1. Reading reference

Read [1] from §2.2.3 through the end of §2.2.6 (page 87). Sections 2.2.7 through the end is the topic for the next class.

Read [1] §2.3.7, paying attenting to Figure 2.60. The “(figure out why)” at the end of page 101 is related to the goal of the Lab 1-B Current Mirror circuit.

Supplemental: read [2] chapter 4N.

2. Big beta is bettah

Parameters and variablesubscripts

Now is a good time to briefly review transistor parameters and our capitalization convention for circuit quantities.

BJT hand analysis rules

  • Guess the transistor’s mode of operation {cutoff, active, saturation}.

  • Substitute the transistor symbol with the appropriate Hand-calculation model.

  • Solve the rest of the circuit.

  • Check that the computed voltages and currents are consistent with the chosen mode.

    • No? Then make a different guess and try again.

2.1. Strategic laziness analysis method

aoe fig2 28 sim
Figure 1. Relying on β for stable bias conditions

Now consider Figure 1.

  • What value of RB is needed so that Vout is 7.5 V?

Remember what β means for a bipolar transistor: \(\beta = i_C / i_B\), and KCL gives: \(i_E = i_C + i_B = (\beta + 1) i_B\)

Process to figure this out:

  1. Find iE first because it is pretty much handed to you.

    Details

    Set Vout to 7.5 V (“assuming the consequent” — we are solving backwards from the desired result).[1]
    Then the current through RE is 1.0 mA,
    which is in series with the Q1 emitter, thus iE also is 1.0 mA.

  2. Easy one: VB =

    Details

    Go back “up” the KVL stack.
    = VE + VBEon (assume 0.6 V)

We now know the voltage across the resistor and continue our aggessive Ohm’s law campaign. …​ But are stopped because we don’t know its current since the resistor value is also unknown.

What to do then? Easy! Declare that this is an impossible problem, throw away your phone and laptop (which contain evil transistors), buy a farm, horse, and cow, and live like Laura Ingalls Wilder’s family.

Notice that the RB current is the same as the Q1 base current because no DC current flows through the Cs capacitor. What is iB, then?

We know iE and how a bipolar transistor works, leaving us to ask:

  • What is the value of β?

Select a strategically lazy value of exactly 99.

We then compute iB =

and RB=

Select the closest value from the E12 series. This is what the ECE department keeps in stock in GEM 166 from 1 Ω through 10 MΩ.

Did the “99” number seem suspicious? Good! You should question numbers chosen for convenience if they don’t match reality. Doing otherwise leaves you at risk for choosing the wrong magic ground symbol ⏚

Look at ON Semiconductor’s 2N2904 datasheet.

What happens when β changes?

3. Bag-of-equations analysis method

Instead of searching for the easiest number to compute at each step (aggressive Ohm’s law), simply write down all of the circuit equations for all of the components. Then combine and solve for the variables of interest.

This is most useful if you keep everything symbolic and only substitute in numbers at the very end. Otherwise, you get to do the algebra all over again when you encounter the circuit with the same form but different numbers.[2]

  • KCL

  • KVL

  • device equations, or “constitutive relations”:

    • resistors: Ohm’s law

    • transistors: equations from Bipolar transistor modes (remember to verify the assumed mode)

    • capacitors: e.g. \(i_C = C \frac{d v_C}{dt}\) or integral version [3]

    • inductors: e.g. \(v_L = L \frac{d i_L}{dt}\) or integral version

    • dependent sources

3.1. Modified Nodal Analysis

You have learned the formal technique of nodal analysis.

  • Label all nodes.

  • Select one to be the reference node (commonly called the “ground” node, but I prefer to avoid the term when teaching because it encourages bad habits and makes it easy to confuse what’s really happening).

  • Write a KCL equation at the N-1 non-referenec nodes.

  • Use device equations to replace the KCL currents to be in terms of the node voltages.

  • Solve the matrix equation for the node voltages.

This has an immediate problem when there are voltage sources in the circuit. Then you use the concept of a “super node” and other bolt-on hacks to make Nodal Analysis work for any arbitrary circuit.

Fortunately, there is a technique called “Modified Nodal Analysis” (MNA) that adds some extra rows and columns to the matrix that gets inverted in a systematic way. This then does not have any exceptions or extra special rules for constructing the matrix and solving.

Therefore, ALL circuit simulators use MNA

Original paper: https://doi.org/10.1109/TCS.1975.1084079

The best 4 descriptions of how MNA works and how to easily create the 4 sub-matrices:

4. Thevenin / Norton equivalent circuits

Do you remember how to compute these for a circuit? Time to review and practice!

Any one port (two-terminal) linear circuit can be replaced by an equivalent circuit

Thevenin {voltage source in series with an impedance}, or
Norton {current source in parallel with an impedance}
Voc

open-circuit voltage (= VTh)
Isc

short-circuit current (= INor)
Rth, nor

Voc/Isc, or set independent sources to zero and find equiv resistance
thevenin box
Figure 2. Linear circuit with a test load
thevenin loadline
Figure 3. Terminal I/V characteristic: load line
The textbook from ECE 264 may be useful. Go to PDF page 154 / book page 140 for Section 3-6. The book walks through three methods for finding RTh — which are merely special cases of solving for the slope if you have any two points on Figure 3. This set of slides also has some nice worked examples: http://garytuttle.ee/circuits/topics/thevenin_norton.pdf

4.1. garytuttle.ee practice

Generate a problem from:
http://garytuttle.ee/circuits/practice/thevenin.php

and find the Thevein and Norton equivalent of the circuit, reporting all of:

  • VTh

  • INor

  • RTh (same as RNor)

Your paper should begin with the circuit schematic with labels and component values. Then sucessively document the process you used to solve the problem, this should include:

  • English words, phrases, or a sentence,

  • re-drawn schematics,

  • math equations and expressions,

  • arrows and boxes helping show the flow of information and calculation,

  • and/or other communication features as appropriate.

4.2. autoCircuits

4.2.1. Two exercises

Generate a circuit to solve from autoCircuits with the following parameters:

  • Get Circuit → Chapters → DC Thevenin-Norton equivalents

  • Numeric — Integer — Medium — No — No — No

  • Open the PDF and zoom in so the circuit fills the screen

Work on the solution as a group and check against the answer in the PDF.

Then generate another problem with these same parameters.

4.2.2. Two more exercises

Generate two more problems with these settings:

  • Get Circuit → Chapters → DC Thevenin-Norton equivalents

  • Numeric — Real — Medium — Yes — No — No

autoCircuits uses conductance for the Norton equivalent.  G = 1 / R

5. References

[1] P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. Cambridge University Press, 2016 [Online]. Available: https://artofelectronics.net

[2] T. C. Hayes, Learning the Art of Electronics: A Hands-On Lab Course. Cambridge University Press, 2016 [Online]. Available: https://learningtheartofelectronics.com

1. This is technically a logical fallacy, but is also a great way to solve a problem for things like a math proof or other design/analysis. See Fallacy Files: Affirming the Consequent and WP: Affirming the consequent.
2. Being strategically lazy will sometimes mean that you work a little harder now to save huge amounts of time later. Knowing when to do this takes some experience and applied wisdom.
3. two things to not forget: initial. conditions. The integral versions explicitly include them while the process of finding the complete answer involves both the general and particular solutions, where latter uses the initial conditions.