Howdy from the homefront!
I’ve had a lot of experience with systems since the beginning of freshman year. I am currently working for Doctor Tolle on control systems, we learned about them in differential equations, derived them in circuits 1, had to create them from scratch in mechatronics, and finally are required to know how to derive transfer functions and make bode plots from circuits in circuits 2.
One thing that is consistent with every professor here is that they do an awful job explaining systems. I don’t know what it is. They can perfectly explain away differential equations and phasors but the second we try to tie the two together, there’s a major gap that’s left.
That’s the purpose of this post is to aid future SDSM&T students in the solving and deriving of systems. I’ll first approach systems from a circuits point of view (since I’m an EE), later I’ll be transitioning into more physical examples.
This tutorial expects you to have taken circuits 1.
Let’s get started:
The circuit below is what is called a low-pass filter. In other words, it only lets low frequencies pass through undistorted and unattenuated.
With this type of filter, we incidentally create a first-order system (but let’s not think of it as such just yet). As a result, our output will always differ from our input in both phase and magnitude.
What we’ve just done was create a voltage divider in the phasor domain.
Practically, we use this configuration for filtering out noise in a DC circuit or smoothing a rectified sine wave into a more DC-like signal.
First, lets model and solve this circuit in the phasor domain. If you’ve taken circuits 1 (from Montoya) this should be second nature and therefore the easiest way to solve. It goes as follows and is exactly the same as voltage division above. This method, also, gives us phase information.
Let’s take the same circuit from above and apply various frequencies (ω) and see what happens to the amplitude and the phase of the output waveform. For this example, we’ll use a 4.7μF capacitor and a 1k resistor. The input waveform will be 15sin(ωt). Pay special attention to the magnitude and the phase shift.
Congratulations! You’ve done a really low resolution frequency response analysis. These are important because now you know how the filter will respond to different input frequencies. You can use this data to help determine the correct resistor-capacitor combination for your filter. And while it isn’t entirely useful to plot this information with first order systems, I’ll use this as a segue into Bode Plots because they’re entirely important for systems greater than 1st order.
As mentioned earlier, the data we collected from the three ω values gave us a low-res view on what the frequency response of that circuit looked like. But what happens if we take a bunch of magnitude and phase information and plot it? Well that’s a bode plot!
As you can see, our low pass filter is doing exactly that, passing the lows! At very low frequencies there is essentially no attenuation or phase distortion (just how we like it) but the high frequencies dramatically drop out of the output signal. While this graph sure is pretty, it doesn’t tell us much that we didn’t already know or could easily find out. They do have their uses, however for higher order systems!
Part 2 will cover a second-order system example. Please let me know what you think or what I should improve on!