# Assignment

```EE 435
Homework 1
Spring 2016
Due Wed Jan 20
Problem 1
Identify one operational amplifier that has been published in one of the
following in the past 5 years:
IEEE Journal of Solid State Circuits
IEEE Trans. On Cirucits and Systems (Part 1 or Part 2)
IEEE International Symposium on Circuits and Systems
Give the circuit schematic, citation information, and briefly summarize the useful
properties that the author claims for this circuit.
Problem 2
Identify one operational amplifier that has been patented in the past 5
years. Give the circuit schematic, patent number, and briefly summarize the useful
properties that the author claims for this circuit.
Problem 3
Consider the following operational amplifier. The goal is to obtain an
expression for the small-signal output voltage in terms of the input variables VIN+ and
VIN-.
a)
b)
c)
d)
Write a complete set of small-signal equations that can be solved to obtain
VOUT. Assume the small-signal parameter go is present in all MOS devices.
Solve these equations by hand for VOUT but, if you do not have a solution at
the end of &frac12; hour, stop, and comment on your progress and the amount of
effort that you believe would be required to finish the solution.
Obtain a parametric (symbolic) solution for the transfer function VOUT/VIN
from this set of equations with MATLAB. This solution will be the ratio of
two polynomials in s. Each coefficient will be comprised of the sum of a
number of product terms. How many total product terms appear in this
solution? In this part, Vin=Vin+-VinSimplify the solution obtained with MATLAB under the assumption that all
gm terms are small compared to go terms
VDD
M3
M4
VOUT
VIN
M1
M2
VIN
CL
VB2
M9
Page 1 of 5
Problem 4
Assume the amplifier can be modeled by an input-output relationship
VIN
VOUT
-A
VOUT   AVIN  VXQ   VYQ
a)
Derive an expression that shows how the closed-loop output relates to the closedloop input in the feedback amplifier shown below. Assume the input VIN is comprised of
a quiescent component (VinQ) and a small signal component (Vinss) and is given by the
expression VIN  VinQ  Vinss
R2
VIN
R1
V1
VOUT
A
b) Show that if A is large, this expression reduces to
VOUT  
R2
R
Vinss  VXQ  2 VXQ  VinQ 
R1
R1
Page 2 of 5
Problem 5
Assume the amplifier shown below is designed in a 0.5&micro; CMOS process.
and that VDD=5V , VSS=-1.5V, and IDQ=4mA.
a)
Analytically determine the W and L needed to establish a quiescent output
voltage of 1V
VDD
IDQ
VOUT
VIN
VSS
b)
c)
d)
Verify the transfer characteristics by Spice simulation.
Analytically determine the dc voltage gain at the Q-point established in a)
Using SPICE, obtain a plot of the small signal voltage gain versus the
quiescent output voltage.
Problem 6
Assume the amplifier in Problem 5 is used as a single-input, single-output
operational amplifier. With this op amp, analytically determine the small signal voltage
gain and the quiescent output voltage for the circuit shown below if R2=20M and R1=5M.
Assume that the dc offset of VIN is 1V. Verify the gain of this amplifier with SPICE
simulation.
R2
VIN
R1
V1
VOUT
A
Page 3 of 5
Problem 7
A transresistance amplifier with a gain RT is shown. Derive an expression
for the voltage gain of the amplifier as a function of the transresistance gain RT and
determine what that reduces to if RT is very large.
R2
R1
VIN
VOUT
RT
Problem 8
(Extra Credit) It was stated in class that all even-ordered distortion terms
introduced by the amplifier vanish in symmetric fully differential amplifiers. Prove this
fact.
(Hint: assume that if an ideal differential sinusoidal signal is applied at the input, one of
the single-input outputs is given by the expression

VOUT 1  t   A1 sin(1t  1 )   Ak sin  k1t  k 
k 2
where ω1 is the frequency of the sinusoidal input and the parameters A2,A3, … and θ2,θ3,
… characterize the distortion introduced by the amplifier.)
Problem 9
(Extra Credit) A model of an operational amplifier used in a basic
feedback amplifier configuration with an inverting gain is shown below.
a) Derive the voltage gain of the feedback amplifier that includes the nonideal
effects of the input impedance RIN and the output impedance RO of the op amp.
b) Show that RIN and RO do not affect the feedback gain if the open loop votage gain
AV is large.
R2
R0
R1
VIN
V1 RIN
Op Amp
Page 4 of 5
AVV1
VOUT
RL
Problem 10
(Extra Credit) The “dead network” of a circuit is obtained by setting all
small-signal inputs to 0. That is, by replacing all ac voltage sources with short circuits
and all ac current sources with open circuits. The β of a feedback amplifier is a
characteristic of the “dead network”. Consider the basic inverting and noninverting
feedback amplifiers shown below. These are widely used as small-signal voltage
amplifiers.
R2
R1
V1
VOUT
AV
VIN
R2
R1
VIN
V1
VOUT
AV
a) Show that they both have the same “dead network”
b) The β of the two amplifiers shown is   R . Show that the gain of the
1
R1  R2
noninverting feedback amplifier can be expressed by the standard feedback
equation
V
AV
A VF  OUT =
VIN 1+A Vβ
c) Take the limit at AV goes to ∞ for the gain derived in part b) and compare with
that derived in EE 230 for the gain of the noninverting feedback amplifier.
d) Derive the gain of the inverting feedback amplifier in terms of AV and β and
comment on why it does not look like the standard feedback equation.
Page 5 of 5
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