Complex numbers

Chapter 4 Complex numbers and electricity
Chapter 4.1 Introduction
The alternating currents and voltages calculations are the mathematical operations on the sinusoids. There is no problem when 
R resistors are used only. The currents and voltages sinusoids phases are the same! The direct currents methods may be used then i.e. Kirchoff, Ohm, Thevenin, Norton and others.
The coils and capacitors in the circuits, i.e. L inductance  and C capacity cause currents amplitude and phase φ changes! All the problem is to calculate these currents parameters. We can do it by common trigonometry but the formulas are very complicated. Meanwhile, complex numbers make these formulas easier and more clear.

Chapter 4.2 Complex number exp(jωt) as a rotating vector
Complex numbers are strongly relevant to sinusoids. Alternating currents are just sinusoids. The complex numbers emerged first in the mathematics and this is obviously. But they emerged with alternating currents in the electricity in XIX century then.
Sinusoids are treated by electricians as rotating vectors or as complex function exp(jωt). See more–>Fourier Series article chapter 3.1.

Chapter 4.3 RL circuit
The simple RL circuit is analysed here. What are and specific data? Not important.
The oscilloscope time process is important only–>Fig 4-2b. The phase displacement is φ=π/6=30º.

Fig. 4-2
circuit when input alternate voltage type
Fig. 4-2a
input voltage U=1sin(1t) in volts
Output current  I=0.5sin(1t-π/6) in ampers.
Fig. 4-2b
Time process
The process is very slow when ω=1 (more strictly ω=1/secT=2π=6.28…sec.
The coil must be really big  because φ=π/6  i.e. 30º by these RL parameters!
Fig. 4-2c
Vectors diagram 
– Green vector – input voltage
Red vector output current
The vector lenghts are voltage or current amplitudes.
Time process and vector diagram with ω given are equivalent of course.
Voltage and current vectors have counterclockwise rotations when ω=1/sec. Fig. 4-2c is rotations photography when t=0.
The vectors Im z projections (yellow dot vertical movements-> Fig-4-1) is a Fig 4-2a time process.
You see the current delay =π/6= 30º acc. to voltage. 
are more clear than time process! φ phase parameter especially.
These vectors are complex numbers:
The vectors or complex numbers operations are easier than trigonometric
That’s why electricians love them.

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