Electronic Circuits ELCT604 (Spring 2020) Chapter 4 ...eee.guc.edu.eg › Courses › Electronics › ELCT604 Electronic Circuits › … · The 3-dB Higher cutoff frequency is the

Electronic Circuits ELCT604

(Spring 2020)

Chapter 4 Amplifiers Frequency

ResponseDr. Eman Azab

Assistant Professor

Office: C3.315

E-mail: [email protected]

Dr. Eman Azab

Electronics Dept., Faculty of IET

The German University in Cairo

1

Introduction In the previous lectures we studied the voltage

amplifiers, without taking into account any capacitances

in the circuit

The voltage gain was calculated while all External capacitors

were Short circuit

The Short circuit is based on the assumption that: the

product of the input signal frequency with the capacitor

value is infinity

However, this is not valid for low frequencies!

Dr. Eman Azab



2

𝑍𝐶𝑒𝑥𝑡 =1

𝑠𝐶𝑒𝑥𝑡≅ 0

Introduction

The physical structure and biasing of the transistors (MOS

or BJT) result in Internal capacitances between its

terminals

However, we have considered their impedances as Open

Circuit

However, this is not valid for High frequencies!

Dr. Eman Azab



3

MOS High-Frequency ModelBJT High-Frequency Model

𝑍𝐶𝑖𝑛𝑡 =1

𝑠𝐶𝑖𝑛𝑡≅ ∞

Introduction Intrinsic Capacitances of MOS Transistor

Enhancement MOS transistor has intrinsic parasitic

capacitances

They are mainly originated from:

1. MOS physical structure (gate and Overlap Caps.)

2. Channel charge

3. Reverse biased PN junctions (Bulk and Drain/Source)

Most of these caps. are nonlinear and voltage dependent

Dr. Eman Azab



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DS

G

B

CGDCGS

CSB CDBCGB

Introduction

MOS Structure Caps

Dr. Eman Azab



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tox

n+ n+

Cross section

L

Gate oxide

xd xd

Ld

Polysilicon gate

Top view

Gate-bulkoverlap

Source

n+

Drain

n+W

Introduction Channel Charge Caps

Dr. Eman Azab



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Introduction

Reverse biased Junction Caps

Dr. Eman Azab



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Bottom

Side wall

Side wall

Channel

SourceND

Channel-stop implantN A1

Substrate NA

W

xj

LS

Objective

We want to study the effect that these capacitances have

on the amplifier’s voltage gain

For this study, we will consider that the frequency spectrum is

divided into three bands

Dr. Eman Azab



8

Frequency Response Analysis

Definitions:

‘AMF’ Mid-band Gain: the amplifier voltage gain while all

capacitors are neglected

‘ALF’ Low-Frequency Band Gain: the amplifier voltage gain

while External capacitors are taken into consideration and

Internal capacitors are neglected

‘AHF’ High-Frequency Band Gain: the amplifier voltage

gain while Internal capacitors are taken into consideration

and External capacitors are neglected

Bandwidth of Amplifier: the difference between the higher

‘fH’ and lower ‘fL’ 3-dB cutoff Frequencies

Dr. Eman Azab



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How to calculate the Lower 3-dB ‘fL’ cutoff frequency?

Assume that, the amplifier has ‘n’ External Capacitor

Each capacitor will generate a pole and a transmission Zero

‘fL’ is the frequency at which the voltage gain drops with 3-dB

from its maximum value

Dr. Eman Azab



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Example: Bode Plot for Amplifier with Three External Capacitors

𝐴𝐿𝐹(𝑠) =𝐴𝑀𝐹

1 +2𝜋𝑓𝐿1𝑠 1 +

2𝜋𝑓𝐿2𝑠 … 1 +

2𝜋𝑓𝐿𝑛𝑠

𝑓𝐿 = 𝑓𝐿12 + 𝑓𝐿2

2 +⋯ .+𝑓𝐿𝑛2


How to calculate the Higher 3-dB ‘fH’ cutoff frequency?

Assume that, the amplifier has ‘n’ Internal Capacitor

Each capacitor will generate a pole

The 3-dB Higher cutoff frequency is the frequency at which the

voltage gain drops with 3-dB from its maximum value

Dr. Eman Azab



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Example: Bode Plot for Amplifier with One Internal Capacitor

𝐴𝐻𝐹(𝑠) =𝐴𝑀𝐹

1 +𝑠

2𝜋𝑓𝐻11 +

𝑠2𝜋𝑓𝐻2

. . . 1 +𝑠

2𝜋𝑓𝐻𝑛

𝑓𝐻 =1

1𝑓𝐻12 +

1𝑓𝐻22 +⋯ .+

1𝑓𝐻𝑛2

Calculating the Lower 3-dB

frequency

Dr. Eman Azab



12

Calculating fL The lower cutoff 3-dB frequency is calculated by NOT considering

the impedances of the External capacitances Short Circuit

The poles external capacitors should be calculated by deriving

the voltage gain

However, deriving the small signal model with ‘n’ external

capacitors is complicated

We will consider each external capacitor at a time while the

other external capacitors will be considered Short Circuit

Repeat for every capacitor and all the poles will be evaluated

The internal capacitors while calculating the lower cutoff 3-dB

frequency are neglected (considered open circuit)

Dr. Eman Azab



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Example

Calculate the Low-frequency band gain for the Common

Source Amplifier

Dr. Eman Azab



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1. Calculate the Mid-band Gain

2. Use the Short-Circuit time Constant method to determine the lower 3-dB

Frequency

Example

1. Calculate the Mid-band Gain

2. Calculate the pole due to CC1 by deriving the gain

(without considering CC1 impedance a short circuit)

Dr. Eman Azab



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𝐴𝑀𝐹 =𝑣𝑜𝑢𝑡𝑣𝑖𝑛

= −𝑔𝑚 𝑟𝑑𝑠 ∕∕ 𝑅𝐿 ∕∕ 𝑅𝐷𝑅1 ∕∕ 𝑅2

𝑅1 ∕∕ 𝑅2 + 𝑅𝑠𝑖𝑔

Short-Circuit Time constant

method Instead of calculating the voltage gain more than once, we

can use the following Short-Circuit time Constant method

Consider one external capacitor at a time while the others are

short Circuit

Each External capacitor ‘Ci’ will generate a pole ωL,i

This pole is given by:

Rtot is the resistance seen by the capacitor while:

1. All other capacitors are neglected (Externals are Shorted, Internal are

Open)

2. Deactivating the AC input signal

3. Replace the capacitor with a battery

4. RThevinen at the battery is Rtot

Dr. Eman Azab



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𝜔𝐿,𝑖 =1

𝐶𝑖𝑅𝑡𝑜𝑡

Example

2. Calculate the low frequency Poles (Cont.)

A. CC1 Pole

Dr. Eman Azab



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𝜔𝐿,𝐶1 =1

𝐶𝐶1 𝑅1 ∕∕ 𝑅2 + 𝑅𝑠𝑖𝑔

Example


B. CC2 Pole

Dr. Eman Azab



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𝜔𝐿,𝐶2 =1

𝐶𝐶2 𝑟𝑑𝑠 ∕∕ 𝑅𝐷 + 𝑅𝐿

Example


C. CS Pole

Dr. Eman Azab



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Neglecting rds

𝜔𝐿,𝐶𝑠 ≅1

𝐶𝑆 𝑅𝑆 ∕∕1𝑔𝑚

Notes on the Example1. The lower 3-dB frequency is calculated using this

formula

2. Each Coupling capacitor generates a Transmission Zero

at s=0

3. Bypassing Capacitor generates a Zero at:

4. Proof: Derive the low-frequency band voltage gain for CS

without using Short-Circuit Time-constant method

(Assignment!)

Dr. Eman Azab



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𝑓𝐿 = 𝑓𝐿,𝐶12 + 𝑓𝐿,𝐶2

2 + 𝑓𝐿,𝐶𝑠2

𝜔𝑍 =1

𝐶𝑆𝑅𝑆

Notes on the Example5. The Low-frequency band gain for this example is given

by:

6. Sketch the Bode Plot for the Low frequency gain

(Assignment!)

Dr. Eman Azab



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𝐴𝐿𝐹(𝑠) =𝐴𝑀𝐹𝑠

2 𝑠 + 𝜔𝑍

𝑠 + 𝜔𝐿,𝐶1 𝑠 + 𝜔𝐿,𝐶2 𝑠 + 𝜔𝐿,𝐶𝑠

Calculating the Higher 3-dB

frequency

Dr. Eman Azab



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Calculating fH The higher 3-dB cutoff frequency is calculated by NOT

considering the impedances of the Internal capacitances Open

Circuit

The internal capacitors poles can be calculated by deriving the

voltage gain

However, deriving the small signal model with more than one

internal capacitor is complicated

We will consider each internal capacitor at a time while the

other internal capacitors will be considered Open Circuit

Repeat for every capacitor and all the poles will be evaluated

The external capacitors while calculating the higher 3-dB cutoff

frequency are neglected (considered short circuit)

Dr. Eman Azab



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Open-Circuit Time constant

method Alternative Solution: USE Open-Circuit time Constant

method:

Consider one Grounded internal capacitor at a time while

the others are Open Circuit, Each capacitor ‘Ci’ will generate

a pole ωH,i

This pole is given by:

Rtot is the resistance seen by the capacitor while:

1. All other capacitors are neglected (Externals are Shorted, Internal are

Open)

2. Deactivating the AC input signal

3. Replace the capacitor with a battery

4. RThevinen at the battery is Rtot

Dr. Eman Azab



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𝜔𝐻,𝑖 =1

𝐶𝑖𝑅𝑡𝑜𝑡

Example

Calculate the High-frequency band gain for the

Common Source Amplifier

Dr. Eman Azab



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1. Calculate the Mid-band Gain (Solved in Previous Example)

2. Use the Open-Circuit time Constant method to determine the Higher 3-

dB Frequency

Example

2. Calculate the High frequency Poles

Equivalent Circuit in High Frequency Band

Note that: Cds=Cdb

Cgd is not Grounded!, thus we have to use Miller’s

Theorem before using Open-Circuit Time Constant

method

Dr. Eman Azab



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Miller’s Theorem

Used in case we have a feedback Capacitor in the

amplifier (Connecting Input and Output Nodes)

Miller replaces the Feedback Element with two

grounded elements, one at each port of the network

Dr. Eman Azab



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Miller’s Theorem

If ‘Z’ is a capacitive impedance, then:

The Equivalent Circuit in High Frequency band is

shown

Dr. Eman Azab



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1 1C C A 2

11C C

A

/ / / /m ds D LA g r R R

Example


We can add together parallel capacitors

Calculate the high frequency poles accordingly

Dr. Eman Azab



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1in gsC C C

2out dsC C C

Example


A. Cin High Frequency Pole

Dr. Eman Azab



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𝜔𝐻,𝐶𝑖𝑛 =1

𝐶𝑖𝑛 𝑅𝑠𝑖𝑔 ∕∕ 𝑅1 ∕∕ 𝑅2

Example


B. Cout High Frequency Pole

Dr. Eman Azab



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𝜔𝐻,𝐶𝑜𝑢𝑡 =1

𝐶𝑜𝑢𝑡 𝑅𝐿 ∕∕ 𝑅𝐷 ∕∕ 𝑟𝑑𝑠

Notes on the Example

1. The Higher 3-dB frequency is calculated using this

formula

2. The high-frequency band gain is given by:

3. Sketch the Bode Plot for the High frequency gain

(Assignment!)

Dr. Eman Azab



32

𝑓𝐻 =1

1𝑓𝐻,𝐶𝑖𝑛2 +

1𝑓𝐻,𝐶𝑜𝑢𝑡2

𝐴𝐻𝐹(𝑠) =𝐴𝑀𝐹

1 +𝑠

𝜔𝐻,𝐶𝑖𝑛1 +

𝑠𝜔𝐻,𝐶𝑜𝑢𝑡

Notes on Drawing Bode plots

Wednesd

ay,

March

25, 2020

Dr. Eman Azab



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Electronic Circuits ELCT604 (Spring 2020) Chapter 4 ...eee.guc.edu.eg › Courses › Electronics › ELCT604 Electronic Circuits › … · The 3-dB Higher cutoff frequency is the