Dynamical Analogies Or Thermodynamics Without Entropy

Thermodynamics without entropy or analogies among dynamical systems • please note the distinction between Static and Dynamic energy storages • small w denotes total energy, capital W denotes energy density • systems can be “lumped parameter” or distributed • Four variable models -- note the “oddball” down in the lower left corner • yes, Virginia, dynamic is really different from static energy storage (you always knew there was something strange about kinetic energy and momentum) • Primary network variables are the ones whose product is power (or power flux) and whose ratio is impedance -- the use of any other variable pair as the primary variables impedes building complicated systems from elementary building blocks; although it can, at times, be thermodynamically advantageous

Copyright 2001 Kevin G. Rhoads

v

q

voltage

charge

Push

stuff

λ

i

flux-turns linkage current

Thud

Flow

Impedance = v / i

C

v d⁄dt

wdynamic = ∫i dλ

wstatic = ∫v dq

q d⁄dt

λ

i L

Power = v i

Uncertainty Pair

Electrical

Copyright 2001 Kevin G. Rhoads

Lumped Element

F

x

Force

position

Push

stuff

p

v

momentum

velocity

Thud

Flow

Mechanical Impedance = F / v

1/k

F d⁄dt

wdynamic = ∫v • dp Uncertainty Pair

wstatic = ∫F • dx

x d⁄dt

p

v m

Power = F v

Linear Mechanical

Variables (i.e., F, x, v, p) may be scalars or vectors.

Copyright 2001 Kevin G. Rhoads

Lumped Element

T L

θ

Torque

position

Push

stuff

velocity

Thud

Flow

(angular)

ω

momentum

Mechanical Impedance = T / ω

1/k

θ

T d⁄dt

wdynamic = ∫ω dL

wstatic = ∫Tdθ d⁄dt

ω

L I0

Power = T ω

Uncertainty Pair

Rotational Mechanical Copyright 2001 Kevin G. Rhoads

Lumped Element

T

Q

?

I[Q´]

Temperature

heat, energy

Push

stuff

heat flow

Thud

Flow

Nameless CThermal

Thermal Impedance = T / I

T

Q

d⁄dt

d⁄dt

?

I ??

Thermal An example of an incomplete analogy. There is nothing to put into the “thud” slot in

Lumped Element

general; although for studies of TTT processing, “soakage” might be a reasonable candidate.

Copyright 2001 Kevin G. Rhoads

E

D

Electric field Intensity

B

H

Magnetic Flux Density

Wdynamic = ∫H • dB Uncertainty Pair, in some extended sense

*

Magnetic Field intensity

Push

stuff

Thud

Flow

ε0

Wave Impedance = |E| / |H| ∂⁄∂t, ∇×

Displacement Flux Density

E

D

B*

H µ0

∂⁄∂t, ∇×

Wstatic = ∫E • dD

Power Flux Density = E × H

Electrical Fields

instead of B one could use A, this does break the

symmetry of the analogy and is not recommended

Copyright 2001 Kevin G. Rhoads

Distributed System

P

Q [V]

Pressure Differential

Volume

p

Q´

Momentum Density

Volume rate of flow

stuff

Thud

Flow

Gas Law

Flow Impedance = P / Q´

P

Wstatic = ∫P dQ

Q

∂⁄∂t

Wdynamic = ∫Q´ dp

Push

∂⁄∂t

p

Q´ ρ

Power Flux Density = P Q´

Gas Flow Fluid Flow Copyright 2001 Kevin G. Rhoads

Distributed System

Pop Quiz (answers ahead) • What is the pneumatic equivalent of “ground bounce” • What is the electrical equivalent of “stiction” • Comparing the Mississippi River and a fire hose • which is higher impedance? • why? • Is a direct analogy between the old style automotive ignition and hammering in a nail reasonable? • If yes, why? • If no, why not?

Copyright 2001 Kevin G. Rhoads

P

ξ

Pressure variations displacement (Intensity)

Push

stuff

p

v

momentum density

Thud

Flow

velocity

Acoustic Impedance = P / v

1/M*

ξ

P ∂⁄∂t

Wdynamic = ∫v dp ? Uncertainty Pair, in some extended sense

*

Wstatic = ∫P dξ ∂⁄∂t

p

v ρ

Power Flux = P v

Acoustic

M is bulk modulus; for isothermal it is (for small strains) equal to the unperturbed pressure

Distributed System

for sound, however, adiabatic is more accurate than isothermal, so multiply by 1.4 for diatomic gases

Copyright 2001 Kevin G. Rhoads

• What is the pneumatic equivalent of “ground bounce” • Pressure rise in the pneumatic return line upon operation • What is the electrical equivalent of “stiction” • One example is crossover distortion in Class B push-pull stages • Comparing the Mississippi River and a fire hose • which is higher impedance? • Fire hose: impedance = Push/Flow • why? • Higher pressure (more push), lower flow • Is a direct analogy between the old style automotive ignition and hammering in a nail reasonable? • If yes, why? • Yes - both involve storing energy for rapid release as dynamic storage is stopped -- it is hard to get similar results with static storage • If no, why not?

Copyright 2001 Kevin G. Rhoads