# Academic Simplifications Produce Meaningless Equations

June 13, 2012
Jerry Twomey cautions designers (and design managers) about the pitfalls of relying on oversimplified equations when you're dealing with real-life circuit elements.
A few years back, I went into a job interview. It was a long day with lots of questions. There were some good questions and a few disagreements about methods, but productive discussions nonetheless. At the end of the day, I met with the vice president of engineering:

VP: “I hear good things in the hallway from the folks that interviewed you already. This is a position that is heavily analog, mixed signal, RF, and semiconductors, so let’s talk equations.”

Me: “Oh, which equations? There are lots of them out there.”

(I picked up a pencil and drew the V-I curve for a forward biased diode.)

Me: “You mean this?”

VP: “Yes. What’s the equation for that?”

Me: “Well, the standard diode equation is an academic simplification, and it can be misleading.”

Me: “It doesn’t include some very important things, so even using the equation for dc bias you can have problems. As you can see here, you have a high current region where the resistance of the junction needs to be considered. Then, in the reverse bias situation you have two things happening: some reverse leakage current and a breakdown condition, which many call the Zener voltage or avalanche breakdown point.”

(Handing the drawing over to the VP and pointing…)

Me: “Even a lot of diode vendors don’t like to publish this kind of plot, because it shows the non-ideal characteristics of the diode. Usually they give you a forward biased plot from 0 to 1 V and give you a set of maximum values so you can’t violate limitations of current and reverse bias voltage. Put a real diode on a curve tracer and this is what you will see.” (Fig. 1)

1. This shows a non-scalar representation of a diode over a wide bias range. The real diode will exhibit reverse bias breakdown, reverse bias leakage, and resistance in the PN junction.

VP: “So what’s the equation?”

Me: “The diode equation isn’t valid except under a limited set of assumptions and conditions, and so I try to not memorize useless equations. Better to get the specifics response curves for the diode you are interested in.”

Oops. The “useless equations” comment struck a nerve and I knew I was in trouble. Right about then I felt like I was in school and the professor had never touched a soldering iron. There’s one old rule of academic survival—give them what they want to hear. So I hurriedly wrote the equation down, the one with current being exponentially related to voltage and all that. If you’re in a hole, it’s a good idea to stop digging, right?

### Know The Situation

People often learn equations and quickly forget the simplifications or assumptions made to make them nice and easy to understand. But that leads you down some treacherous paths. The derivation of the diode equation leaves out important performance details to make it simple to derive. Also, many engineers love to eat, sleep, and breathe math, and being able to put things in a neat equation with that nice equals sign in the middle seems to be comforting.

Does this sort of thing happen a lot? Unfortunately, it does a little too often. And in the era of “proof of concept by simulation alone,” it’s sometimes an expensive path. Another example was a journal article on communication systems. Modern communication systems are heavily centered in mathematical models, but this one actually survived peer review (Fig. 2).

2. Mathematically, an ideal set of VCOs do the exact same thing. In reality, the two devices will not phase match and oscillate at different frequencies.

The article described the use of two identical signals underneath a common control. The signals were presumed identical in amplitude, phase, and frequency. They were open-loop voltage-controlled oscillators (VCOs) in two different locations, not tightly controlled inside some form of control system! Real-world issues like (not so) perfect matching weren’t considered. Even with perfectly matched VCOs, there was no way to guarantee the phase relationship. It works on paper, right? In reality, not so much.

One of the earliest things taught in any class on analog CMOS design is the “square law” for the drain current in a MOSFET (Fig. 3). The premise is that the drain current changes as the gate voltage squared, after you subtract the threshold voltage. Is that valid? Well, sort of. It’s a simple equation that can be fit to the device physics, again with a lot of simplifications and assumptions. The reality is a little different, and the bottom plot is a lot closer to what the actual transistor does for small-geometry CMOS devices.

3. The “Square Law” equation defines the drain current of an ideal MOSFET and behaves like a high-impedance current source. A short channel MOSFET will exhibit some impedance slope and at higher voltages break away from the desired current source behavior.

You can find a limited area in the middle where the square law can be roughly applied, but the equation does not clearly define where it’s a valid region. Get a bias curve set and use it to set up your device bias. At least then, you will know if you are in the “sweet spot” for good dynamic range bias. With the equation, you won’t know if you are suffering from short channel effects, on the right side of the plot, or close to looking like a resistor, on the left side.

### Another Look

What about that diode equation again? If the equation was real, then each diode in the pair should have the exact same current (Fig. 4a). In truth, they will be slightly different, and ignoring that difference can lead to some interesting results with high-power devices. The same relationship should hold true for a large bipolar transistor (Fig. 4b), with a distributed base-emitter junction.

4. Ideal diode matching (a) of currents doesn’t happen. An improperly designed high-current power transistor (b) can exhibit thermal runaway where one junction gets warmer and progressively takes more current. Including small amounts of emitter resistance (c) compensates for this and balances the current across the transistor.

If the equation holds, all the currents should balance, and they don’t. What happens when you power this device up is called “thermal runaway,” where one of those junctions takes more current and gets warmer. That warming causes the –2-mV/°C change in the junction shown in Figure 1. There’s a warmer junction, smaller junction voltage, and more current. Pretty soon, one junction has most of the current.

In a power device with enough current available, the junction fails. The current then goes through the device with the next lowest junction voltage, and it fails. This domino effect causes the whole device to self-destruct, and the results can be spectacular in a smoky sort of way. Plastic encapsulated chips can explode if the inside gets hot enough. If you really like equations, look up Boyle’s law of gases for the equation that created the smoke show. You can avoid the explosion by adding small amounts of resistance to the emitter (Fig. 4c).

Equations can be a blessing and a curse. Some engineers want to have everything reduced to a bunch of math, and in their zeal to do so, they often simplify important things out so they can get there. Doing so looks nice and neat, but it’s often not well connected with the reality of what the circuitry actually does. I prefer things like bias curves taken from real silicon, or a more complex spice model based in empirical testing of silicon. Simulators have their problems too, but that’s another story.