# Beyond Control: Maths Of A Control System

Control systems are all around us, and understanding them is going to make you much better at hardware design. In the last article — Beyond Control: The Basics of Control Systems — we looked at an overview of what a control systems are in general with the example: “everything in between water and time is a control system”. We also observed control systems in nature, where I described my keen ability to fill a glass of water without catastrophic results. That discussion involved the basic concept of a block diagram (without maths) and we expanded that a bit to see what our satellite dish example would look like (still without maths).

I promised some big ugly maths in this article, and we’ll get to that in a bit, never you fear. First let’s have a look at how some basic elements: resistors, inductors, and capacitors are defined in the time domain. Don’t let these first few definitions turn you off. No matter how you feel about calculus, you don’t necessarily need to fully understand each equation. What’s more important is how the equations themselves combine to solve the circuit. Also important is that I will do everything possible to get out of doing difficult math. So stick with me through the article and you’ll learn that agony-saving trick for yourself!

A quick recap on transfer functions before we get going might be beneficial. A control system is used to define electromechanical behavior. For example: our satellite dish (from the previous article) at some point will need to be moved from one position to another position and as control engineers it is our job to determine just how this action will take place. I’m not talking about setting the mood for the big emotional robotic rotation, more like: not damaging the equipment or any people that might be nearby when moving the dish. For many reasons the dish would need to be moved with extreme care and in a very precise manner. The control system is the mathematical definition of that movement. Often the maths of the definition are nasty differential equations, (remember I’m avoiding any math that can be avoided, right?) so, instead of using differential equations to define the system, the transfer function will define the system with algebra, relating the output of the system to the input.

# 3D Printed RC Servo To Linear Actuator Conversion

RC servos are handy when you need to rotate something. You can even modify them to rotate continuously if that’s what you need. However, [Roger Rabbit] needed linear motion, but wanted the simple control afforded by an RC servo. The solution? A 3D printed housing that converts a servo’s rotation into linear motion.

The actuator uses five different parts, a few screws, and a common RC servo. The video shows the actuator pushing and pulling a 200g load with a 6V supply. There’s some room for adjustment, so different servos should work.

# Electrolytic Capacitor Repair

If you’ve ever worked on old gear, you probably know that electrolytic capacitors are prone to failure. [Dexter] undertook a repair of some four-decade-old capacitors in a power supply. He didn’t replace them. He fixed the actual capacitors.

The reason these units are prone to fail is the flip side of what people like about electrolytics: high capacitance in a small package. In a classic parallel plate capacitor, the capacity goes up as the distance between the plates shrinks. In an electrolytic, one plate is a rolled up spiral. The other plate is conductive fluid. The insulator between (the dielectric) is a very thin layer of oxide that forms on the spiral. Over time, the oxide degrades, but this degradation repairs itself when using the capacitor. If the capacitor isn’t powered up for an extended period, the oxide will degrade beyond the point of self-repair.