DC circuits
The load that receives the maximum power is the one that equals the internal resistance, and right there the efficiency is 50%. That is why in radio frequency you match, and in a power system you never do.
01Current and Ohm's law
Current is charge crossing a cross-section per unit of time. Its conventional direction is that of positive charges, even though in a metal what moves are electrons and they go the other way: it is an inherited convention that is better accepted than fought.
In a 1 mm² copper conductor carrying 1 A, the drift velocity of the electrons is about 0.07 mm per second: they would take four hours to travel one meter. What travels fast is not the electrons but the field, which is established at an appreciable fraction of the speed of light. That is why the lamp lights up instantly even though the electrons barely creep along.
02Kirchhoff's two laws
Solving any DC circuit comes down to writing down two things, and both are conservation principles in disguise:
| Law | Statement | What it conserves |
|---|---|---|
| Node law | \( \sum I = 0 \) at each node | Charge: it does not accumulate |
| Mesh law | \( \sum V = 0 \) around each closed loop | Energy: the field is conservative |
With those two and Ohm's law any resistive network can be solved, however large. And from there come the two dividers that you end up using from memory:
03Real sources and maximum transfer
No real source holds its voltage when loaded: it has an internal resistance. Thévenin's theorem says that any linear network, however complicated, behaves seen from two terminals just like an ideal source in series with a resistance.
A source with internal resistance feeding a variable load. The three curves are voltage, power and efficiency. Look for the maximum of the power and see where the efficiency ends up at that point.
Matching the load to the internal resistance delivers the greatest possible power, and that is why it is done in radio frequency, where what matters is that signal reaches the antenna. But in a power system it would be absurd: a matched power grid would throw away half of what it generates heating its own lines. There the opposite is sought, \( R_i \ll R_L \), where the delivered power is not the maximum possible but the efficiency approaches 100%.
04Measuring without disturbing
Every instrument becomes part of the circuit it measures, and so it disturbs it. The criterion is that its resistance be negligible compared with that of the circuit, or enormous, depending on how it is connected.
| Instrument | Connected | Its resistance must be | Typical value |
|---|---|---|---|
| Voltmeter | In parallel | As high as possible | 10 MΩ |
| Ammeter | In series | As low as possible | Less than 1 Ω |
| Ohmmeter | Across the isolated component | — | Injects a known current |
From the second row comes the most expensive mistake in the lab: connecting an ammeter in parallel with a source is putting a short circuit across it. And from the third, that an ohmmeter only measures well with the component disconnected: on the board it measures the component and everything in parallel with it.
05In the lab
Measure the voltage of a battery at no load and with several known loads. Plot \( V \) against \( I \), and obtain \( \varepsilon \) from the y-intercept and \( R_i \) from the slope. Repeat with a depleted battery and compare.
With the same source and a potentiometer as the load, measure \( V \) and \( I \) for about ten values. Plot the power against \( R_L \), locate the maximum and verify that it falls where \( R_L = R_i \).
Build a two-mesh network with three resistors and two sources. Predict all the currents by solving the system, measure them and compare. Discuss whether the differences are explained by the tolerance of the resistors.
06Common mistakes
- Believing Ohm's law always holds. A diode, a lamp and an arc are not ohmic.
- Inverting the current divider. It has the resistance of the other branch in the numerator.
- Designing a power system for maximum transfer. That fixes the efficiency at 50%.
- Connecting the ammeter in parallel. It is a short circuit, and it usually ends in a blown fuse.
- Measuring resistance with the component on the board. You measure it together with everything in parallel with it.
- Thinking electrons move fast. They drift at tenths of a millimeter per second; what moves fast is the field.
07Self-assessment
What does the node law conserve?
Charge: what enters a node leaves it, because it does not accumulate.
When is the power delivered to the load maximum?
When \( R_L = R_i \), and there it equals \( \varepsilon^2/4R_i \). The efficiency at that point is 50%.
Why is a power grid not matched?
Because it would throw away half the energy heating its lines. There the goal is \( R_i \ll R_L \).
What resistance should a voltmeter have?
As high as possible, because it is connected in parallel and must not divert current.
What does Thévenin's theorem say?
That any linear network seen from two terminals is equivalent to an ideal source in series with a resistance.
At what speed do electrons drift in a wire?
At tenths of a millimeter per second. What travels fast is the electric field.
08Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). Current, resistance, EMF, DC circuits and Kirchhoff's rules.
- Edward M. Purcell and David J. Morin. Electricity and Magnetism. 3rd ed., Cambridge University Press, 2016 (Spanish edition, “Electricidad y magnetismo”). Where conduction comes from and why Ohm's law is a property of the material and not a general law.
- Paul Horowitz and Winfield Hill. The Art of Electronics. 3rd ed., Cambridge University Press, 2015. The first chapter: Thévenin, dividers and source loading, with the practical criterion of the designer.