### Complex numbers

**Chapter 4 Complex numbers and electricityChapter 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

**R**and

**L**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**

RLcircuit when input

RL

**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/sec**)

**T=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.**is

**Fig. 4-2c****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.**

Vectorsare more clear than

Vectors

**time process**!

**φ**phase parameter especially.

These vectors are

**complex numbers:**

1+0j

0.433-0.25j

The vectors or

1+0j

0.433-0.25j

**complex numbers**operations are easier than t

**rigonometric**

That’s why electricians love them.