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Unit 5: Transformers; Complex Numbers

Most of the electronic devices in your home contain transformers. Transformers have several different uses. Their most common use is to change the size of an ac voltage. For example, the sinusoidal voltage waveform that’s present at an electrical outlet in your home has an effective value of about 120 V rms. That is far larger than the voltage required by most electronic devices. Therefore, these devices contain transformers to reduce the voltage from 120 V to the voltage that they require (maybe 5 or 10 V rms). As we’ll see, a transformer can change not only voltage, but also current and the apparent size of a resistor.

Like inductors, transformers rely on the close relationship between electricity and magnetism. In fact, if you looked at a transformer’s internal construction, you’d see that it consists basically of two inductors wound on a single core.

In this unit we’ll learn some simple but useful equations related to transformers.

We’ll also look at some math that is very useful for analyzing AC circuits. This math involves what are called complex numbers. You may have learned about complex numbers in a high school or college math course. If so, much of this unit will be a review for you. But I’m not going to assume that you already know this material, so even if you’ve never seen complex numbers in a math class, you’ll be able to pick up what we need pretty easily. A lot of students are confused or scared by the names “complex number” and “imaginary number.” Don’t let those names bother you. There’s nothing particularly complicated (complex) about this material, and you don’t need to have a great imagination to understand imaginary numbers.

Another thing that confuses students is the idea of taking the square root of a negative number. If this were a math class, we’d have to spend some time thinking about how that’s possible. But for what we want to do in electronics, we don’t need to get bogged down on this point. The main thing that we need is to learn some simple rules for how to work with these complex numbers. You can learn those rules without getting into deep philosophical discussions about what a complex number really is.

Unit 4 Review

Typical Transformers
Primary & Secondary
Could This Be Magic?

Turns Ratio
Voltage Transformation
Step-up Transformer
Step-down Transformer
Current Transformation
Power in an Ideal Transformer

Input Impedance
Impedance Transformation and Reflected Load
Impedance Matching

Multiple-Secondary Transformers
Tapped Secondaries

Transformers are for AC Only

Real Numbers Versus Imaginary and Complex Numbers
Imaginary Unit,  j
Complex Numbers
The Complex Plane

Rectangular & Polar Forms
Converting from Rectangular Form to Polar Form
Converting from Polar Form to Rectangular Form

Mathematical Operations 
Adding Complex Numbers
Subtracting Complex Numbers
Multiplying Complex Numbers
Dividing Complex Numbers

Complex Numbers on Your Calculator

This concludes our mathematical review of complex numbers. Now let’s take a quick look at how complex numbers are used in electronics.

Representing Phasors with Complex Numbers

Many Uses for Phasors
Using Phasors to Add Sinusoids
Procedure for Adding Sinusoids

Unit 5 Review

Congratulations! You’ve completed the e-Lesson for this unit.