university of michigan eecs 522: analog integrated ... · pdf fileuniversity of michigan eecs...

University of Michigan EECS 522: Analog Integrated Circuits

Winter 2009

Final Exam

April 28, 2009

NAME: ________________________________________________ Honor Code: I have neither given nor received unauthorized aid on this examination, nor have I concealed any violations of the Honor Code. Signature _____________________________________

Problem Points Score Initials

1 22

2 22

3 22

4 22

5 12

Total

Initials: ____________

of 17

Problem 1 (22 Points): Potpourri. This problem has 3 unrelated parts. a) Derive an expression for the conversion gain of the mixer shown below. Assume the LO

amplitude is much larger than the RF amplitude, and that the diodes are ideal and switch instantaneously.

of 17

b) Use the non‐inverting amplifier below. Assume the opamp is ideal and noiseless. Derive an expression for the noise factor of the amplifier. Simplify your expression so that it is in terms of resistor values only.

vout

vs

RS

R2R1

Initials: ____________

of 17

c) The following schematic is the equivalent tank for a Colpitts oscillator. Derive an expression for the equivalent parallel resistance for the tank below at self resonant frequency . Assume ’s are high.

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Problem 2 (22 Points): Use the following VGA for all parts of this problem. Assume the FET is in the linear region (no channel length modulation) and the opamp is ideal. a) Derive a large signal expression for in terms of . Your answer should be in the form

.

R

vin

vout

VB

Initials: ____________

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b) Derive expressions for the and terms where . The general expression for the Taylor Series given below, where is the nth derivative of , evaluated at .

!

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c) Derive an expression for the input‐referred 2nd‐order intercept point IIP2. You answer may be in terms of and . The IIP2 is defined for a two‐tone test as the input amplitude at which the intermodulation amplitude (term at sum/difference frequency) is equal to the fundamental amplitude.

Initials: ____________

of 17

Problem 3 (22 Points): The following schematic is the half‐circuit of a biquadratic cell, a basic building block that can be used to realize arbitrary active filters. This particular biquad benefits from reduced in‐band noise. Use this schematic to answer the following parts. Ignore body effect, channel length modulation, and parasitic capacitance. Assume ( ). a) Derive an expression for the transfer function / of

the biquad. Using the following general expression for a 2nd‐order filter, solve for and of the biquad.

1

1 1 1

iin

C2

-1

ac

ac

iout

C1

M1

M2

of 17

b) Assume , / , and 1 . Derive the transfer function from drain thermal noise in to the output. In the space below, sketch the spectral density of the mean‐square noise at the output.

iin

C-1

ac

ac

iout

C

M1

M2

Initials: ____________

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c) Assume , / , and 1 . Derive the transfer function from drain thermal noise in to the output. In the space below, sketch the spectral density of the mean‐square noise at the output.

iin

C-1

ac

ac

iout

C

M1

M2

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Problem 4 (22 Points): The following schematic is a non‐inverting amplifier with a gain of / 1 / . Ignore body effect, channel length modulation, and parasitic

capacitances. is a DC current source with infinite output resistance (infinite resistances can sometimes produce infinite gains). a) Draw the low‐frequency small‐signal circuit for the amplifier and derive the transfer

function / . Do not include .

IBIAS

M1

M2

vout

R2

R1

vin

Initials: ____________

of 17

b) Consider only drain thermal noise in , does transistor contribute noise to the output? If so, derive an expression for /∆ from only. Do not include .

IBIAS

M1

M2

vout

R2

R1

vin

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c) Derive expressions for /∆ from thermal noise in the two resistors, and . Do not include .

IBIAS

M1

M2

vout

R2

R1

vin

Initials: ____________

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Problem 5 (12 Points): The following parts are multiple choice. Circle only one answer for each part. Refer to the circuit below for all parts. is a DC bias voltage, and and are large values that do not impact the RF performance. a) As the value of is decreased, in order to maintain an input match at the same target

center frequency the value of / will:

Increase | Decrease | Stay the same Briefly explain why. b) As the value increases while keeping fixed, the gain of the amplifier will:

Increase | Decrease | Stay the same Briefly explain why. c) As the value increases while keeping all else fixed, the value of / will:

Increase | Decrease | Stay the same

Briefly explain why.

LG

LS

CMIM

CGS+CMIM

fixed

RS

vs

vout

LL CL

RB

CB

VB

W/L

of 17

d) As the value increases while only and adjust to maintain input match and center

frequency, the value of IIP3 will:

Increase | Decrease | Stay the same e) As the value decreases while only and adjust to maintain input match and center

frequency, neglecting correlating gate noise, noise factor will:

Increase | Decrease | Stay the same Briefly explain why. f) As the value decreases while , , and total capacitance remain

constant, in order to maintain the same center frequency and input match the value of will:

Increase | Decrease | Stay the same

and the value of current density / will:

Increase | Decrease | Stay the same Briefly explain why.

LG

LS

CMIM

CGS+CMIM

fixed

RS

vs

vout

LL CL

RB

CB

VB

W/L

Initials: ____________

of 17

(Space for additional work)

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