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6.002 Spring 2020 Lecture 18 1

6.002 CIRCUITS ANDELECTRONICS

Lecture 18 - Non-linear components, Method of assumed states

April 21, 2020

Contents:1. Non-linear components2. Circuits containing non-linear components3. Graphical analysis4. Diode models5. Method of assumed states to solve diode circuits6. Half- and full-wave rectifier circuits

Reading Assignment:Agarwal and Lang, Ch. 4 (§§4.1-4.4); Ch. 16 (§16.1-16.3)

Handouts:Lecture notes

Announcements:Lecture is being recorded and it will be posted in the password-protected part of the 6.002 website.

Quiz 2 scheduled for this Wednesday (4/22)

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6.002 Spring 2020 Lecture 18 2

What do all these devices have in common?

Light-emitting diode (LED)

Solar cell

Laser diode

Photodetector

MoS2 Rectifier

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They are all “diodes” and they are used in many applications!

Power electronics

LED displays

Photovoltaic arrays

Fiber optic communications

RF Energy Harvester

Solid state lighting

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Demo

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Key question for today…

• How do we deal with circuits that contain components with non-linear i-v characteristics such as diodes (or transistors)?

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1. Non-linear components: diode

• Diodes follow an exponential law:

IS ≡ saturation current (typically in pA – nA range)q ≡ electron charge (1.6x10-19 C)k ≡ Boltzmann constant (1.4x10-23 J/K)T ≡ temperature (K)kT/q ≡ thermal voltage (26 mV at 300 K)

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-

iD

vD

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6.002 Spring 2020 Lecture 18 8

2. Circuits containing non-linear components

• Consider simple resistor-diode circuit:

• Example: fiber optic transmitter driving laser• Want to know iD and vD for different values of V, R.• Try node method.

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iD

vD+-

R

V

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6.002 Spring 2020 Lecture 18 9

• Node method.

• Node equation:

• Transcendental equation: no closed-form solution.• Need better techniques:

– Numerical analysis– Graphical analysis– Approximations

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iD

vD+-

R

V iR

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3. Graphical analysis• Not very accurate, but provides useful physical intuition.• Consider same example again:

• Strategy:– isolate non-linear component and graph i-v characteristics– graph i-v characteristics of rest of circuit on the same chart/axis– Solution is intersection of both sets of characteristics.

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• i-v characteristics of diode:

• i-v characteristics of rest of circuit (use same variables: iDand vD):

• Graphical representation:

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iD

vD

iD

+-

R

V

iD

vD00

V

V

R

1R

diodeequation

equation forrest of circuit(”load line”)

solution!-

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iD

vD

iD

+-

R

V+

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iD

vD

iD

+-

R

V

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Graphical technique provides good physical intuition:• Effect of changing resistance:

• Effect of changing voltage:

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iD

vD00

V

V

R

iD

vD00

V

V

R

R

V

iD

vD00

V

V

R

iD

vD00

V

V

R

R

V

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4. Diode models• Not all the problems need the same level of accuracy/complexity in

the way we represent a diode… Different models.

• Exponential model (model 1):

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• Ideal diode model (model 2):

Model is quite crude: diode never dissipates power à problem in power electronics circuits

Piecewise linear model: device described by different linear branches that apply in

different regimes

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• A more accurate diode model (model 3):

Typical values of vg ~ 0.5-0.7 V (although it depends on the specific diode)

This model now accounts for power dissipation in diode

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• An even more accurate diode model (model 4):

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• Models typically have trade-offs between accuracy and complexity:

• Think of models as tools in a toolbox: àchoose the most suitable model for the job at hand

• Simpler models are “piece-wise”: – different regions of the i-v characteristics described by different

equations– discontinuous derivatives at boundaries

• How do we work with piece-wise models?

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5. Method of assumed states to solve diode circuits

1. Draw equivalent circuits for each of the assumed states of the diode.

2. For each subcircuit:• Solve for output• Identify range of inputs for which subcircuit applies

3. Assemble complete solution by “piece-wise” splicing partial solutions.

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• Example:

With

• Use model 3 for diode:

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1. Draw equivalent circuits for each of the assumed states of the diode.

Diode has two states: – open (reverse bias) or OFF– short (forward bias) or ON

diode opendiode short

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2. For each subcircuit:• Solve for output• Identify range of inputs for which it applies

• Diode in short state:

For short state to apply, iD ³ 0 (forward bias):

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• Diode in open state:

Voltage across ideal diode:

For open state to apply, vDi<0 (reverse bias):

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3. Assemble complete solution by “piece-wise” splicing partial solutions.

diode short diode open

Demo

Half-wave

rectifier c

ircuit

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6. Full-wave Rectifier

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Demo

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Summary

• Circuits with non-linear devices often result in transcendental equations.

• Graphical technique: approximate method that yields valuable physical intuition.

• A given device can be described by different models.• Models typically have trade-off between accuracy and

complexity: important to select the most suitable model for each situation.

• Method of assumed states: technique to solve circuits involving non-linear elements described by piece-wise model.

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