Electronics and Signal Processing

Lab 2 Formative

Inductance and Internal Resistance

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Before the lab, read the Background section below and answer Question 1. The error propagation rules you will need are in the Lab Guide.

L2.1 Goals

L2.2 Background

Measuring inductance and internal resistance

In Lab 1, we measured the capacitance \(C\) of a capacitor by placing it in series with a resistor of resistance \(R\) and measuring the time constant \(\tau_\text{RC} = RC\). We did this by using a wave generator to drive the circuit with a square wave, and by using the oscilloscope to measure the fall time \(\Delta t\) of the voltage across the resistor between 80% and 20% of its maximum value. Using the theoretical form of this curve, we found (Question 1 of Lab 1) that \(\tau_\text{RC}\) could be determined from \(\Delta t\). Knowing \(R\) from a multimeter measurement then gave us \(C\).

In this experiment, we will use a similar setup. This time, however, we will use an RL circuit to measure the internal resistance of the wave generator, using the rated value of the inductance \(L\) of the inductor.

Just as before, we will use the wave generator to apply a square-wave voltage to the circuit, and record the voltage across the resistor during the charge and discharge cycles using the oscilloscope. Now, however, the current through the resistor rises when the voltage steps up (see the RL circuit, (5.1), in RL, RLC and AC Circuits). The voltage is \[\begin{equation} \tag{L2.1} V_\text{R}(t) = V_0\left(1 - e^{-Rt/L}\right), \end{equation}\] so that the time constant is \[\begin{equation} \tag{L2.2} \keyeq{\displaystyle \tau_\text{RL} = \frac{L}{R}} \end{equation}\] By measuring the rise time \(\Delta t\), the time it takes for the voltage to rise from 20% to 80% of its maximum value, we can determine \(\tau_\text{RL}\) and then compute \(R\) using the rated value of \(L\).

Question 1: Compute \(\tau_\text{RL}\) from the rise time \(\Delta t\). How does your answer compare to the relation between \(\tau_\text{RC}\) and \(\Delta t\) that you found in Lab 1? Insert your answer below.

\(\tau_\text{RL} = {}\) 

Because \(R\) appears in the denominator of (L2.2), a large value of \(R\) makes \(\tau_\text{RL}\) very short. Since commercially available inductors usually have \(L\) in the millihenry range, we need to keep \(R\) as small as possible so that \(\tau_\text{RL}\) stays within a reasonable range for our scope.

Unfortunately, this means that the total resistance of the circuit is dominated not by the resistor itself but by the internal resistance of the wave generator. This resistance is set to around \(600\,\mathrm{\Omega }\) but is not known exactly, and it is already large enough to make \(\tau_\text{RL}\) fairly small. (It was also present in the RC circuit of Lab 1, but was negligible next to the \(100\,\mathrm{k\Omega }\) resistor.) Because the inductor itself also has some resistance, the total resistance of the circuit is \[\begin{equation} \tag{L2.3} \keyeq{R = R_R + R_L + R_\text{int}} \end{equation}\] where \(R_R\) is the resistance of the resistor, \(R_L\) the (DC) resistance of the inductor, and \(R_\text{int}\) the internal resistance of the wave generator.

In this experiment, we will determine \(R_\text{int}\) by first finding \(\tau_\text{RL}\) from a measurement of \(\Delta t\), and then measuring \(R_R\) and \(R_L\).

L2.3 Materials

L2.4 Instructions

rl circuit

Figure L2.1. RL circuit driven by a square-wave generator. The oscilloscope measures the voltage across the resistor (between Scope Probe and Scope GND).

rl circuit photo

Figure L2.2. The RL circuit of Figure L2.1 on a breadboard.

L2.5 Analysis

Show your derivations for all calculations. Every number you report should come with an error! When computing an error for a derived quantity, use the rules in Error propagation in the Lab Guide. The natural places to propagate errors in this lab are marked below.

Keep a record of these results. You will use them for the Signals Lab!