**Complex Numbers**

**Chapter 4 Complex numbers in electrical engineering**

**Chapter 4.1 Introduction**

The calculation of **alternating currents** and **voltages** is a mathematical operation on **sine waves**. There is no problem when the electrical circuit consists only of **resistances**. Then the **current** **sine** wave is** in phase** with the **voltage sine** wave. You only need to calculate the** current amplitude** by **DC** methods. Some **Kirchoffs, Thevenins, Nortons, eyelets, etc**…

The introduction of inductance **L** or capacitance **C** into the electric **circuit** changes not only the **current amplitude** but also **shifts** its **sine** wave by the appropriate **angle-phase φ**. The trick is to calculate the **amplitude** and** phase** of the **current**. This can be done using **trigonometry**, but very complicated formulas result! Meanwhile, **complex numbers** make **calculus** much easier.

**Chapter 4.2 The complex function exp(jωt) as a rotating vector****Complex numbers** are very much related to the **sine wave**, or **harmonic motion**. Alternating currents are **sine** waves. Therefore, it is not surprising that **complex numbers** appeared first in mathematics, which is obvious, but then in **electrical engineering** of the **19th** century, together with alternating currents. **Sine** waves are treated by electricians as **rotating vectors**, or what is almost the same as **complex numbers**. For more on this, see **3.1. ****Fourier Series, Chapter**.

** ****Chapter 4.3 RL Circuit**

Let’s consider a simple example with resistance and **inductance,** i.e. with the **RL** circuit. What are the specific values of **R** and **L**? Never mind. Only the **time charts** with these parameters on the oscilloscope is important – **Fig. 4-1b**. The phase shift **φ=π/6=30º** appears.**Fig. 4-1****RL** circuit with a variable **voltage input****Fig. 4-1a****– input** sinusoidal voltage **U=1sin(1t)** in **volts****–output** sinusoidal current** I=0.5sin(1t-30º)** in **amperes**.**Fig. 4-1b**

Time charts

The waveforms here are very slow with a pulsation **ω=1** (more precisely** ω=1/sec**) which corresponds to the period **T=2π=6.28…sec**.

That coil **L** must be huge! With these RL parameters, the sine wave of the current is shifted in phase by the angle** φ=π/6, i.e. by 30º**.**Fig. 4-1c**

Phasor charts**– Green** input **voltage** indicator**– Red** output **current** indicator

The phasors l**engths** are the **voltage** or **current amplitudes**.**Timing** and** phasor plots** with given pulsation **ω** are of course equivalent.

Both phasor-vectors **rotate counter-clockwise** with pulsation** ω=1/sec. Fig. 4-1c** is a photo of rotating vectors at** t=0**. Their projections as a function of time on the **vertica**l axis are exactly the waveforms in** Fig. 4-1b**. The **phasor diagram** shows more clearly the lag of **φ=30º** of the current in relation to the **voltage** and the **amplitude** as the **length** of the **phasors**.

Vectors are here a **complex numbers**:**1+0j****0.433-0.25j**

Just in this case, adding sinusoids or phasors describing the circuit in **Fig. 4-1a** does not make sense. You can’t add **current** to **voltage** any more than you can add **apples** to **submarines**! Only **currents** or only **voltages** can be added, and this is the basis of** electrical engineering**. Adding two **vectors** is much easier than **trigonometric** adding **two sine** waves!

And **vectors** are almost **complex numbers.**

This is why **electricians** love complex **numbers**!