In automation design, the choice between a Stepper Motor and a Servo Motor is often decided by budget. But looking at the price tag alone is a mistake that leads to machine failure. Steppers are excellent for holding loads stationary (high holding torque). Servos are kings of high-speed motion. If you choose a stepper for a high-speed application, it will lose torque and "miss steps." If you choose a servo for a simple low-speed application, you have wasted $500. This guide explains the physics behind the choice. Table of Contents 1. Open Loop vs. Closed Loop (The Risk) 2. The Torque Curve: Speed Kills Steppers 3. Inertia Mismatch 4. Selection Summary Advertisement 1. Open Loop vs. Closed Loop (The Risk) The biggest difference is not the motor itself, but how it is controlled. Figure 1: Steppers run "blind" (Open Loop). Servos use an encoder to verify position (Closed Loop). ...
Figure 1: Partial Pivoting involves swapping rows to avoid zero pivots.
The "Fatal Flaw" in Basic Code
In our previous post, Gauss Elimination (Part 2), we derived the mathematical foundation of the algorithm. However, if you implement that raw math directly into code (VBA, MATLAB, or C++), your program will eventually crash.
Why? Because the basic algorithm assumes the diagonal element (the pivot) is never zero. In the real world of engineering simulations, zeros happen frequently.
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The Problem: Division by Zero
Let's look at a standard solvable system of equations:
The first step of the algorithm is to normalize the first row by dividing by the coefficient of x1 (which is 5). Since 5 is not zero, this works perfectly.
But what if the equations were rearranged?
Mathematically, the order of equations doesn't matter. But computationally, it changes everything. Look at this arrangement:
Here, the coefficient of x1 is 0. The computer attempts to divide by zero, resulting in a runtime error.
The Hidden Danger: Small Numbers
Even if the number isn't zero, but is very small (e.g., 0.000001), it causes Round-off Errors. Dividing by a tiny number creates a massive result, which can cause the computer to lose precision in the remaining digits. This is known as an "Ill-conditioned system."
The Solution: Partial Pivoting
To solve this, we use a technique called Partial Pivoting. Before performing the elimination step on a column, the algorithm must:
- Search: Look down the current column (below the diagonal).
- Identify: Find the row with the largest absolute value.
- Swap: Switch the current row with that row.
By ensuring the largest possible number is always in the pivot position, we avoid division by zero and minimize round-off errors.
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Next Part: The Final Code
Now that we understand the logic and the safety features required, we are ready to build the robust Excel VBA tool.
Continue to Part 4:
Solving System of Equations using Gauss Elimination Method (Part 4: The Final Program)
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