# Lumped Element Abstraction

Node Boundaries

Circuit Boundaries

The rate of change of magnetic flux linked with any portion of the circuit must be zero for all time.

The rate of change of the charge at any node in the circuit must be zero for all time. A node is any point in the circuit at which two or more element terminals are connected using wires.

The signal timescales must be much larger than the propagation delay of electromagnetic waves through the circuit.

S1E1.5 Simple Power

V=RI

P=IV

I=10/50=.2

Pr=.2*10=2 Watts

Ps=-Pr=-2 Watts

S1E2: Power

R=8 Ohms

P=11.0 Watts

V=P//I=P(V/R)=PR/V

V=sqrt(PR)

V=9.38

I=V/R

I=9.38/8=1.1

-I=-1.17 Amps

S1E3: AC Power

R=110.0 Ohms

Vpeak=120*sqrt(2)*cos(2*pi*60*0)=120*sqrt(2)=169.705627 Volts

P=V^2/R=(169.705626^2)/110.0=261.81 Watts

Vrms=Vpeak/sqrt(2)=169.705627/sqrt(2)=120 Volts

P=V^2/R=(120^2)/110.0=130.90 Watts

V=120 Volts

P=V^2/R=(120^2)/110.0=130.90 Watts

•  Replace Maxwell Differential Equations with Algebra
•  Sum of V in a lumped lopp = 0 (KVL)
•  Sum of I in a lumped node = 0 (KCL)

S1E5: KVL-0

v1=1.4 Volts

v2=0.9 Volts

v3-v1+v2=0

v3=v1-v2=1.4 Volts - 0.9 Volts=0.5 Volts

v3=0.5 Volts

S1E5: KV3

V-v1-v3=0

v3=V-v1

V+v4-v2=0

v4=-V+v2

V-v1+v5-v2=0

v5=v1+v2-V

S1E7: KCL-0

i1+i2+i5=0

i2=-i1-i5=-(-0.7 Amps )-i5

i5=i4-i3=0

i5=i4+i3=1.3 Amps +3.0 Amps = 4.3 Amps

i5=4.3 Amps

i2=-(-0.7Amps) - 4.3 Amps=-3.6 Amps

i2=3.6 Amps

S1E8: KCL

i4+i3-i2=0

i4=i2-i3

i5+i4+i1=0

i5=-i4-i1=-(i2-i3)-i1

i5=-i1-i2+i3

i6+i3-i5=0

i6=-i3+i5=-i3-i1-i2+i3=-i1-i2

i6=-i1-i2

S1E9: Battery Model

v1+vR1+vR2-v1=0

vR1+vR2=0 