is there any physical interpretation for complex variables?

Question

I'm a student in Electronics.

Answer 1

A meaningful physical interpretation of a complex number depends on the specific type of physical vector it represents.

Complex numbers are convenient algebraic expressions of vector quantities which have both a magnitude and a direction. Complex numbers enable us to manipulate algebra on equations based on rectangular coordinates, rather than sketching vectors on polar coordinate.

For example, the impedance of an electronic part on polar coordinate intuitively is a vector having the length corresponding to the resistance and an angle corresponding to the phase shift that the part exerts to an applied voltage. Mapping this vector from polar to rectangular coordinates gives a complex impedance number that looks some thing like a+jb, where the resistance is (a^2 + b^2)^0.5, and the reactance (the phase shift) is arctan(b/a). This complex number is no longer intuitive to us, but it can be manipulated by a computer.

Complex numbers should have been called "vectorial numbers" to be less intimidating.

Answer 2

I am an engineer, but I got me an "A" in Complex Variables CALCULUS in the School of Math (with Prof. Demko) for my example of physical interpretation of complex variables.

If you have one known solution of a Laplace (=potential equation) (an electrostatic field for you EEs), any complex variable function of that solution is automatically also a solution. It is like having a master key that will open all the doors in the world.

Basically, you can plot the electromagnetic field of an infinite flat plate, take a (some) complex variable function of that and calculate the electromagnetic field outside a square

Problem is (there is always a problem) you don't immediately know what the new solution is a solution of.

The Schwarz-Christoffel transformation (see sources), which to me is the third most beautiful equation in the world (Euler e^iPI+1=0 is 1st, Einstein's E=mc^2 is 2nd) says how to map polygonal areas

Other potential (Laplace) field physics phenomena for which this works are of course conduction heat transfer, gravitational field, magnetic field, and, in fluid dynamics, potential flow (constant density, irrotational, zero viscosity; ie, low pressure gases:, ie: airfoil design).

Complex variable calculus, the calculus of a+ib, imaginary numbers can and solve problems in all these fields. See PV1990_127.pdf below for example of airfoil design with complex variables. Next time you get on a plane, it is the complex variables which keep the wing flying. How's that for an example of complex variables at work? You can relax and enjoy your flight knowing it is all those imaginary numbers a+ib keeping you airborne.

Answer 3

Yes, if you have an unknown complex impedance, you can consider the real part is a resistance, the complex part is an inductance with ositive sign, and a capacitance if its a negative sign.

the complex numbers are used to represent real oscillatory functions. by eulers formula.

Answer 4

Hi,
I want to know what kind of variable you want to interpret?
If you study books on Transducers, sensors, instrumentation you can understand almost every thing can be measured or controlled by some means of instrumentation technology.
-Mithuna

Answer 5

Yes, it usually relates to the phase of the AC signal.