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Complex Numbers in Electrical Engineering

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- Complex Numbers
- FOIL & Conjugate Review
- Voltage, Resistance, and Current
- Multiplication using V=I•R
- Division using V=I•R

**Complex Numbers**

A complex number is a number in the form , containing both a real part and an imaginary part. The imaginary part is followed by i, which is the imaginary unit,

One application of complex number is in electrical engineering (as well as other engineering and scientific fields). Complex number occur in calculations involving electrical currents, which will be explored in the examples below. Depending on the situation, we will need to either multiply or divide two complex numbers. During these processes, we use FOIL and complex conjugates to find our solutions. Let's briefly review the FOIL process and complex conjugates.

**FOIL & Complex Conjugate Review**

FOIL stands for First, Outside, Inside, Last, and refers to the terms that are multiplied together to form individual addends to the product. Here is an example of using FOIL in binomial multiplication:

When using FOIL with two complex numbers, one of our terms will be an term. This simplifies to a real number because

When dividing two complex numbers, we use the denominator's complex conjugate to create a problem involving fraction multiplication. A complex number and its conjugate differ only in the sign that connects the real and imaginary parts. Here is a table of complex numbers and their complex conjugates.

We use the denominator's complex conjugate to create a fraction equivalent to 1. As we will see in our division example, this eliminates all imaginary numbers from the denominator.

**Voltage, Current, and Resistance**

When working with electrical circuits, electrical engineers often apply the following formula to relate voltage, current, and resistance:

Voltage is measured in volts, current is measured in amps, and resistance is measured in ohms.

The notation engineers use for complex numbers is a bit different than what we may be used to seeing. There are generally two big differences:

- Engineers commonly use instead of , so as not to confuse the imaginary unit with the variable for current. So keep in mind in these examples that whenever we see , this represents our imaginary unit, and has a value of
- In addition to using , this variable is also often written before its coefficient, rather than after. For example the complex number might be written as

**Multiplication using V = I • R**

An electrical circuit has a current of amps, and a resistance of ohms. What is the voltage of the circuit?

To find the voltage, we need to multiply the current by the resistance, giving us the equation:

We can find the product of current and resistance by using FOIL:

We can express the voltage as:

**Division with V = I • R**

An electrical circuit has a voltage of volts, and a resistance of ohms. What is the circuit's current?

Here, we will need to divide the voltage by the resistance in order to get an expression for the current:

To solve complex number division problems, we multiply the fraction by another fraction equivalent to 1, with the denominator's complex conjugate as the numerator and denominator of the second fraction:

Let's multiply the numerators and denominators separately:

Finally, we can simplify the entire fraction:

The circuit has a current of