It’s exponentials all the way down

1. What you know and can do

1.1. Analysis

Circuits containing a single inductor.

  • Find the initial current Iinitial before any changes, treating the inductor as a short-circuit.

  • Immediately after the change, the inductor current remains the same, because current through an inductor cannot change instantaneously.

  • Find the circuit voltages at the instant after the change by treating the inductor as an open-circuit.

  • After a long time, the inductor again looks like a short-circuit. Find the (new) resting current Ifinal.

  • Find the time constant \(\tau = L / R_{eq}\), where \(R_{eq}\) is the resistance as seen by the inductor.[1]

Then the inductor current or voltage waveforms will have the following shape

\[i_L(t) = I_{\mathit{initial}} + \biggl(I_{\mathit{final}} - I_{\mathit{initial}}\biggr) \left\lbrack 1 - \exp\left(\dfrac{-t}{\tau}\right) \right]\]
\[v_L(t) = V_{\mathit{initial}} + \biggl(V_{\mathit{final}} - V_{\mathit{initial}}\biggr) \left\lbrack 1 - \exp\left(\dfrac{-t}{\tau}\right) \right]\]

1.2. Real inductors

A real inductor is nothing more than a coil of wire, sometimes wrapped around a ferrous (magnetic) material that enhances the magnetic fields generated from current flowing through the wire.

You do know that wire is not a perfect conductor (short circuit). Therefore, a real inductor is best modeled, for the purposes of circuit analysis, as an ideal inductor of value L in series with a resistor RL. Typical real inductors are the least ideal of all of the circuit components, so it is generally a bad idea to ignore this series resistance when matching circuit analysis results to the operation of a physical circuit with inductors.

inductor model
Figure 1. Simple model of a real inductor

1.3. Simulation

  • Use PSpice to construct a circuit with one (or more) inductors, sources, and resistors.

  • Setup initial conditions for inductor current if necessary.

  • Perform a transient simulation to show the circuit’s voltage and current waveforms.

  • Measure waveform characteristics using cursors on the waveform plots.

1.4. Lab equipment

Check it out: you have NEW MULTIMETERS!

All ECE labs are now equipped with the same model as those in GEM 166 “the junior lab.”

These meters are both accurate and precise to at least 5 digits, which is way more precision than necessary in most bench work. We generally report measured numbers to 3 digits of precision (roughly 1%).

From previous laboratory activities, you know how to:

  • Use a multimeter to measure DC characteristics of devices (resistance) and voltage differences and branch currents in a circuit.

  • Setup a function generator to output waveforms (square, sine, etc.) of known frequency, duty cycle, and amplitude.

  • Display circuit voltage waveforms on an oscilloscope, taking care to setup the trigger system to stabilize the display.

  • Use cursors to measure voltage and time intervals.

Nearly all function generators, including all of the models in the various ECE labs, have a Thevenin-equivalent output resistance of 50 Ω. The circuit of Figure 2 shows how to account for this when analyzing a circuit that includes a function generator.

function generator
Figure 2. Equivalent circuit of the bench-top function generators

2. What you therefore can do now

By these powers combined, you have all of the tools required for this laboratory’s activity!

Given a random physical inductor, find its inductance L and series resistance RL.

You can use a multimeter, function generator, oscilloscope, and various known resistors to determine the values of both RL and L for your inductor.

Design a test procedure and then use that procedure to figure out the characteristics of a few provided inductors.

  • Write out your strategy and process, including relevant schematics. Have your instructor check this before proceeding.

  • Obtain an unknown inductor and collect the necessary supplies.

  • Construct your test circuit and perform your measurements.

  • Keep a log of your process and the resulting measurements.

  • Verify your values against the measured values of the unknown inductor as measured by the LCR meter.

Write a short report that includes your procedure, measurements, calculations, and results.

1. Zero all independent sources, then find the resistance between the two inductor terminals (similar to finding Thevenin/Norton equivalent resistances).