1. Introduction

This assignment uses mathematics (gasp!) to compute some rules of thumb that are helpful when designing circuits with pn junctions, including diodes and bipolar junction transistors (BJTs). It can be easy to get mired in the mathematical details and lose sight of why and how these equations affect circuit design decisions.

All electronic designs need to take into account that the circuit will likely be used at various ambient temperatures. Knowing the general behavior of semiconductor voltages and currents when temperature changes increases the chance of design success and avoids costly troubleshooting and re-designs.

Using the results of the second question in the hand-calculation phase of design can avoid the need for temperature simulations at least 50% of the time.

2. Questions

2.1. When is the diode I/V approximation valid?

\[\label{eq-exact} i_D = I_S \left[\exp\left(\dfrac{v_D}{\eta V_T}\right) - 1\right]\]
\[\label{eq-approx} i_D(\mathrm{approx}) = I_S \exp\left(\dfrac{v_D}{\eta V_T}\right)\]

The approximation is never equal to the full Shockley equation, but we need to get a sense of the magnitude of the approximation error. Re-state the question for you to answer as:

When does the iD approximation of equation \(\eqref{eq-approx}\) differ from the exact equation \(\eqref{eq-exact}\) by less than x%, for 1 % and 10 % ?

Show your derivation work, and state the result as: vD ≥ y mV for each case. Assume T = 300 K and η = 1.8.

2.2. What is the diode forward voltage’s temperature coefficient?

ECE 340 Lab (3) - Temperature Effects : Section 2. Theory derives the expressions used in this topic. This question considers temperature effects in forward-bias.

What is the temperature coefficient of the diode’s forward voltage (vD) with respect to changes in temperature at a constant 1 mA current?

Assumptions:

  • silicon pn junction

  • η = 1.8

  • at T=300 K, IS = 1 nA

  • iD is held constant at 1 mA

  • use equation \(\eqref{eq-is}\) to include the temperature-dependence of IS

\[\label{eq-is} I_S(T) = C \cdot T^4 \exp\left(\frac{-E_G}{\eta k_B T}\right)\]

First find the forward voltage, vD under these conditions. (consider your results from the first question!)

  • If shown in a plot, this relationship would have vD on the vertical axis and temperature (K) on the horizontal axis.

  • The slope is not a constant. We are only interested in the point-slope at 300 K, or the first Taylor series coefficient.

  • Consider whether to use equation \(\eqref{eq-exact}\) or \(\eqref{eq-approx}\) for the diode current/voltage relationship — your choice.

  • Don’t forget that VT also changes with temperature~

  • This will be a negative, single-digit number with units mV / °C, report your final result to two significant digits.

Use any means or tools available to you: mathematical, programming, experimental, etc.

2.3. Razavi problem 2.19

Report your values in a table using 3 significant figures in engineering units (mV, uA, nA, etc.). Pay attention to how much V_D changes compared to V_X. Use whatever means you wish for finding the numerical solution.

You will find these documents full of useful strategies for actually solving the equations generated from your circuit analysis:

2.4. Read AoE §1.6

The Art of Electronics section 1.6 (pages 31—​40) is pretty great.

Read the section! We will cover only a few of these applications as an official part of this course, but all of them are relatively common.

There is nothing to turn in to Blackboard for this “question.”

3. Submission

Submit a single PDF file with all your work.

Scan your neat handwritten work and briefly describe your solution and process for the second question in a reasonable format.

4. References

This series of posts exploring the ubiquitous 1N4148 diode is worth your time to read in context of the course.