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6.002 Spring 2020 Lecture 9 1

6.002 CIRCUITS ANDELECTRONICS

Lecture 9 – Energy processing; PWM

March 5, 2020

Contents:1. Energy Efficient Delivery2. Pulse Width Modulation3. Time Averaging4. Audio Example

Handouts:Lecture 9 notes

Announcements:Quiz 1: 3/11 from 7:30 to 9:30pm in 32-123 (everything except C’s and L’s)Quiz 1 review: 3/8 from 1 to 3pm in 4-370Quiz 1: Close books but calculators and a single two-sided page of notes allowed

6.002 Spring 2020 Lecture 9 2

All these solutions need Efficient Energy Delivery!

Energy Processor+-Supply Load

What goes here?Challenges:

- Varying Supply- Varying Load- Varying Load Demand

Efficiency Benefits: - Reduced energy use or cost / higher autonomy- Simplified Thermal management

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6.002 Spring 2020 Lecture 9 3

A Simple Solution

+-VS, Supply

(Load)

• Design min{vS} ≥ max {vL=RLiL}

• Choose vcontrol to provide iL and vL

RL+vL-

+-vcontrol

+vT-

(Transistor)

6.002 Spring 2020 Lecture 9 4

Energy Efficiency

We are only efficient at full load!!

Can we do better than this?

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6.002 Spring 2020 Lecture 9 5

The Problem Revisited

+-

vS

(Load)

• What about non-dissipative devices?• Capacitors• Inductors• Switches (via transistors)

RL+vL-

This “device” must experience the voltage vS-RLiL,

and so must dissipate the power(vS-RLiL)iLif it is dissipative…

?

6.002 Spring 2020 Lecture 9 6

An Alternative Solution

+-

vS

(Load)RL+vL-

M1

M2

M1 close; M2 open à Charge inductor, power loadM1 open; M2 close à Discharge inductor, transfer power to load

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6.002 Spring 2020 Lecture 9 7

Resistor-Inductor Networks (a quick review)

+-

vM (Load)RL+vL-

iLLM

M L

First-order, linear differential equation

Solution:

vM=0, t≥0iL(0) given 𝑖𝐿 𝑡 = 𝑖𝐿(0)𝑒!"/$

vM≠0, t≥0iL(0) given 𝑖𝐿 𝑡 = 𝑖𝐿(0)𝑒!"/$+%!

&(1 − 𝑒!"/$)

t=L/RL

6.002 Spring 2020 Lecture 9 8

Resistor-Inductor Networks (a quick review)

𝑖𝐿 𝑡 = 𝑖𝐿(0)𝑒#$/&

𝑖𝐿 𝑡 = 𝑖𝐿(0)𝑒#$/&+'!((1 − 𝑒#$/&)

t=L/RL

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Coming back to the Energy Processor…

6.002 Spring 2020 Lecture 9 9

6.002 Spring 2020 Lecture 9 10

Cyclic operation

+-

vM(t) (Load)RL+vL-

iLvM

t

M1 OnM2 off

M1 OnM2 off

VS

DT T

T = Switching periodD = Duty cycle (# from 0 to 1)

0

0 DT T t

vS/RL

iL

t=L/RL

• What is the role of D?• What is the average current?• What is the current ripple?• What is the dynamic behavior?

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6.002 Spring 2020 Lecture 9 11

Cyclic operation – Role of Duty Cycle

Large D à Large iL Small D à Small iL

t t

iL(t) iL(t)

VS/RLVS/RL

• In steady-state the output signal needs to be periodic.• For a given D, there is only one point for which the current increase during the

M1 on state is equal to the current drop during the M1 off state

6.002 Spring 2020 Lecture 9 12

Cyclic Steady-State Operation

KVL à

Average à 𝑥 ≡1𝑇."!1

"𝑥 𝑠 𝑑𝑠 for any t

Average KVL à

𝐿 23)(")2"

+ RLiL(t) = vM(t)

41 ∫"!1

" 𝐿 23)(5)2"

+ RLiL(s) ds=41 ∫"!1

" vM(t) 𝑑𝑠

61𝑖𝐿 𝑡 − 𝑖𝐿(𝑡 − 𝑇 ) + RL𝑖𝐿 =𝑣𝑀

In steady-state…𝑖𝐿 𝑡 = 𝑖𝐿 𝑡 − 𝑇 for every t

Finally … 𝑖𝐿 = ?!&)

= %*@&)

PWM (Pulse Width Modulation) control variable

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6.002 Spring 2020 Lecture 9 13

Current peak-to-peak ripple underCyclic Steady-State Operation?

M1 on

M2 onDiL

M1 On à

M2 On à

𝑣𝑆 ≈Δ𝑖𝐿𝐷𝑇

𝐿 + 𝑅𝐿𝑖𝐿

0 ≈−Δ𝑖𝐿

(1 − 𝐷)𝑇𝐿 + 𝑅𝐿𝑖𝐿

Difference à 𝑣𝑆 ≈𝐿∆𝑖𝐿𝑇

1𝐷+

11 − 𝐷

=𝐿∆𝑖𝐿

𝑇𝐷(1 − 𝐷)

∆𝑖𝐿 ≈?*1@(4!@)

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Minimize ripple:

- Maximize L … expensive- Maximize 1/T … switching losses

6.002 Spring 2020 Lecture 9 14

Dynamics

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6.002 Spring 2020 Lecture 9 15

Demo

R

C

6.002 Spring 2020 Lecture 9 16

Demo

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6.002 Spring 2020 Lecture 9 17

Summary

• Efficient energy processing is key for many electronic circuits• Non-dissipative elements can be used to temporarily store energy and,

approach, in this way 100% efficiency in energy processing.• PWM and the duty cycle are widely used methods to control power transfer• When analyzing energy processing circuits, it is important to impose

periodicity conditions and study the steady-state behavior