Circuit theorems
Seen from two terminals, any linear circuit is a source with a series resistance, and no experiment can tell them apart. Why an antenna is matched at 50 Ω and a loudspeaker is not comes from there.
01Superposition
In a linear circuit with several independent sources, any voltage or current is the sum of those that each source would produce acting alone. To “turn off” a source it is replaced by what it is when it delivers zero: a voltage source by a short circuit; a current source by an open circuit.
Superposition holds for voltages and currents, which depend linearly on the sources. Power depends on the square: if one source alone drives 1 A through a resistor and the other alone also 1 A, together 2 A flows and the power is four times as much, not two. Adding partial powers is one of the most repeated mistakes of the course.
Superposition is, moreover, the underlying justification for everything done with harmonics: each Fourier component is a source, and the total response is the sum of the responses.
02Thévenin and Norton
Seen from two terminals, any linear circuit —however complicated inside— behaves exactly like a voltage source with a series impedance (Thévenin) or like a current source with a parallel impedance (Norton). For whoever connects something to those terminals, no experiment can distinguish the original circuit from its equivalent.
The practical usefulness is enormous: it lets you analyze over and over what happens with different loads without solving the rest of the circuit again. The whole idea of the “output impedance” of an amplifier or the “internal resistance” of a battery is a Thévenin equivalent.
03Maximum power transfer
If the load is very small, the current is large but the voltage across it is almost zero; if it is very large, the opposite happens. In between there is a maximum of delivered power, and it falls exactly at the load equal to the Thévenin resistance.
Real circuit: 12 V source, a divider of 1 kΩ and 3 kΩ, and 250 Ω in series out to the terminals. Its equivalent: 9 V with 1 kΩ. You move the load: the panel calculates the current by both paths —complete circuit and equivalent— and the curve shows power and efficiency.
04Other useful theorems
| Theorem | States | What it is for |
|---|---|---|
| Reciprocity | In a passive network, swapping the source and the meter gives the same reading | An antenna receives with the same pattern with which it transmits |
| Millman | Several parallel branches with a source and a resistor: \( V = \sum (V_k/R_k) / \sum (1/R_k) \) | The midpoint of a resistive summing network, in one line |
| Substitution | A branch with known voltage and current can be replaced by a source of that value | Splitting a large circuit into parts that are solved separately |
| Tellegen | The sum of \( v_k i_k \) over all the branches is zero | Conservation of energy, proved using only Kirchhoff’s laws |
05Where it shows up in electronics
| Case | Is it matched for maximum power? |
|---|---|
| Antenna and RF transmitter | Yes: everything is designed at 50 Ω so as not to waste the signal and to avoid reflections |
| Output of an audio amplifier to a loudspeaker | No: the amplifier has an output impedance of hundredths of an ohm compared with the speaker’s 8 Ω, in order to control it well |
| Power grid and a house | No: with matching, half the energy would be lost in the cables; what is sought is efficiency, not maximum power |
| Solar panel | Almost: the maximum power point tracker (MPPT) continuously adjusts the load to the point of greatest output |
The maximum power rule has a price that the lab shows: at that point the efficiency is exactly 50 %. When what matters is the signal —an antenna receiving microvolts— it is worth it. When what matters is the energy, it is not.
06In the lab
Build the lab’s circuit. Measure the open-circuit voltage and the short-circuit current with a multimeter, calculate \( R_{th} \) and compare with 1 kΩ.
Measure the voltage of an AA battery with no load and with a 10 Ω resistor. Calculate its internal resistance. Repeat with a worn-out battery and compare.
Add to the circuit a 2 mA current source between the terminals and ground, and calculate the output voltage by superposition. Verify that the power does not add up.
07Common mistakes
- Superposing powers. Voltages and currents are superposed; the power is calculated at the end.
- Turning off a current source with a short circuit. A turned-off current source is an open circuit.
- Believing that matching is always good. Maximum power means 50 % efficiency.
- Applying Thévenin to a nonlinear circuit. It holds only for linear circuits, or circuits linearized around an operating point.
- Forgetting dependent sources when calculating \( R_{th} \). They are not turned off; you have to drive the terminals and measure.
08Self-assessment
How is \( R_{th} \) obtained from two measurements?
By dividing the open-circuit voltage by the short-circuit current.
With what load is the maximum power delivered?
With \( R_L = R_{th} \); in AC, with \( Z_L = Z_{th}^* \).
What is the efficiency at maximum transfer?
50 %: half of the power is dissipated in the Thévenin resistance.
Why is an audio amplifier not matched to the loudspeaker?
Because the aim is to control the loudspeaker with an almost ideal voltage source, not to extract the maximum power at the cost of losing half of it.
Why can powers not be superposed?
Because power depends on the square of the current or the voltage, and the square of a sum is not the sum of the squares.
09Further reading
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 7th ed., McGraw-Hill, 2021. The chapter on circuit theorems, with superposition, Thévenin, Norton and maximum transfer.
- Mac E. Van Valkenburg. Network Analysis. 3rd ed., Prentice Hall, 1974 (Spanish edition, “Análisis de redes”, Limusa). Reciprocity and Tellegen treated with the rigor of network theory.
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. Conjugate matching taken to the field where it matters most: radio frequency.