Design Build Measure package 2

1. Introduction

Goals

  • Relate how transistor parameters (e.g. Β and VBE) affect the DC bias stability of a circuit.

  • Explore the common biasing circuits for behavior and sensitivity to transistor and temperature variations.

Objectives

  • Generate symbolic circuit analysis equations for the DC solutions of three bias schemes.

  • Simulate how the DC solution varies when transistor Β and circuit temperature changes.

  • Find ranges of these variations that keep the bias conditions within acceptable limits.

  • Relate measurements of circuits using multiple physical transistors.

2. Elements

The three bias circuits to consider:

  1. day08: Figure 1

  2. Lab 2: Figure 1 (a)

  3. Lab 2: Figure 1 (b)

    hand analysis

    Generate symbolic circuit analysis equations to express the collector current as a function of resistor values, VBE, and β. Select resistor values so that the collector current in each of the three circuits are about equal (within 10% on paper). Round to E12 series values as much as possible to make it easier to match with your built circuits.

    simulation

    Use LTspice or CircuitLab or both to find the DC solutions. These should match your hand calculations to within 10% — change either the simulation or your hand calculation numbers until you get a good match.

    physical build and measurements

    Construct these three circuits and measure the DC node voltages and branch currents (especially IC). These should also match both the hand calcuation and simulation results within 10%.

All three results must mutually match!

On occasion, a small number of variables (typically collector voltage) is quite sensitive to small changes in either assumptions (like hand-calculation VBE or β) or temperature. In those cases, your report must demonstrate that you know the cause of the discrepency by doing something like an example simulaton that varies the parameter and yields a match.

Simply saying "it varies a lot" is not sufficient.

2.1. Specific set of conditions

"Level 1" is to get all three (hand/sim/build) to match for one set of resistors. This can be the values from lab2.

Write up your results and show that they match in your report.

State how close this match is. Do not require your reader to manually flip around your document to extract this information and assemble themselves.

2.2. Circuit degrees of freedom

"Level 2" is about comparing the three circuits for sensitivity to part tolerances.

There is not one set of resistor values to give a specific desired collector current (for example 1.0 mA) and a reasonable collector DC voltage for any of these circuits.

  • So how do you decide which resistors to use when there are an "infinite number" of choices?

Pay attention to how the resistor values affect the result. This is why we do the hand analysis. Specifically, you should pay attention to

  • Which have the most effect in setting a desired current or voltage.

  • How sensitive the result is to a change in a specific value.

Sometimes, such as with voltage divider type circuits, the resistor ratio is the direct result, but the total value (R1+R2) is not specified. It is helpful to see how the "vertical" current through the divider compares to the (base) current "leaking" out the side of the divider. A large ratio of vertical or standing current to the base current would be described as a stiff divider.

Translated to technical terms, a "stiff" divider is one whose Thevenin equivalent resistance is small. Such a circuit behaves closer to a voltage source and whose "output" voltage is not influenced by the current drawn from the middle node.

Sometimes the relationship is between voltages. Consider how the voltage across the emitter resistor in these circuits compares to the transistor’s VBE. Specifically, Lab 2 Figure 1 (b) base voltage and the KVL stack of VBE + VRe2 — mispredicting VBE has less of a consequence for collector current in circuit (b) compared to circuit (a). What if the voltage across Re2 is small compared to VBE, or about the same, or much larger?

Of course, the R1,R2 voltage divider ratio for the base voltage and the value of Re2 need to change to keep the same collector current; the intuition is on the relationship and sensitivity to tolerances.

Your report must discuss how these relationships affect how stable the circuit’s DC solution is to changes in part tolerance.

Imagine this is a design for a product. You need the collector current to be within X% for every unit manufactured. But you also know that a parameter like transistor β flops all over a wide range.

Discuss and use some supporting examples of how to deal with these situations with an eye towards the DC solution of a circuit.

3. Related

4. Resources

Sensitivity

the relative change in out to relative changes in x.

\[{\Large S}^{out}_{x} = \dfrac{\left(\dfrac{\Delta out}{out}\right)}{\left(\dfrac{\Delta x}{x}\right)} \text{ as } \Delta x \rightarrow 0\]
\[{\Large S}^{out}_{x} = \frac{x}{out} \frac{\partial out}{\partial x}\]
Examples

  • \({\large S}^{I_C}_{R1} = -1.0\) means a +5% change in R1 results in a −5% change in IC. (perhaps not a surprise)

  • \({\large S}^{I_C}_{\beta} = +0.20\) means a ±30% change in transistor β results in only a ±6% change in IC. (this is good!)

  • \({\large S}^{V_E}_{V_B} = 1\) means that the emitter voltage changes in (about) the same way as the base voltage. (this should be obvious)

Some recommended articles about sensitivity: