words, in terms of small signals,the same signal is applied to f() in both Fig. 1(a) and(b). Determine the Taylor series of g1() which is thenonlinear function relating Vo to Vi in Fig. 1(b). Asbefore, compute it up to the third order and comparethe results to those obtained for Fig. 1(a).The gainA in Fig. 1(c) is the same as the small signalgain from Vi to Vo in Fig. 1(a). Determine the non-

linear voltage Vnl,(in terms of Vi and the small sig-nal parameters of Fig. 1(a)) which should be addedto the input so that (i) The part proportional to V 2i isthe same as in Fig. 1(a), (ii) The part proportional toV 3i is the same as in Fig. 1(a).Repeat the above computations of Vnl to make thenonlinear output components in Fig. 1(c) to be thesame as that in Fig. 1(b).

+-

u dt(k-1)R

R

Vo

Vf

VeVi

+-

u dt(k-1)R

R

Vo

Vf

Ve

Vi

(a)

(b)

Figure 2: Problem 1

2. Fig. 2(a) shows the amplifier studied in class.Fig. 2(b) shows the same system with the input ap-plied at a different place. Calculate the dc gain, the-3dB bandwidth, and the gain bandwidth product ofthe system and compare them to the correspondingquantities in Fig. 2(b). Also compare the loop gains.Remark on conventional wisdom such as constantgain bandwidth product, closed loop bandwidth =unity gain frequency/closed loop dc gain. What isthe reason for the discrepancy?

Draw an equivalent block diagram of Fig. 2(b) suchthat the classical form of feedback (sensed error in-

1

2tegrated to drive the output) is clearly obvious (Hint:compute the error voltage Ve).

+

+Vi Vi

R

(k-1)R

R

kR

(a) (b)

Vo Vo

Figure 3: Problem 2

3. Fig. 3(a) and Fig. 3(b) shows amplifiers which re-alize gains of k and k respectively with idealopamps. Compare the following parameters of thetwo circuits. Model the opamp as an integrator u/s.

(a) Input impedance(b) Bandwidth(c) Differential (V+(s) V(s)) and common

mode ((V+(s) + V(s))/2) input voltages ofthe opamps

Assuming that the sign of the gain is unimportantin your application, what would make you chooseone over the other? Is there any reason to chooseFig. 3(b) at all?

+

ViVo

+

Ii

Rin RoutRin

Vo

(k-1)R

R

+

Vi Rin Rout

R

+

vopa

vopa

Io

load

Rout

+

Rin Rout

R

+

vopa

vopa

Io

loadIi

(k-1)R

(a)

(c)

(b)

(d)

R

Figure 4: Problem 3: (a) VCVS, (b) CCVS, (c) VCCS,(d) CCCS

4. Fig. 4 shows the four types of controlled sourcesusing an opamp. Model the opamp as an integra-tor u/s. For each of these, calculate the trans-fer ratio (output/input), input impedance, and output

impedance at (a) dc, and (b) an arbitrary frequency. For (b), set Rout = 0 when calculating the inputimpedance and Rin = while calculating the out-put impedance. What happens to these three quanti-ties at high frequencies in each case?

5. The loop gain L(s) of a system with N extra polesand M < N extra zeros is given by

L(s) =u,loops

Mm=0 bms

mNn=0 ans

n

b0 = a0 = 1. What does the loop gain step re-sponse (inverse laplace transform of L(s)/s) looklike after the initial transient period? Give youranswer in terms of the poles of the additional fac-tor (Hint: Split L(s) into a sum of two parts, one ofwhich is u,loop/s; This problem doesnt require alot of algebra, but requires reasoning and basics ofpolynomials)

6. Assume that an opamp has two poles at the originand a zero elsewhere, i.e. its transfer function isgiven by

A(s) =uz1s2

(1 +

s

z1

)

If this opamp is placed in unity feedback, determinethe natural frequency and the damping or quality fac-tor. Determine the location of the zero z1 for criticaldamping. Sketch the loop gain magnitude and phaseplots of such a system and compare it to the othergood cases that you are already familiar with.