6.002.1x Circuits and Electronics 1
Lumped Element Abstraction Consider V The Big Jump from physics to EECS Suppose we wish to answer this question: What is the current through the bulb? Reading: Skim through Chapter 1 of A&L 7
We could do it the Hard Way Apply Maxwell s V I? Differential form Integral form Faraday s Continuity Others E J E B = t ρ = t = l l l ε ρ 0 φb E dl = t q J ds = t q E ds = ε 0 l l l 8
Instead, there is an Easy Way First, let us build some insight: Analogy I ask you: What is the acceleration? You quickly ask me: What is the mass? I tell you: You respond: Done!!! 9
Instead, there is an Easy Way In doing so, you ignored l the object s shape l its temperature l its color l point of force application l Point-mass discretization F a? 10
The Easy Way Consider the filament of the light bulb. A B We do not care about l how current flows inside the filament l its temperature, shape, orientation, etc. We can replace the bulb with a discrete resistor for the purpose of calculating the current. 11
The Easy Way A Replace the bulb with a B discrete resistor for the purpose of calculating the current. 12
The Easy Way A B A + V B In EECS, we do things the easy way I R V I = R R represents the only property of interest! Like with point-mass: replace objects with their mass m to find a = F m 13
V-I Relationship A + V I R and V I = R B R represents the only property of interest! R relates element V and I V I = R called element v-i relationship 14
R is a lumped element abstraction for the bulb. 15
Lumped Elements Lumped circuit element described by its vi relation Power consumed by element = vi Resistor + i v - i v Voltage source + v - i +" " V i v 16
Demo only for the sorts of questions we as EEs would like to ask! Lumped element examples whose behavior is completely captured by their V I relationship. Demo Exploding resistor demo can t predict that! Pickle demo can t predict light, smell 17
Not so fast, though A B Bulb filament Although we will take the easy way using lumped abstractions for the rest of this course, we must make sure (at least for the first time) that our abstraction is reasonable. In this case, ensuring that V I are defined for the element 18
I must be defined. A I + V B black box 19
I must be defined. True when SB I into S A = I out of q True only when = 0 in the filament! t I A I B S A S B q I A = I B only if = 0 t We re engineers! So, let s make it true! So, are we stuck? 20
V Must also be defined. A I + V So φ VAB defined when B = 0 t VAB = E dl outside elements AB see A & L Appendix A.1 B So let s assume this too! Also, signal speeds of interest should be way lower than speed of light 21
Welcome to the EECS Playground The world The EECS playground Our self imposed constraints in this playground φb t q t = 0 = 0 Outside Inside elements Bulb, wire, battery Where good things happen 22
Lumped Matter Discipline (LMD) Or self imposed constraints: More in Chapter 1 of A & L φ B t q t = 0 = 0 outside inside elements bulb, wire, battery Connecting using ideal wires lumped elements that obey LMD to form an assembly results in the lumped circuit abstraction 23
So, what does LMD buy us? Replace the differential equations with simple algebra using lumped circuit abstraction (LCA). a For example: V+" " b R 1 R 3 d R 4 R 2 R 5 What can we say about voltages in a loop under the lumped matter discipline? c Reading: Chapter 2.1 2.2.2 of A&L 24
What can we say about voltages in a loop under LMD? a V+" " b R 1 R 3 d R 4 R 2 R 5 c Kirchhoff s Voltage Law (KVL): The sum of the voltages in a loop is 0. Remember, this is not true everywhere, only in our EECS playground 25
What can we say about currents? a V+" " b R 1 R 3 d R 4 R 2 R 5 c 26
What can we say about currents? Ica S a Ida I ba Kirchhoff s Current Law (KCL): The sum of the currents into a node is 0. simply conservation of charge 27
KVL and KCL Summary KVL: KCL: 28
Summary Lumped Matter Discipline LMD: Constraints we impose on ourselves to simplify our analysis φ B t q t = 0 = 0 Outside elements Inside elements wires resistors sources Also, signals speeds of interest should be way lower than speed of light Allows us to create the lumped circuit abstraction Remember, our EECS playground 29
Summary i + v - Lumped circuit element power consumed by element = vi 30
Summary Review Maxwell s equations simplify to algebraic KVL and KCL under LMD. KVL: j ν loop j = 0 This is amazing! KCL: j i j = 0 node 31