Separate sinusoidal peak, rms and transformed power
A rotating coil in a uniform field has sinusoidally changing flux. The supplied ideal generator has50 turns, area .02 m² and field .1 T, so peak emf is NABω=.1ω with ω=2πf. One complete cycle has zero signed mean voltage, but nonzero mean squared voltage. A resistor therefore can absorb positive mean power even though the voltage alternates sign.
Learning outcome: Compare frequencies10,20,40 Hz at one turns ratio, then change turns ratio at one frequency. Retain peak, rms and signed mean separately. Explain why doubling secondary voltage across the same resistor quadruples power and changes the required primary current.
Starting recap
Use a right-handed coordinate system, oriented surface normal, signed charge/current and declared energy boundary. Separate an entry force from a trajectory, emf from current, and rms from signed average.
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At f=20 Hz, generator peak emf=4π V and primary rms=2√2π≈8.885766 V. With turns ratio1.5, secondary rms=3√2π≈13.328649 V. Secondary mean power is18π²/100≈1.776529 W, and ideal primary mean power matches.
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Zero signed mean sinusoidal voltage across a resistor implies:
Experiment
Compare frequencies10,20,40 Hz at one turns ratio, then change turns ratio at one frequency. Retain peak, rms and signed mean separately. Explain why doubling secondary voltage across the same resistor quadruples power and changes the required primary current.
Compare sinusoidal peak, rms and ideal transformed power
Worked teaching model · synthetic values · Ideal rotating 50-turn coil area .02 m² in B=.1 T provides sinusoidal source. Ideal transformer drives a purely resistive 100 Ω secondary load. No winding/core losses, saturation or magnetizing current. Fixed rotation maintained externally; no mains equipment claim.
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One sinusoidal generator cycle
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: maintained rotation (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for One sinusoidal generator cycle
maintained rotation
| time (s) | instantaneous emf (V) |
|---|---|
| 0 | 0 |
| 0.00125 | 1.96581 |
| 0.0025 | 3.88322 |
| 0.00375 | 5.70501 |
| 0.005 | 7.38633 |
| 0.00625 | 8.88577 |
| 0.0075 | 10.1664 |
| 0.00875 | 11.1967 |
| 0.01 | 11.9513 |
| 0.01125 | 12.4117 |
| 0.0125 | 12.5664 |
| 0.01375 | 12.4117 |
| 0.015 | 11.9513 |
| 0.01625 | 11.1967 |
| 0.0175 | 10.1664 |
| 0.01875 | 8.88577 |
| 0.02 | 7.38633 |
| 0.02125 | 5.70501 |
| 0.0225 | 3.88322 |
| 0.02375 | 1.96581 |
| 0.025 | 1.53894e-15 |
| 0.02625 | -1.96581 |
| 0.0275 | -3.88322 |
| 0.02875 | -5.70501 |
| 0.03 | -7.38633 |
| 0.03125 | -8.88577 |
| 0.0325 | -10.1664 |
| 0.03375 | -11.1967 |
| 0.035 | -11.9513 |
| 0.03625 | -12.4117 |
| 0.0375 | -12.5664 |
| 0.03875 | -12.4117 |
| 0.04 | -11.9513 |
| 0.04125 | -11.1967 |
| 0.0425 | -10.1664 |
| 0.04375 | -8.88577 |
| 0.045 | -7.38633 |
| 0.04625 | -5.70501 |
| 0.0475 | -3.88322 |
| 0.04875 | -1.96581 |
| 0.05 | -3.07787e-15 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Angular frequency | 125.664 | rad/s |
| Generator peak emf | 12.5664 | V |
| Primary rms voltage | 8.88577 | V |
| Secondary rms voltage | 13.3286 | V |
| Secondary rms current | 0.133286 | A |
| Primary rms current | 0.19993 | A |
| Mean secondary resistor power | 1.77653 | W |
| Mean primary electrical power | 1.77653 | W |
| Cycle-mean signed generator emf | 0 | V |
Zero cycle-mean voltage does not mean zero rms voltage or zero mean resistor power.
Peak emf=NABomega. For a sine, Vrms=Vpeak/sqrt2; a purely resistive load has mean power Vrms²/R. An ideal transformer gives Vs/Vp=Ns/Np and Ip/Is=Ns/Np, conserving mean power while changing voltage/current. The mechanism requires changing linked flux; steady DC does not produce the stated continuous secondary sinusoid. Ideal lossless assumptions do not certify real equipment behavior.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Zero cycle-mean voltage does not mean zero rms voltage or zero mean resistor power.
Peak emf=NABomega. For a sine, Vrms=Vpeak/sqrt2; a purely resistive load has mean power Vrms²/R. An ideal transformer gives Vs/Vp=Ns/Np and Ip/Is=Ns/Np, conserving mean power while changing voltage/current. The mechanism requires changing linked flux; steady DC does not produce the stated continuous secondary sinusoid. Ideal lossless assumptions do not certify real equipment behavior.
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