dr. eman azab microelectronics elct 703 (w19) lecture 1

MICROELECTRONICS ELCT 703 (W19) LECTURE 1: ANALOG MULTIPLIERS

Dr. Eman Azab

Assistant Professor

Office: C3.315

E-mail:

[email protected]. EMAN AZAB

ELECTRONICS DEPT., FACULTY OF IET

THE GERMAN UNIVERSITY IN CAIRO

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COURSE OVERVIEWCourse Team

Lecturer

Dr. Eman AzabE-mail: [email protected]

Office: C3.315

Office hours: Via E-mail

Teaching

Assistant

Eng.: Radwa KhairyE-mail: [email protected]

Office: C3.307

Office hours: Via E-mail

DR. EMAN AZAB



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Teaching Method Location

One Lecture per Week

(Thursday 2nd Slot)H6

One Tutorial per Week

(Tuesday 1st/3rd)

Check Your

Schedule

Evaluation Method Percentage %

Assignments 10

Quizzes 15

Project 10

Mid-Term 25

Final 40

COURSE GUIDELINES Please follow GUC regulations for attendance

Course Prerequisites:

Semiconductors

Electronic Circuits

Radio Frequency

Course Objectives: Analog Signal Processing Circuits Design

Analog Multipliers: Differential Amplifier

Op-amp Circuit design, non-idealities, Linear and Non-linear applications

Active RC Filters

Switched capacitor circuits and applications

Operational Trans-conductance Amplifiers Circuit design and Applications

DR. EMAN AZAB



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TENTATIVE COURSE SCHEDULELecture # Lecture Description

1Revision: Differential Amplifiers & Analog

MultipliersAnalog Multipliers Transistor level design

2 &3 Feedback and Power Amplifiers Feedback Theory in Electronic Circuits and Power amplifiers

4 & 5 Op-amp Circuit design & non-idealities Op-amp Circuit design on the transistor level & Non-ideal

characteristics of Op-amp Circuit realization

6 Compensation TheoryCompensation theory: How to design Stable closed loop systems

using Op-amp?

7 & 8 Op-amp Circuit Applications Linear and Non-linear Closed-loop Applications using Op-amps

9 Active-RC Filters Op-amp based Filters

10 & 11 Switched Capacitors Circuits SC Circuits Design Concept and Applications

12Operational Trans-conductance Amplifiers

Circuit design & ApplicationsOTA CMOS Transistor Level Circuit Design

DR. EMAN AZAB



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TENTATIVE COURSE EVALUATION SCHEDULE

Assignment Quiz Project

Milestone 1: CMOS Op-

amp Modeling (Verilog A)

Quiz 1: Feedback and

Compensation Theory

Milestone 2: CMOS Op-

amp Circuit Design

(Simulation)

Milestone 3: Op-amp

Application (Simulation)

Assignment 1: Feedback and

Compensation Theory

Mid-Term

Milestone 4: PCB for Op-

amp Application

(Hardware)

Assignment 2: Research Paper

Part 1

Quiz 2: Op-amp

nonlinear Applications

Assignment 3: Research Paper

Part 2

Quiz 3: OTA-C/Switched

cap Filters

DR. EMAN AZAB



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COURSE GRADING RULES

Grading scheme is based on GUC Regulations

Copies will be graded as ZERO

This is applicable for Assignments, quizzes and Projects

Stick to the office hours for questions

Send an e-mail for urgent questions

Attend the lectures and take notes!

All the Course material will be available on the website

DR. EMAN AZAB



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REFERENCES1. “Analysis and Design of Analog Integrated Circuits”, Gray,

Hurst, Lewis & Meyer

2. “ Fundamentals of Microelectronics”, Razavi

3. “ Design of Analog CMOS Integrated Circuits”, Razavi

4. “Analog Integrated Circuit Design”, Johns & Martin

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ANALOG MULTIPLIERS Transistor level Circuit

Design

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ANALOG MULTIPLIERS: INTRODUCTIONAnalog Multiplier is a circuit that takes two analog inputs (I/V) andgenerate an output proportional to their Product

K is the multiplication gain factor

DR. EMAN AZAB



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𝑋𝑜𝑢𝑡 = 𝐾 𝑋𝑖1 × 𝑋𝑖2

Multiplier

(K)

Xi1

Xi2

Xout

ANALOG MULTIPLIER

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Amplitude Modulation Illustration

The circuit is used in communication systems for modulation/demodulation

ANALOG MULTIPLIER

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Analog Multipliers can be one, two or four quadrant multipliers

This classification depends on the polarity of the input signals

Multiplier

(K)

Xi1

Xi2

Xout

Multiplier Xi1 Xi2

Single (one) Quadrant Unipolar Unipolar

Two Quadrant Unipolar Bipolar

Four Quadrant Bipolar Bipolar

ANALOG MULTIPLIER: EMITTER COUPLED CIRCUIT

Emitter Coupled Circuit can beused as a two Quadrant analogmultiplier

It is formed with two matchedBJT with their emitters connectedtogether

Assume Q1 and Q2 are Active and βF

is large

DR. EMAN AZAB



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𝐼𝐸1 + 𝐼𝐸2 = 𝐼

𝐼𝐶1 = 𝐼𝑠𝑒𝑉𝐵𝐸1𝑉𝑇 𝐼𝐶2 = 𝐼𝑠𝑒

𝑉𝐵𝐸2𝑉𝑇

𝐼𝐶1 ≅ 𝐼𝐸1 & 𝐼𝐶2 ≅ 𝐼𝐸2

Where Is is the reverse saturation current and VT is the thermal voltage

(25mV @ room temperature)


The input voltage Vi1 changesthe collector currents

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𝑉𝑖1 = 𝑉𝐵𝐸1 − 𝑉𝐵𝐸2𝐼𝐶2𝐼𝐶1

= 𝑒−𝑉𝑖1𝑉𝑇

𝐼𝐶1 + 𝐼𝐶2 = 𝐼𝐶1 + 𝐼𝐶1𝑒−𝑉𝑖1𝑉𝑇 ≅ 𝐼

𝐼𝐶1 =𝐼

1 + 𝑒−𝑉𝑖1𝑉𝑇

𝐼𝐶2 =𝐼

1 + 𝑒𝑉𝑖1𝑉𝑇


The ECC differential output current orvoltage is related to the input voltage“Vi1” and biasing current “I” as follows:(Prove that:)

DR. EMAN AZAB



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∆𝐼𝐶 = 𝐼𝐶1 − 𝐼𝐶2

∆𝐼𝐶 = 𝐼 tanh𝑉𝑖12𝑉𝑇

∆𝑉𝐶 = −𝐼𝑅𝐶 tanh𝑉𝑖12𝑉𝑇

∆𝑉𝐶 = 𝑉𝐶1 − 𝑉𝐶2

Note: tanh(x) ≈ x for x<<1


To use ECC as a multiplier, the followingcondition must be satisfied:

The input voltage Vi1 value must be lessthan 50mV (2VT)

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Note: tanh(x) ≈ x for x<<1

for Vi1<<2VT

∆𝑉𝐶 ≅ −𝑅𝐶2𝑉𝑇

𝐼 𝑉𝑖1

∆𝐼𝐶 ≅1

2𝑉𝑇𝐼 𝑉𝑖1∆𝐼𝐶 = 𝐼 tanh

𝑉𝑖12𝑉𝑇

∆𝑉𝐶 = −𝐼𝑅𝐶 tanh𝑉𝑖12𝑉𝑇


The ECC output is proportional to themultiplication of the differential inputvoltage “Vi1” and the biasing current “I”

ECC is a two quadrant multiplier as thecurrent “I” is unipolar

If the current “I” becomes negative, Q1 and Q2will not operate in the active mode, then theexponential equation is not valid anymore

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for Vi1<<2VT


𝐼 𝑉𝑖1∆𝐼𝐶 ≅1

2𝑉𝑇𝐼 𝑉𝑖1

𝐾𝑉 = −𝑅𝐶2𝑉𝑇

𝐾𝐼 =1

2𝑉𝑇 Note: tanh(x) ≈ x for x<<1


The ECC can be modified to be a twoquadrant voltage multiplier by replacingthe biasing current source “I” with thecircuit shown

The circuit has another condition on “Vi2”to work as a multiplier

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for Vi1<<2VT

𝐼 =𝑉𝑖2 − 𝑉𝐵𝐸,𝑜𝑛

𝑅


𝑉𝑖2 − 𝑉𝐵𝐸,𝑜𝑛𝑅

𝑉𝑖1

for Vi1<<2VT & Vi2>>VBE,on

∆𝑉𝐶 ≅ −𝑅𝐶2𝑉𝑇𝑅

𝑉𝑖2𝑉𝑖1 𝐾𝑉 = −𝑅𝐶2𝑅𝑉𝑇

Figure from Gray & Mayer, “Analysis and Design of Analog Integrated Circuits”

John Wiley & Sons, inc.

ANALOG MULTIPLIER: GILBERT CELL

The ECC is used as a basicbuilding unit for a four quadrantmultiplier: Gilbert Cell

The circuit is formed with threecross coupled ECCs as shown inFigure

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∆𝐼𝐶 = 𝐼𝐶3−5 − 𝐼𝐶4−6

∆𝐼𝐶 = 𝐼𝐶3 + 𝐼𝐶5 − 𝐼𝐶4 + 𝐼𝐶6

∆𝐼𝐶 = 𝐼𝐶3 − 𝐼𝐶4 + 𝐼𝐶5 − 𝐼𝐶6

𝐼𝐶3 − 𝐼𝐶4 = 𝐼𝐶1 tanh𝑉𝑖12𝑉𝑇

𝐼𝐶5 − 𝐼𝐶6 = 𝐼𝐶2 tanh −𝑉𝑖12𝑉𝑇

= −𝐼𝐶2 tanh𝑉𝑖12𝑉𝑇

Figure from Gray & Mayer, “Analysis and Design of Analog Integrated Circuits”



Gilbert Cell analysis (Cont.)

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∆𝐼𝐶 = 𝐼𝐶3 − 𝐼𝐶4 + 𝐼𝐶5 − 𝐼𝐶6

𝐼𝐶1 − 𝐼𝐶2 = 𝐼 tanh𝑉𝑖22𝑉𝑇

∆𝐼𝐶 = 𝐼𝐶1 − 𝐼𝐶2 tanh𝑉𝑖12𝑉𝑇

∆𝐼𝐶 = 𝐼 tanh𝑉𝑖12𝑉𝑇

tanh𝑉𝑖22𝑉𝑇

for Vi1 & Vi2 <<2VT

∆𝐼𝐶 ≅𝐼

2𝑉𝑇2𝑉𝑖1𝑉𝑖2 𝐾𝐼 =

𝐼

2𝑉𝑇2


Gilbert Cell is a four quadrantmultiplier

The differential output current isproportional to the multiplication ofthe voltages “Vi1” and “Vi2”

the voltages “Vi1” and “Vi2” mustbe less than 50mV (2VT)

The output current can beconverted to a voltage signal

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∆𝑉𝐶 ≅ −𝐼𝑅𝐶2𝑉𝑇

2𝑉𝑖1𝑉𝑖2 𝐾𝑉 = −

𝐼𝑅𝐶2𝑉𝑇

2


To remove the constraint of the input voltage to be less than 50mV, thefollowing Circuit can be used

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𝑉𝑖1′ ∝ tanh−1 𝑉𝑖1

𝑉𝑖2′ ∝ tanh−1 𝑉𝑖2

∆𝐼𝐶 = 𝐼 tanh𝑉𝑖1′

2𝑉𝑇tanh

𝑉𝑖2′

2𝑉𝑇


To remove the constraint of the inputvoltage to be less than 50mV, thefollowing Circuit can be used

Assume Q7 and Q8 Active and matched withlarge β

The Differential voltage-to-current Converterhas the output current given as:

DR. EMAN AZAB



22Figure from Gray & Mayer, “Analysis and Design of Analog Integrated Circuits”


𝑉𝑖1′ = 𝑉𝐵𝐸8 − 𝑉𝐵𝐸7

𝑉𝑖1′ = 𝑉𝑇𝑙 𝑛

𝐼𝑐8𝐼𝑐7

= 𝑉𝑇𝑙𝑛𝐼2𝐼1

𝐼2 = 𝐼𝑜1 +1

𝑅1𝑉𝑖1 𝐼1 = 𝐼𝑜1 −

1

𝑅1𝑉𝑖1


The Differential voltage-to-current Converterhas the output current given as:

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𝐼𝑐8𝐼𝑐7

= 𝑉𝑇𝑙𝑛𝐼2𝐼1

𝐼2 = 𝐼𝑜1 +1

𝑅1𝑉𝑖1 𝐼1 = 𝐼𝑜1 −

1

𝑅1𝑉𝑖1


𝐼𝑜1 +1𝑅1

𝑉𝑖1

𝐼𝑜1 −1𝑅1

𝑉𝑖1

= 𝑉𝑇𝑙𝑛1 +

𝑉𝑖1𝑅1𝐼𝑜1

1 −𝑉𝑖1𝑅1𝐼𝑜1

𝑙 𝑛1 + 𝑥

1 − 𝑥= 2 tanh−1 𝑥


1 +𝑉𝑖1𝑅1𝐼𝑜1

1 −𝑉𝑖1𝑅1𝐼𝑜1

= 2𝑉𝑇 tanh−1

𝑉𝑖1𝑅1𝐼𝑜1

Note that:


Now the Output differential Current of Gilbert Cell is:

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∆𝐼𝐶 = 𝐼 tanh𝑉𝑖1′

2𝑉𝑇tanh

𝑉𝑖2′

2𝑉𝑇

𝑉𝑖1′ = 2𝑉𝑇 tanh

−1𝑉𝑖1𝑅1𝐼𝑜1

𝑉𝑖2′ = 2𝑉𝑇 tanh

−1𝑉𝑖2𝑅2𝐼𝑜2

∆𝐼𝐶 = 𝐼 tanh2𝑉𝑇 tanh

−1 𝑉𝑖1𝑅1𝐼𝑜1

2𝑉𝑇tanh

2𝑉𝑇 tanh−1 𝑉𝑖2

𝑅2𝐼𝑜22𝑉𝑇

∆𝐼𝐶 =𝐼

𝑅1𝑅2𝐼𝑜1𝐼𝑜2𝑉𝑖1𝑉𝑖2 𝐾𝐼 =

𝐼

𝑅1𝑅2𝐼𝑜1𝐼𝑜2


Circuit Implementation of Differential voltage to differential CurrentConverter:

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𝐼𝑜1 =𝑉𝐶𝐶 − 𝑉𝐵𝐸13,𝑜𝑛

𝑅

𝐼1 = 𝐼𝑜1 +𝑉𝑖1𝑅1

𝐼1 = 𝐼𝑜1 −𝑉𝑖1𝑅1


Assignment: Draw the complete Gilbert CellMultiplier (Transistor Level)

DR. EMAN AZAB



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dr. eman azab microelectronics elct 703 (w19) lecture 1

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