Solving circuits with differential equations is hard. If we limit ourselves to sinusoidal input signals, a whole new method of AC analysis emerges. Created by Willy McAllister.

Here's a preview of how AC analysis is going to work. To get ready we need to review some of the ideas from trig and complex numbers. Created by Willy McAllister.

We break a sinusoidal input voltage into two complex exponentials. Using superposition, we can recover the complex output signals and reassemble them into a real sinusoidal output voltage. Created by Willy McAllister.

We can solve circuits by the direct application of the fundamental laws: Ohm's Law and Kirchhoff's Laws. (Part 1 of 2). Created by Willy McAllister.

We solve the equations created by direct application of the fundamental laws: Ohm's Law and Kirchhoff's Laws. (Part 2 of 2).

The capacitor current-voltage equation in derivative form and integral form. Created by Willy McAllister.

Definition of circuit terms: element, component, node, branch, mesh, and loop . Created by Willy McAllister.

A closer look at the complex exponential term in Euler's Formula. We see that it represents a complex number, a distance of 1 from the origin of the complex plane. Created by Willy McAllister.

When we put time in the exponent of a complex exponential, the complex number it represent rotates in a circle on the complex plane. You can think of it as a spinning number! Created by Willy McAllister.

Multiplying a real or complex number by the imaginary unit j corresponds to a rotation by +90 degrees. This is the key feature of j that makes it such a useful number. Created by Willy McAllister.

By convention, we define positive direction of current to be in the direction a positive charge would move. Electrons (with their negative charge) move in the opposite direction of the positive current arrow. Created by Willy McAllister.

The positive sign for current corresponds to the direction a positive charge would move. In metal wires, current is carried by negatively charged electrons, so the positive current arrow points in the opposite direction the electrons move. This has been the sign convention for 270 years, ever since Ben Franklin named electric charges with + and - signs. This convention came about 150 years before the discovery of the electron. Created by Willy McAllister.

Introducing the diode as a circuit element. A diode conducts current in one direction but not the other. Created by Willy McAllister.

We solve a diode circuit graphically by plotting a diode i-v curve and resistor to find the intersection. Graphical solution methods are a common way to work with non-linear components like diodes. Created by Willy McAllister.

Remember the relationship between current and voltage in an inductor and capacitor with the help of ELI the ICE man. Created by Willy McAllister.

Current is the flow of charge. We measure current by counting the amount of charge passing through a boundary in one second. Created by Willy McAllister.

A cosine wave emerges from Euler's Formula. Music, no narration. Animated with d3.js. Created by Willy McAllister.

Euler's formula relates the complex exponential to the cosine and sine functions. This formula is the most important tool in AC analysis. It is why electrical engineers need to understand complex numbers. Created by Willy McAllister.

A sine wave emerges from Euler's Formula. Music, no narration. Animated with d3.js. Created by Willy McAllister.

Feedback is the design technique where a part of the amplifier output "feeds back" to the input of the amplifier. The overall effect creates a very stable gain determined by resistor ratios. Created by Willy McAllister.