In this lab, we will use measurements to determine whether various circuit components (a resistor, light bulb, and diode) follow Ohm's law, V=IR.^{Hoveroverthese!}

Resistors

The voltage across a resistor is determined by the current flowing through it and a property of the resistor known as the resistance using Ohm's Law:¹

$$V=IR\label{Ohm}$$

In general, circuit components for which this property are called Ohmic. It is possible to have something which acts "like" a resistor, in the sense that voltage is dependent on current through it, but the resistance depends on voltage; such components are called non-Ohmic.

In reality, like Hooke's law, "being Ohmic" is always an approximation that is only valid over a certain range, in this case a certain range of voltages. Nevertheless, the range is sufficiently large, for appropriate components, that the approximation is often useful.

In general (to an appropriate approximation), the resistance of a (cylindrical) resistor is determined by the length of the resistor \(L\), the cross-sectional area \(A\), and the "resistivity" \(\rho\):

$$R=\frac{L\rho}{A}$$

The resistivity \(\rho\) is not a property of the geometric shape of the resistor, just of the material the resistor is made from. However, it can be influenced by the thermodynamic state of the material (i.e., the temperature, pressure, etc. of the material). This can be important if your temperature changes significantly (as it does in, e.g., a light bulb, which has the filament become very hot in order to glow).

Diodes

An interesting non-Ohmic component is a diode. An ideal diode acts like a "one-way infinite resistor": one way, no current passes through (so we say it has "infinite" resistance in that direction); in the other direction, it has no resistance (so it's like it's not even there).

A more realistic diode has a resistance which varies exponentially with voltage. This makes it so a small "forward" voltage causes the resistance to decay to almost zero, whereas a small "backward" voltage causes the resistance to explode to infinity. This is concisely represented by the Shockley diode equation:

$$I=I_S\left(e^{\frac{V}{nV_T}}-1\right)$$

Here, there are a few constants characterizing the diode, but the most important one is \(V_T\), the thermal voltage. This is ~26mV at room temperature, hence much smaller than the voltages we will deal with in this lab. The others are \(n\simeq 1\), known as the quality factor, and \(I_S\), known as the saturation current.

When we go backwards with a large voltage, \(V\ll -V_T\), we have the exponential being approximately zero, hence our current becomes an (approximately constant) \(-I_S\). We then have a measurement of resistance that looks like \(R\simeq\frac{|V|}{I_S}\rightarrow \infty\) as \(V\rightarrow -\infty\).

Conversely, when we go forwards with a large voltage, we can neglect the \(-1\), and so the current grows exponentially with voltage. Practically speaking, this means that other resistances in the circuit are more likely to determine the resistance and limit the current. Thus, in the limit of large voltage (compared to the very-small \(V_T\)), we can (relatively speaking) neglect the voltage across the diode.

Click here for advice on NOT burning out the fuse in your ammeter.

The simpler power supplies only have one knob, which will be what you control your voltage with in this experiment. There are also three-knob power supplies, which are not much more complicated, but you have to know what knob does what.

Click here if you have a three-knob power supply.

Part I: Measuring the Resistance of a Single Resistor

Over the course of the lab, your TA will be going around measuring resistances of your resistors with a multimeter (or they may be provided ahead of time).

First, choose a resistor to measure. Read the resistor code (the colors on the side of the resistor), and record the resistance. Also record the value measured with a multimeter.

Click here for instructions on how to read resistor codes.

Then, wire the circuit up as shown below:²

Click here for step-by-step instructions for setting up your circuit for Part I.

Once everything is wired up, set your power supply to 1V or so. Measure voltage and current.

Slowly turn up your voltage to 10V in units of 1V (again, approximately). Measure voltage and current at each step.

Now, we want to also test negative input voltages to our resistor. The easiest way to do this is to flip the wires at your power supply (so red is wired to black and vice versa).

Part II: Light Bulb

Replace the resistor with the light bulb in your circuit, and swap the power supply back to "normal."

Set your power supply to 5V or so; your light bulb should now light up. If it does not, consult your TA.

Now, turn your voltage down low, where the bulb does not light up (much). Take ten measurements of current and voltage between 0V and 2V.

Flip your orientation (as before), and take ten measurements between -2V to 0V.

Part III: Diode (Optional)

Revert the power supply back to "normal." We'll now be dealing with the diode+resistor combination; note that you should only hook the power supply up to the ends of this combination, not the middle peg. (The middle peg can short out your ammeter!)

The following diagram shows you (in terms of the physical appearance of the diode) which way current is "allowed" through the diode.

In accordance with that picture, connect the "high" (red) end of the power supply to the tail of this arrow, and the "low" end of the power supply to the head (on the two ends of the diode-resistor pair - again, nothing to the middle peg). However, some boards have a diode which conducts in the opposite direction! Use the hand-drawn arrows on the board as a guide.

Set your power supply to 1V, and check that you get a nonzero current. Then, take ten measurements of voltage and current from 0-10V.

Swap the direction of the power supply, and take 0V to -10V. (Are your results the same?)

Finally, using the color bands (as in Part I), record the resistance of the resistor (i.e., without the diode) on your data sheet.

Part I: Resistor

Make a plot of voltage vs. current. We violate the usual convention for such plots here, which would be to put the independent variable (voltage) on the x-axis and the dependent variable, current, on the y-axis. We do this, simply put, to make the analysis easier.

Determine the resistance (and uncertainty) based on this plot. Answer the question on the data sheet about whether your result agreed with expectation.

Part II: Light Bulb

Make a plot of voltage vs. current for the light bulb, without a best fit line. Look at the qualitative behavior of this plot.

If your data is linear with no intercept (as you should have obseved in part I), then the light bulb acts like an ohmic resistor (voltage is directly proportional to current). If not, then the resistance depends on the current.

State your results on your data sheet by answering the stated questions.

Part III: Diode (Optional)

As with the previous two circuit components, make a current vs. voltage plot, again without a best fit line. Is it linear? What does ths imply about whether or not the diode is ohmic (overall; i.e., with both directions considered together)?

Now, the "wrong" direction behavior should be pretty clear just by looking at your data; let's look at the "right" direction behavior (where you see nonnegligible currents). Make a separate current vs. voltage plot for just this part of your data, and add a best fit line this time.

From this second plot, calculate the resistance of the resistor+diode combination. Is this compatible with the expected resistance in this direction? (Consider what the diode's resistance in the "forward direction" should be, and how the the resistances of the diode and resistor combine in series.)

Your TA will ask you to discuss some of the following points (they will tell you which ones to answer):

• Non-ideal Voltmeter/Ammeter: In this experiment, we make an assumption of an "ideal voltmeter" with infinite resistance, and an "ideal ammeter" with zero resistance. Let's see what happens if we don't make this assumption.

  Suppose our components were non-ideal: suppose the ammeter instead had a small (say, 3Ω) resistance \(R_A\), and the voltmeter had a large but still finite (say, 1MΩ) resistance \(R_V\). These resistances can be incorporated by making the following replacements in the relevant circuit diagrams:

  

  I.e.: a realistic voltmeter is actually more like a combination of an ideal voltmeter and a large resistor in parallel; a realistic ammeter is more like a combination of an ideal ammeter and a small resistor in series. (In other words, our "better assumption" is just that the voltmeter accurately measures the voltage across it and that the ammeter accurately measures the current through it.)

  • Impact of Non-Ideal Components: Draw the new circuit diagram with the non-ideal voltmeter and ammeter. If we take the voltage across the voltmeter and divide by the current through the ammeter (as, effectively, was our procedure in this experiment), what resistance do we expect to actually measure? Does the voltmeter's non-infinite resistance cause a systemic error here? Does the ammeter's non-zero resistance?
  • Alternative Wiring: Suppose now that we had used instead a slightly different wiring setup, where the voltmeter was instead wired across both the ammeter and resistor (instead of just across the resistor). Draw the new circuit diagram, first with an ideal voltmeter/ammeter and then assuming they are realistic (with the above replacements). Now, what resistance do we actually measure? Again, does the voltmeter's non-infinite resistance cause a systemic error here? Does the ammeter's non-zero resistance?
  • Comparison of Alternative Wiring: Take your resistor's listed value for Part I, and compute the "actual" measured resistance for each wiring setup (the one we used, and the one we didn't), assuming the values for \(R_A\) and \(R_V\) suggested above. For which one is the measured resistance closer to the resistance we want to measure?
• Analyzing the Light Bulb: You should have noticed that the light bulb doesn't have a single well-defined "resistance," since the current vs. voltage plot is nonlinear. Nevertheless, one can define a "voltage-dependent resistance" as \(R(V)=\frac{V}{I(V)}\) as the ratio of voltage to current.¹
  • Basic Behavior: According to your data, does this resistance increase or decrease with voltage? A reasonable (and correct) thought is that the impact is really with temperature, as the light bulb heats up with more power going into it. How does your data imply resistance varies with temperature?
  • Thermal Expansion: One hypothesis you might have is that the reason is that the resistor expands slightly with increased temperature (since most materials do), and hence the cross-sectional area and length of the resistor change. Supposing the resistor increases in size by the same factor in every direction, what direction does the resistance change? (I.e., does the resistance get larger or smaller?) Is this the direction that you expect based on your answer to the previous part?²
  • Resistivity Changes: Another hypothesis would be that the resistivity itself changes with temperature. In order to understand why, we need a model of resistivity; one often-used simple model is the Drude model,³ wherein the resistivity can be calculated as \(\rho=\frac{m}{nq^2\tau}\). Clearly, neither the mass \(m\) nor charge \(q\) of the electron will depend on temperature. Thus, for the resistivity to vary with temperature, either the number density of "conduction electrons," \(n\), or the time between electron collisions with ions, \(\tau\), will change with temperature. What is a plausible reason that one of these factors might change with temperature? Does the direction of that change result in a change in resistivity that matches what you expect? (I.e.: would the temperature-dependent effect you note cause the resistance to increase or decrease? Does that match what you expect?)
• Diode Circuit: Why do we do this extra work of adding a resistor to the diode circuit, then subtracting it out? Why do we not just measure voltage and current across the diode alone to measure its resistance?
• Four-Point Resistance Measurement: In our circuitry directions, we advised using new alligator clips for each connection (in particular, when you connect the voltmeter), rather than "stacking" banana cables into each other. This helps to reduce a problem known as contact resistance: there is some resistance in the connection between the alligator clips and the resistors. Why would using more alligator clips help this problem? (Hint: draw out the circuit diagram with both configurations, with this "contact resistance" in the appropriate place in the circuit for each configuration. Then, remember: in some places, extra resistance is problematic; in other places, it is not.)⁴³

Guide to Uncertainty Propagation & Error Analysis (Quick Reference)

PHY 121/122 Plotting Tool