Complex Numbers and Operations: Apply It 1

  • Understand the difference between imaginary numbers and complex numbers
  • Learn how to plot a complex number on a special graph called the complex plane
  • Perform calculations with complex numbers and visualize how these operations change their position and size when graphed
  • Find the points where a quadratic equation crosses the x-axis, including both real and complex solutions

Complex numbers might seem abstract at first, but they’re actually crucial tools in electrical engineering, especially when working with alternating current (AC) circuits. When electrical engineers analyze circuits, they use complex numbers to represent impedance (Z) – a measure that combines regular resistance (R) with reactance (X), which is how circuit components resist changes in current.

Imagine you’re an electrical engineering student working on analyzing various circuit components. Let’s apply your knowledge of complex numbers to solve some practical problems that electrical engineers face every day. Just like a multimeter helps you measure voltage and current, complex numbers will help you understand how components behave in AC circuits.

Now let’s look at a real-world application of how impedance affects current in an AC circuit. When you plug a device into a wall outlet, the voltage from the power supply needs to push current through the circuit’s impedance. Understanding this relationship helps engineers design safe and efficient electrical systems – from small electronics to large industrial equipment.
Let’s move from circuit analysis to control systems, where complex numbers help us understand how systems behave. Control systems are everywhere – from the cruise control in your car to the temperature control in your home. Engineers use transfer functions to describe how these systems respond to inputs, and finding the poles of these functions helps predict system stability and performance.