Learning view
A view selects a representative measurement; it does not change the source amplitude or frequency, the cursor angle θ, or the fixed initial phases.
Cursor angle and initial phase: different roles
One time coordinate for all voltages
θ is the electrical angle elapsed since t = 0. Moving the cursor selects another instant on the same waveforms.
An initial phase for each voltage
The constants φU, φV, and φW are the sine phases at t = 0. They do not change when the cursor moves.
Instantaneous phase: add the two angles
The sine argument of vV is θ + φV. For vU it equals θ because φU = 0; that does not make every voltage's phase equal to the cursor angle.
These initial phases use the chosen sine convention and time origin. A different time origin would change their values, not their differences; moving the cursor does not change that origin. Arguments written with π use radians; readouts and examples marked ° use degrees.
Example: V when the cursor is at 90°
For a nonzero amplitude, at θ = 90°, φV remains −120°. The instantaneous phase is 90° − 120° = −30° and vV = −Vm/2. The 90° cursor position is not the initial phase of V.
Phase voltages · Fundamentals
Three phases, one frequency
The three voltages are measured relative to the same neutral N. They have the same peak Vm and frequency f, and are separated by 120° (2π/3 rad). The fixed initial phases are φU = 0, φV = −2π/3, and φW = +2π/3; θ is the common time angle.
Track the positive maxima: U, then V, then W. The 120° spacing is between matching points of different phases—not between any two neighboring zero crossings.
Why does this happen?
The equivalent expression vW = Vm sin(θ − 4π/3) is identical to Vm sin(θ + 2π/3): the arguments differ by one complete turn, 2π rad. These are equivalent ways to write the fixed initial phase of W, not different cursor positions.
Line voltages · Potential differences
A line voltage is a difference—not a new source
For vUV, put the positive probe at U and the negative probe at V. Subtract both instantaneous potentials at the same instant. The resulting initial phases are φUV = +π/6, φVW = −π/2, and φWU = +5π/6. They are constant, unlike θ.
vUV, vVW, and vWU lead vU, vV, and vW, respectively, by 30°. Their peak is √3Vm; the line voltages still differ from one another by 120°.
Derive √3 and the 30° shift
U and V are 120° apart, not 180°. Their positive and negative peaks do not occur together. Subtracting the two rotating vectors gives a vector √3 times as long, rotated 30° ahead of U.
Reference polarity · Terminal order
Changing the terminal order changes the sign
vUW, vWV, and vVU are the reversed-polarity counterparts of vWU, vVW, and vUV, respectively. The source is unchanged: reversing a measurement pair does not reverse the U–V–W phase sequence. Their fixed initial phases are φUW = −π/6, φWV = +π/2, and φVU = −5π/6.
For this U–V–W source, vUW, vWV, and vVU lag vU, vW, and vV, respectively, by 30°. Reversing one fixed pair—UV to VU—changes its phase by 180°, not 30°.
Check the three polarity identities
The meter reports the potential of its positive terminal minus that of its negative terminal. Swapping the probes negates every sample. A negated sine wave has the same peak and RMS and is shifted by 180°.
The complete line-voltage set
Six polarities, three distinct phase pairs
Display all six line voltages over two periods. Opposite polarities are exact negatives. The six positive peaks occur at 60° intervals.
Positive-peak order as the cursor θ advances from 0°: WV → UV → UW → VW → VU → WU → WV. These are peak positions, not initial phases or a new phase sequence.
The three distinct phase pairs are UV, VW, and WU; they are not three algebraically independent voltages: vUV + vVW + vWU = 0.
From the overlay to a maximum–minimum envelope
At any instant, the largest positive line voltage is obtained by subtracting the lowest phase potential from the highest. Select “Show maximum–minimum envelope” to make that relationship visible. This optional extension is a mathematical construction, not a simulation of a rectifier power circuit.
For the balanced source, 1.5Vm ≤ vspan ≤ √3Vm. Two phase potentials tie at each change of the maximizing pair. No diode conduction, load current, or commutation is being calculated.
Connection to ideal diode rectifiers with an R load
For the balanced phase sources shown here, zero source impedance, ideal uncontrolled diodes and a purely resistive load:
Half-wave, with neutral return (three pulses per source period):
Full-wave bridge (six pulses per source period):
In each R-load case, . The highest phase forms the three-pulse output; the maximum–minimum envelope displayed here forms the six-pulse output. These are ideal voltage constructions: they do not calculate diode drops, source-inductance overlap, SCR firing or capacitor charging currents. Select R load and ideal diodes when comparing with the rectifier laboratories.
Open the three-phase uncontrolled half-wave laboratory.
Open the three-phase uncontrolled full-wave bridge laboratory.
Shared source and cursor
Equivalent phase peak:
Move θ to choose another instant. The time origin and all initial phases stay fixed; this is not an initial-phase adjustment.
The animation advances 30° per real second for observation. It is not a real-time reproduction of the electrical frequency. Manual angle changes pause it. The animation loops the two-period viewing window; initial phases stay fixed.
Initial didactic values: 325 V phase peak and 50 Hz. These are examples for practice, not mandatory ratings of a three-phase system.
Waveforms at the same instant
Click or tap the plot to move the cursor. The slider and number field provide the same operation by keyboard. On narrow screens, scroll only the plot sideways.
Both phase and line voltages share one vertical scale, which follows the chosen phase peak. Read the axis values when changing the amplitude. The horizontal window always spans two periods; changing frequency changes the time labels, not the angular spacing.
Angles of the selected voltage:
RMS values are calculated over a full period of the sinusoidal source. They are not the current cursor values.
Voltages, fixed initial phases, and phases at the cursor
| Quantity | Instantaneous (V) | Peak (V) | Initial phase φk (°) | At cursor θ + φk (°) |
|---|
CSV: UTF-8, comma separator, decimal point. 1,441 uniform samples from 0° to 720° inclusive, every 0.5°. Time is in seconds, voltages in volts. The final endpoint repeats the start of the periodic signal; omit it when averaging. Lines beginning with # contain provenance and settings; the data header follows them.
The plot and numerical table share one voltage unit, selected from the source amplitude. Standalone readings may use a different, explicitly shown SI prefix. Numerical inputs remain in the stated units; CSV voltages always remain in volts.
Columns: theta_deg, t_s, U_R_V, U_S_V, U_T_V, U_RS_V, U_ST_V, U_TR_V, U_RT_V, U_TS_V, U_SR_V, U_span_V, U_probe_V. N is always 0 V. U_probe_V uses the selected probe terminals; U_span_V is the optional mathematical envelope, not a simulated rectifier output. For compatibility, the existing column names retain their original notation: R→U, S→V, T→W. Thus U_R_V, U_S_V, and U_T_V record vU, vV, and vW; paired subscripts follow the same mapping. U_span_V and U_probe_V record vspan and the selected instantaneous voltage. Display-name metadata records this mapping; numerical values, polarities, and units are unchanged.
theta_deg is the sampled time angle ωt, not a voltage's initial phase; cursor_deg records the current cursor position in the metadata. The fixed initial phases and the extra on-screen angle column are not new CSV columns. Voltage samples keep exactly the documented convention.
Model consistency checks
The implementation checks the instantaneous phase sum, the loop identity vUV + vVW + vWU = 0, and the opposite-polarity identities. Tiny residuals can come from floating-point arithmetic. These checks do not validate a different physical model.
Peak vectors at the cursor
Vertical projection = instantaneous voltage; vector length = peak voltage. Its current direction is θ + its initial phase, not the initial phase alone. This uses the sine/vertical-projection convention.
The dashed construction starts at the positive-terminal vector tip and adds the negative of the other vector. The resultant starts at the origin. N is the zero vector.
The arc measures a fixed phase difference, not θ. It runs from U to W (120°) in the phase view, or from the first-terminal phase to the selected line voltage (±30°) in the next two views. Both vectors rotate together; their separation stays fixed.
Virtual voltmeter
Both probes are on the same terminal: the measured difference is zero.
Ideal, non-loading measurement of a potential difference. This is a reference-polarity diagram, not a simulated load circuit. N is the common zero-potential reference.
For the balanced source, 1.5Vm ≤ vspan ≤ √3Vm. Two phase potentials tie at each change of the maximizing pair. No diode conduction, load current, or commutation is being calculated.
Keep the conventions straight
Notation and physical meaning
In this lesson, vU, vV, and vW denote instantaneous phase-to-neutral voltages. Vm is their common peak; an “rms” subscript explicitly identifies an RMS value. U, V, W, and N are upright terminal names, not variable indices. The rotating vectors use peak values. The positive phase sequence is U–V–W, corresponding to the original R–S–T names in the source and unchanged technical data keys. The sine function is written sin in all three languages.
The chosen origin is t = 0 at the upward zero crossing of vU, so φU = 0. The cursor is θ = ωt; it is not an adjustable initial phase. Initial phases are defined relative to this same origin and use the sine convention. Te = 1/f; the phase named W is not the period. Changing the origin would be a different operation, which this control does not perform.
Peak, instantaneous, and RMS are different
Teaching addition derived from the sinusoidal model. “ph” means phase-to-neutral and “LL” means line-to-line. For a sine wave, its RMS value is its peak divided by √2. Compare RMS with RMS, or peak with peak—not one of each.
Example: a 230 V phase RMS value gives about 398.372 V line RMS, 325.269 V phase peak, and 563.383 V line peak. “230/400 V” would be a rounded nominal description, not an exact equality used by this model.
Investigate before revealing the explanation
Each investigation has one question to test. Load its starting point, write a prediction, make the proposed change, and compare your observation with your explanation. Opening a solution does not overwrite your notes.
01120° is an angle, not a fixed time
Start at θ = 90°, f = 50 Hz. Which phase reaches its positive maximum next? Predict how the time to that maximum changes if the frequency is doubled.
The test change sets f to 100 Hz, keeping θ = 90°. Compare the 120° time interval, then move the cursor to the next positive phase maximum. The waveform window always spans two periods. As you move the cursor, compare the two angle columns in the table: initial phases stay fixed; instantaneous phases advance.
Check your explanation
V peaks next, at 210°. The angular interval stays at 120°, but Δt = 1/(3f) falls from 6.667 ms to 3.333 ms. Doubling the rate of rotation halves the time for the same angular displacement. Neither moving the cursor nor this frequency change alters φU, φV, or φW. At the same θ, all instantaneous sine arguments are unchanged too.
02Does the line peak occur at the phase peak?
Use Vm = 100 V and the UV measurement. Start at 90°, where U is at its positive peak. Predict vUV there and whether it is already at its own maximum.
Apply the test change to move to 60°. Compare both phase potentials and their difference at the two angles.
Check your explanation
At 90°, the potentials are 100 V and −50 V, so vUV = 150 V. At 60°, they are approximately 86.603 V and −86.603 V, giving the line peak 173.205 V. The largest difference is reached before the U peak; the phase voltages cannot simultaneously be +100 V and −100 V. Here φUV stays at +30°: the peak occurs at θ = 60° because θ + φUV = 90°.
03Reverse the probes, not the source sequence
Start with Vm = 100 V, θ = 0°, and the UV measurement. Predict the signed reading after exchanging the probes. Is the phase change 30°, 120°, or 180°?
The test change selects VU and displays the reversed-polarity line-voltage set. Keep the angle fixed, then examine other cursor positions.
Check your explanation
vUV is approximately +86.603 V and vVU is −86.603 V. Every sample changes sign; the peak and RMS are unchanged. This is a 180° shift of the same pair voltage. The 30° statement compares a line voltage with a phase voltage, so it is a different comparison. The cursor θ did not move. The initial phases of UV and VU differ by 180° modulo 360°.
04Follow all six positive peaks
Use a 1 V phase peak. Starting at 0°, record the line voltage with the positive peak at each successive 60° position. Predict when the pattern will repeat.
The test change moves from 0° to 60°. Continue with two +30° steps at a time. Compare the six line peaks with the three phase peaks.
Check your explanation
The order is WV, UV, UW, VW, VU, WU, then WV again at 360°. The line peaks are √3 V, whereas the phase peaks are 1 V. The original three direct line voltages are 120° apart; including their opposite polarities interleaves three more positive peaks.
05At a boundary, two answers can be correct
Optional extension. Start at 0° with Vm = 100 V and the maximum–minimum envelope visible. Predict which phases will be highest and lowest at 30°. Will one line voltage uniquely define the maximum?
Apply the test change to 30°. Then inspect 29° and 31° with the numerical angle field. Record the highest/lowest phases and the envelope value.
Check your explanation
At 30°, U and W both equal +50 V, while V equals −100 V. Therefore vUV = vWV = 150 V, the lower limit of the envelope. Just before the boundary, WV maximizes the difference; just after it, UV does. The voltages remain continuous. A voltage tie alone does not calculate a real diode commutation.
Check what you can explain
One correct answer per question. Submit your own answer before revealing a solution. First attempts, retries, and solutions viewed before answering are recorded separately. This is self-assessment, not a secure examination. The questions use their own stated conditions, independently of the current controls. Assume a nonzero source amplitude unless a question states otherwise.
The handout includes all four teaching views, investigations, your notes, and quiz solutions. It uses the current source settings and a light theme. Printing solutions does not record a quiz attempt or change your saved history.
Model, limits, and teaching reference
This laboratory calculates the steady periodic voltages of an ideal balanced three-phase sinusoidal source with fixed U–V–W sequence, equal phase peaks, a common frequency, and an ideal neutral reference. Line voltages are calculated by instantaneous subtraction. The source amplitude and frequency are prescribed, not solved from a generator or a network.
Not modeled: unbalance, harmonics, negative-sequence sources, source impedance, loads, phase or neutral currents, power, star/delta load connections, semiconductor states, or commutation transients. The √3 and ±30° relations shown here apply to the stated balanced sinusoidal source and the stated voltage ordering. They must not be applied indiscriminately to other systems. Numerical input limits: phase peak 0 V or 10⁻⁹–600 V; frequency 1–100 Hz; cursor 0–720°. Values smaller than the supported nonzero voltage range are rejected, not rounded silently to zero.
Teaching reference: Recordatorio Red Trifásica. This lesson retains its U–V–W sine convention, terminal ordering, and phase/line relationships. The interactive diagrams are redrawings. No accompanying presentation is needed to use the lesson.
| Concept | Relationships covered |
|---|---|
| Phase voltages | Three 120°-spaced phase voltages, time traces, rotating vectors, and phase equations. |
| Line voltages | Direct line voltages, √3 peak factor, and 30° lead. |
| Reverse polarities | Reversed-polarity line voltages vUW, vWV, and vVU; their 30° lag relative to vU, vW, and vV, respectively. |
| Six line voltages | Two-period phase overlay and the complete six-line-voltage overlay. |
Explicit teaching additions: the virtual meter, derivations and convention notes, RMS conversions, numerical examples, linked cursor, optional maximum–minimum envelope and tie handling, investigations, and self-assessment. These are derived teaching aids, not additional claims attributed to the teaching reference.
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