In 1970, when Apollo 13’s oxygen tank exploded, the first explanations were not enough to tell NASA exactly why a mission designed with careful engineering had suddenly turned into a life-or-death emergency. Engineers had to dig deeper than the original assumptions and rebuild the problem with more faithful hardware analysis and testing, tracing the disaster to a chain of real physical effects in the tank’s wiring, heating, and ground-test history. That shift from a rough picture to a higher-fidelity model exposed the root cause and guided the redesign that made later Apollo missions safer. (NASA, “Report of Apollo 13 Review Board, https://ntrs.nasa.gov/api/citations/19700078913/downloads/19700078913.pdf)
Modeling Diodes
When I was a professor teaching at Calvin University, one of my favorite courses to teach was ENGR 204 “Introduction to Electronics”. This was a sophomore level class taken by all engineering disciplines (not just the electrical engineering students). For much of the semester, we studied basic concepts of voltage and current and simple linear devices such as resistors that related voltage to current by a constant proportion:
voltage = current * resistance
Whether the current was big or small, the voltage is always a fixed multiple of the current.
Later in the semester we briefly examined a more challenging device, the diode. Although the diode was one of the simplest of the semiconductor devices, it was still a challenge – because it was non-linear. In this case, current is no longer related to voltage by a simple scaling factor.
You can see the non-linearity in the diagram. Watch what the current (vertical scale) does as the voltage increases going to the right. At first, increasing the voltage has no effect on the current, it remains zero. But as the voltage further increases, suddenly at around 0.7 volts, the current starts rising. But it does not rise in proportional to voltage – it takes off with a vengeance! This is the non-linearity in action. Such unusual behavior made it difficult to use the tools for circuit analysis that the students used thus far in the course.
To cope with this oddly behaving electronic device, I then gave a lecture on modeling. I do admit we actually had been modeling the electronics all along. Even a resistor is not precisely, mathematically linear. But it is close enough for most purposes. Now we needed to explicitly recognize that the model was not the reality.
In my lecture on modeling of diodes, I presented four different models:
- Ideal (switch) model
- Battery (constant V) model
- Battery + R (linear V-I) model
- Shockley Diode model
The first model is very simple, but rather inaccurate. Each model in the list gets progressively more difficult to apply, but it also gets progressively more accurate. The last model, is a non-linear equation and rather difficult to apply:
What I hoped they learned was that choice of model is a trade-off. More fidelity is more work. The simplest model is very easy - one can analyze diode circuits on the back of a napkin. The better models take more work.The higher the fidelity (the closer the model represents reality), the harder it is to calculate. But the higher cost comes with a payoff – we get closer to the complex truth of the semiconductor circuit behavior.
Modeling Computer Hardware
Modeling is not just for students. When I was a computer engineer working at Boeing, I designed software for the flight computers. Safety critical software like this must be thoroughly tested. Software developers from other industries are often astonished at the level and rigor of testing to ensure the software is correct.
Although there are some tools to check software without running it, usually we need to run the software in order to test it. The challenge is often that there is not enough computer hardware to do the testing. This is not as big of an issue when writing software for a laptop or mobile device. However, for devices that are behind the scenes or under the hood, the specialized hardware is often in short supply before the product goes to full production.
Even with my iPhone app development, sometimes I want to run a bunch of tests in parallel, but I only have one iPhone. Thus we turn to simulated hardware. Modern laptops are powerful enough that they can simulate the operation of a computer processor so that we can run our software virtually in order to test it.
However, these simulations are models of the hardware, they are not the real thing. In safety-critical industries, this difference means certain tests *must* be run on real hardware. Which tests? The tests where we cannot be absolutely certain that the simulated behavior will match the real behavior.
One example of such an area is where we need a deterministic worst-case timing response from a multicore processor. Each core might be running a different application. One of the cores running an important calculation may need to finish within a certain time frame in order to maintain safety. However, the other cores can sometimes interfere because cores share memory and certain other features of the hardware. It turns out that even the best simulation and virtualization tools today do not fully capture the nuances of those interactions between cores. We thus turn to the more expensive testing on the real hardware in these cases – because we need to be absolutely sure.
Modeling Christ
Modeling is not just for engineers. As followers of Jesus, we are repeatedly told to imitate (i.e., model) him.
“I have set you an example that you should do as I have done for you.” (John 13:15)
“Whoever claims to live in him must live as Jesus did.” (1 John 2:6)
“In your relationships with one another, have the same mindset as Christ Jesus:” (Philippians 2:5)
It is also clear from scripture that this will not be easy. The highest fidelity requires a lot of work and even sacrifice. Peter tells us this: “To this you were called, because Christ suffered for you, leaving you an example, that you should follow in his steps.” (1 Peter 2:21) In his letter to the church in Ephesus, Paul also says so: “Be imitators of God, therefore, as dearly loved children and live a life of love, just as Christ loved us and gave himself up for us as a fragrant offering and sacrifice to God.” (Ephesians 5:1-2 NIV)
Take Away
Engineers learn to “get it done”. We find the most efficient way to accomplish the requirements. We are not looking for the easy way out so much as we are finding the way that costs the least. We try to avoid making a mountain out of a molehill.
In that electronics course, I taught my students to use the model that fits the need, not spending more time on a harder model if you did not need that level of fidelity. But this thinking must not unduly influence our spiritual walk. When we model Christ, we must not seek the least costly solution. We must be willing to do the hard work of discipleship. We reach the highest fidelity, and most closely emulate Christ when we put the sins in our life to death and live our new life in him.
Isn’t it ironic that we image bearers – the imago dei, made in the likeness of God – need to be reminded to bear the image? “For those God foreknew he also predestined to be conformed to the image of his Son, that he might be the firstborn among many brothers and sisters.” (Romans 8:29) We are children of the King – start acting like it!
