Circuit Intuitions Ali Sheikholeslami Dept. of Elec. & Comp. - - PowerPoint PPT Presentation

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Circuit Intuitions Ali Sheikholeslami Dept. of Elec. & Comp. - - PowerPoint PPT Presentation

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Circuit Intuitions Ali Sheikholeslami Dept. of Elec. & Comp. Engineering University of Toronto, Canada ali@ece.utoronto.ca sponsored by SSCS Distinguished Lecture Program August 23, 2019 University of California, San Diego Ali

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SLIDE 1

Circuit Intuitions

Ali Sheikholeslami

  • Dept. of Elec. & Comp. Engineering

University of Toronto, Canada ali@ece.utoronto.ca sponsored by SSCS Distinguished Lecture Program August 23, 2019 University of California, San Diego

Ali Sheikholeslami Circuit Intuitions 1 of 39

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SLIDE 2

Snapshots from Magazine

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SLIDE 3

Outline

Ali Sheikholeslami Circuit Intuitions

  • Why Circuit Intuitions?
  • Overview of Articles Series
  • Looking into a Node

n Use of Thevenin and Norton Equivalent Circuits

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SLIDE 4

Why Circuit Intuitions?

  • Turning circuit design/analysis into a fun game!
  • Gain intimate understanding of how circuits behave
  • Making things simple, obvious!
  • A gateway to innovation!

Ali Sheikholeslami Circuit Intuitions 4 of 39

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SLIDE 5

Circuit Intuitions Series

  • Methods of Analysis

n Looking into a Node n Source Degeneration; Bandwidth Extension n Miller’s Theorem; Miller’s Approximation

  • Fundamental Concepts

n Process Variation and Pelgrom’s Law n Why Sinusoids? Reinventing the Wheels, Random Walk n Capacitor Analogy (3 articles) n Norton and Thevenin Equivalent Circuits (3 articles)

  • Special Circuits

n Negative Cap.; Chopper Amp.; Capacitor as a Resistor

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SLIDE 6

Focus of this Talk

  • Methods of Analysis

n Looking into a Node n Source Degeneration; Bandwidth Extension n Miller’s Theorem; Miller’s Approximation

  • Fundamental Concepts

n Process Variation and Pelgrom’s Law n Why Sinusoids? Reinventing the Wheels n Capacitor Analogy (3 articles) n Norton and Thevenin Equivalent Circuits (3 articles)

  • Special Circuits

n Negative Cap.; Chopper Amp.; Capacitor as a Resistor

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SLIDE 7

Linear Time Invariant (LTI) Circuits

  • Output is a linear combination of inputs!

n Simple Case: If there are no storage elements (C or L) 𝑧 = ℎ$𝑦$ + ℎ'𝑦' + ℎ(𝑦( + ⋯

  • Superposition holds!

𝑧 = 𝑧$ + 𝑧' + 𝑧( + ⋯

  • Output is sum of contributions from individual inputs!
  • Thevenin/Norton Theorem: consequence of superposition

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SLIDE 8

Thevenin/Norton Theorems

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SLIDE 9

An Intuitive Proof of Thevenin

Ali Sheikholeslami Circuit Intuitions

𝑤+,- 𝑢 = 𝑤+/ 𝑢 − 𝑆23𝑗5(𝑢)

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SLIDE 10

An Intuitive Proof of Norton

Ali Sheikholeslami Circuit Intuitions

𝑗5 𝑢 = 𝑗8/ 𝑢 − ⁄ 𝑤+,-(𝑢) 𝑆23

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SLIDE 11

Small-Signal Model of NMOS/PMOS

Assumption: All transistors are in Saturation Region!

  • gm is the short-circuit transconductance of the transistor
  • gmb is additional gm due to non-zero vbs
  • ro is the transistor output resistance

Ali Sheikholeslami Circuit Intuitions

G D S B G S D B

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SLIDE 12

Circuits with More Transistors

  • Analysis becomes too cumbersome very quickly
  • Practice of KVL/KCL, leaves no room for intuition

Ali Sheikholeslami Circuit Intuitions

RL vin M2 M1 vd1 vout vd1

ro gmbvbs gmvgs ro gmbvbs gmvgs

vout M1 M5 vout M2 M3 M4 vd5 vin

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SLIDE 13

Basic Premise

  • Assume low-frequency analysis in this talk!
  • That is, capacitors do not show up!
  • In small-signal, transistors behave like LTI systems
  • Superposition holds!
  • Every circuit has a Thevenin/Norton equivalent circuit

n Open-circuit voltage source (voc) in series with Req n Short-circuit current source (isc) in parallel with Req n When there is no input signal, they are just resistors!

  • We rely on Thevenin/Norton equivalent circuits ONLY!

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SLIDE 14

“Library Elements”

  • We call these “library elements”
  • Commonly-Used Configurations

n Looking into the gate and the drain n Looking into the source n Diode-Connected transistor n Looking into the drain with source degeneration n Looking into the source with load at the drain n Thevenin/Norton Equivalent looking into the drain n Thevenin/Norton Equivalent looking into the source

  • Let us build this library; one element at a time

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SLIDE 15

Looking into the Gate

  • Looking into the gate, we see infinite resistance

Ali Sheikholeslami Circuit Intuitions

Req = ∞

①

Req ro gmbvbs gmvgs g g Req=?

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SLIDE 16

Looking into the Drain

  • Looking into the drain, while gate and source are

grounded, we see ro!

Ali Sheikholeslami Circuit Intuitions

Req = ro

②

Req d ro gmbvbs gmvgs g d ro g Req=?

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SLIDE 17

Looking into the Source

  • gme = gm + gmb= 1.1-1.2 gm
  • Looking into the source, while gate and drain are

grounded, we see a resistor whose value is 𝑆23 = 𝑠

+||1/𝑕F2 ≈ 1/𝑕F2

Ali Sheikholeslami Circuit Intuitions

Req = ro || 1/gme

③

Req ro

  • gmbvs
  • gmvs

g s ro gmevs ro 1/gme s s Req=?

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SLIDE 18

Diode-Connected Transistor

  • A diode-connected transistor is a resistor whose

value is 𝑆23 = 𝑠

+||1/𝑕F ≈ 1/𝑕F

Ali Sheikholeslami Circuit Intuitions

Req = ro || 1/gm Req

④

g ro gmvin vin ro vin 1/gm Req=?

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SLIDE 19

Looking into the Drain with RS

  • Also known as Source Degeneration
  • Effective way to increase the output impedance
  • Multiplying the source resistance by gmero

Ali Sheikholeslami Circuit Intuitions

Req = Rs + ro + gme roRs

⑤

RS Req ro gmevs s

RS

ro

RS

vs gmerovs ro

RS

gmeroRS RS Req=?

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SLIDE 20

Looking into the Source with RD

  • If RD<<ro, then 𝑆23 ≈ 1/𝑕F2
  • RD is divided by (1+gmero) and appears at the source

Ali Sheikholeslami Circuit Intuitions

⑥

RD Req 𝑆23 = 𝑆K + 𝑠

+

1 + 𝑕F2𝑠

+

ro gmevs RD Req=? RD RD ro gmerovs vs vs RD ro

  • gmerovs

vs

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SLIDE 21

Looking into the Drain with Input

Two Methods:

  • Find the Thevenin Equivalent Circuit:

n

  • pen-circuit voltage voc in series with Req
  • Find Norton Equivalent Circuit:

n short-circuit current isc in parallel with Req

  • Note that voc=isc x Req

In this circuit, easier to find voc first à vs=0, no body effect, voc=-gmrovin à isc=voc/Req Norton more intuitive due to high Req

Ali Sheikholeslami Circuit Intuitions

⑦

RS vin voc isc 𝑤+/ = −𝑕F𝑠

+𝑤LM

𝑗8/ = 𝑤+//𝑆23

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SLIDE 22

Looking into the Source with Input

Circuit Intuitions

⑧

RD vin voc 𝑗𝑡𝑑 = 𝑕𝑛𝑤𝑗𝑜𝑠𝑝 𝑠𝑝 + 𝑆K 𝑤𝑝𝑑 = 𝑗𝑡𝑑𝑆𝑓𝑟 isc

  • Find Norton Equivalent Circuit first:

n short-circuit current isc in parallel with Req

  • Shorting output to ground à vs=0

ro gmvin RD isc isc à Use current division à 𝑗8/ = 𝑕F𝑤LM

UV UVWXY

à Given small Req, Thevenin is more intuitive

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SLIDE 23

Putting It All Together (1 to 4)

Ali Sheikholeslami Circuit Intuitions

Req = ∞

①

Req Req = ro Req

②

Req = ro || 1/gme

③

Req Req = ro || 1/gm Req

④

  • Looking into one terminal while the other two grounded
  • Looking into the drain of a diode-connected transistor
  • No additional resistor in the circuit

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SLIDE 24

Putting It All Together (5 to 8)

Ali Sheikholeslami Circuit Intuitions

Req = Rs + ro + gme roRs

⑤

RS Req

⑥

RD Req 𝑆23 = 𝑆K + 𝑠

+

1 + 𝑕F2𝑠

+

⑦

RS vin voc isc 𝑤+/ = −𝑕F𝑠

+𝑤LM

𝑗8/ = 𝑤+//𝑆23

⑧

RD vin voc 𝑗𝑡𝑑 = 𝑕𝑛𝑤𝑗𝑜𝑠𝑝 𝑠𝑝 + 𝑆K 𝑤𝑝𝑑 = 𝑗𝑡𝑑𝑆𝑓𝑟 isc

  • Adding resistors and input to the circuit

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SLIDE 25

Finding Voltage of “any” Node

Method 1: Use Norton equivalent circuit

  • Short the node to ground; find isc
  • Find Req (using library elements)
  • Multiply the two: vout = isc x Req

Method 2: Use Thevenin equivalent circuit

  • Open the load; find voc
  • Find Req (same as in Method 1)
  • Use voltage divider rule to find vout

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SLIDE 26

Example: Common-Source Amplifier

Ali Sheikholeslami Circuit Intuitions

Finding vout: Use isc x Req: à isc = - gm vin à Req = RL || ro à vout = - gm (RL || ro) vin RL vin vout isc Req

  • What is the intuition here?

n How to increase the gain? How to increase vout ? n Output voltage comes from isc x Req n à either increase isc or Req

  • How can we increase these two?

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SLIDE 27

Example: Cascode Circuit

Ali Sheikholeslami Circuit Intuitions

RL M2 M1 vd1 vout vin To find the output voltage

  • Use isc x Req:
  • Find out how isc and Req change

compared to those for CS amplifier

  • Gain intuition on why the gain

changes from one to the other.

  • Also, find intermediate voltages

(such as vd1) for added insight

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SLIDE 28

Cascode Circuit: Find any Voltage (1)

Ali Sheikholeslami Circuit Intuitions

Finding vd1: Use Thevenin: ② à Req1 = ro1 ⑦ à voc1 = - gmro1vin ⑥ à Req2 = (ro2 + RL)/ (1+ gme2 ro2) à vd1 = - gmro1vin (ro1 || Req2) Finding vout: Use Req2 to find the load current first : à iL = vd1/Req2 à vout = iLRL = vd1 RL /Req2 RL voc1, Req1 M2 M1 vd1 vout RL Req1 M2 Req2 vd1 vout voc1 Method 1: vin

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SLIDE 29

Cascode Circuit: Find any Voltage (2)

Ali Sheikholeslami Circuit Intuitions

Finding vd1: (as in previous slide) à vd1 = - gmro1vin (ro1 || Req2) Treat vd1 as ideal source: Find isc and Req at the output: à isc = vd1/(1/gme||ro) à Req = ro||RL (not Req of original cct) à vout = iscReq RL voc1, Req1 M2 M1 vd1 vout RL M2 Req vd1 vout vd1 Method 2: vin

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SLIDE 30

What Is the Intuition here?

Ali Sheikholeslami Circuit Intuitions

  • isc remained the same
  • Req increased by gmro
  • Voltage gain increased by gmro

RL vin vout isc Req RL M2 M1 vd1 vout vin

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SLIDE 31

Example: Differential Pair (1)

Ali Sheikholeslami Circuit Intuitions

M1 M5 isc1, Req1 vout M2 M4 Req2 Req5 vd5 Finding vd5: Use the Norton Equivalent at the source of M1: ④ à Req3 = ro3 || 1/gm3 ⑥ à Req1 = (Req3+ro1)/(1+gme1ro1) ⑧ à isc1 = gm1 vin ro1/(ro1+Req3) ② à Req5 = ro5 ④ à Req4 = ro4 || 1/gm4 ⑥ à Req2 = (Req4+ro2)/(1+gme2ro2) à vd5 = isc1 (Req1 || Req5 || Req2) Finding vout: à vout = (vd5/Req2) Req4 Req3 Req4 M3 vin Method 1:

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SLIDE 32

Example: Differential Pair (2)

Ali Sheikholeslami Circuit Intuitions

vout M2 M4 Treat vd5 as ideal source: (from previous slide) à vd5 = isc1 (Req1 || Req5 || Req2) Finding vout: Find isc and Req at the output: à isc = vd5 /(ro2||1/gme2) à Req = ro2||ro4||1/gm4 Note: not equal to Req of original circuit à vout = isc Req vd5 Method 2:

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SLIDE 33

Example: Diode-Connected Cascode

Ali Sheikholeslami Circuit Intuitions

M2 M1 Req

  • Not one of the standard library elements
  • Apply a voltage, measure current
  • Use Superposition to find current
  • Ratio of test voltage to current is Req
  • Intuition?

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SLIDE 34

Diode-Connected Cascode

Ali Sheikholeslami Circuit Intuitions

M2 M1 vx ix M2 M1 vx ix vx

  • The two circuits are

equivalent

  • Treat two sources as

independent

  • Use Superposition

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SLIDE 35

Diode-Connected Cascode

Ali Sheikholeslami Circuit Intuitions

vx M2 M1 vx ix1 M2 M1 vx ix2

  • 𝑗Z = 𝑗Z$ + 𝑗Z'
  • 𝑆23 = 𝑆23$|| 𝑆23'
  • 𝑆23$ ≅ 𝑕F2'𝑠+$𝑠+'
  • 𝑆23' ≅ 1/𝑕F$
  • 𝑆23 ≅ 1/𝑕F$

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SLIDE 36

Summary

  • In small-signal, all transistor circuits (assuming

saturation region) can be treated as LTI circuit

  • Transistor circuits without a signal source are resistors

n This is for low-frequencies only n At high-frequencies, we should add capacitors n See other circuit intuition articles for high frequencies

  • Transistor circuits with input signal are represented by:

n Their Thevenin equivalent circuits, or n Their Norton equivalent circuits

  • There are different ways to arrive at the solution
  • Use the method that adds intuition!

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SLIDE 37

References (1 of 2)

All in Solid-State Circuit Magazine

  • [A20] Circuit Intuitions: Transfer Resistor, Winter 2019.
  • [A19] Circuit Intuitions: Thevenin and Norton Equivalent Circuits, Part 3, Fall 2018.
  • [A18] Circuit Intuitions: Thevenin and Norton Equivalent Circuits, Part 2, Su. 2018.
  • [A17] Circuit Intuitions: Thevenin and Norton Equivalent Circuits, Spring 2018.
  • [A16] Circuit Intuitions: Random Walk in a Ring, Winter 2018.
  • [A15] Circuit Intuitions: Reinventing the Wheel, Fall 2017.
  • [A14] Circuit Intuitions: Capacitor as a Resistor, Summer 2017.
  • [A13] Circuit Intuitions: Why Sinusoids?, Spring 2017.
  • [A12] Circuit Intuitions: A Capacitor Analogy, Part 3, Winter 2017.
  • [A11] Circuit Intuitions: A Capacitor Analogy, Part 2, Fall 2016.
  • [A10] Circuit Intuitions: A Capacitor Analogy, Part 1, Summer 2016.
  • [A9] Circuit Intuitions: Chopper Amplifier, Spring 2016.

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SLIDE 38

References (2 of 2)

All in Solid-State Circuit Magazine

  • [A8] Circuit Intuitions: Offset Cancellation, Winter 2016.
  • [A7] Circuit Intuitions: Miller's Approximation, Fall 2015.
  • [A6] Circuit Intuitions: Miller's Theorem, Summer 2015.
  • [A5] Circuit Intuitions: Bandwidth Extension, Spring 2015.
  • [A4] Circuit Intuitions: Process Variation and Pelgrom's Law, Winter 2015.
  • [A3] Circuit Intuitions: Negative Resistance, Fall 2014.
  • [A2] Circuit Intuitions: Source Degeneration, Summer 2014.
  • [A1] Circuit Intuitions: Looking into a Node, Spring 2014.

n See also Correction to Looking Into a Node, Summer 2014. Ali Sheikholeslami Circuit Intuitions 38 of 39

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SLIDE 39

Acknowledgement

I would like to thank my undergraduate students at the University of Toronto, who have been the inspirations behind writing these articles.

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