Unfortunate for technical people, but good for 'normal' customers.
Everyone knows what a day is, and we all get billed in KW-hours. I
doubt that very many people could tell you that a KW-hour/day is a hair
under 42 watts, and even fewer could do it in a blink without a calculator.
As usual Tim is right. I would like to be more explicit about it's
use in control systems. It's what allows you to "close the loop" on
the block diagram; for instance the plant is something like degC1/
Volt, the transducer is something like Volts/degC2 and the controller
is Volts/Volts. When multiplied this give degC1/degC2 as the gain and
the units to measure during verification. Of course a real diagram
has a lot more terms and different units; but the loop product must
balance. Inserting the open loop process variables in proper units
degC1/degC2 into the equations answers many questions.
One additional point is that the form is not as specialized as it
seems, but carries over to generalized Differential Geometry as vector/
covector spaces. Applying dimensional analysis in this realm has
allowed me to make sense of a lot of formulas. The units changes
(volts->degC) correspond to mapping from one vector space to another
(and back); the reduction to a gain scalar is the contraction of a
covector and vector; and so forth.
Tim Wescott wrote:
You raise some good points.
One of the problems with the modern unit system (Pascals, Siemens,
etc) is that the fundamental meaning of the unit is lost. For
example: Pressure X Area = Force.
Pascals X Square meters = Pascal Meter^2. So what? You must first
convert Pascal to its fundamental definition, which is Newtons/Square
Meter. Now dimensional analysis makes sense: (Newtons/square meter) X
(Square meter) = Newtons, which is consistent with a unit of force.
Another example is characteristic impedance of a transmission line.
We've all learned that the equation is R = SQRT(L/C), where L inductance/unit length, C = capacitance/unit length. Resistance (Henries / Farads)^.5? You need to get back to the fundamental
relationships among voltage, current and time for inductance, and the
fundamental relationships among current, voltage and time for
capacitance. v = Ldi/dt.
i = C dv/dt. From these fundamental equations, you can get the
fundamental units of L and C: L=VoltSeconds/Ampere, C= AmpereSeconds/
Volt. Now the equation for characteristic impedance makes sense, in
terms of its fundamental units: R = SQRT([(VoltSeconds/Amps)/
R = SQRT(Volts^2/Amps^2) = Volts/Amps = Ohms.
Capacitance is in cm, resistance in s/cm, inductance in s**2/cm.
Capacitance per unit length is dimensionless, inductance per
unit length s**2/cm**2, so sqrt(L/C) is s/cm, just like
I used to be able to do it in MKS and CGS units about equally,
and sometimes Heaviside-Lorentz units. (Similar to CGS, but
without so many 4pis around.)
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