Transformer phasor diagrams
From no-load through unity, lagging, and leading power factors — the whole steady state in one picture.
Step 1 — Reference axes: V₁ horizontal, E₁ = −V₁
Reference notes
Use Next → on the narrator above to step through six configurations of the phasor diagram, from no-load through unity, lagging, and leading power factors.
What a phasor is
A phasor is the complex-amplitude representation of a sinusoid. Each AC quantity at frequency f — voltage, current, flux — is represented by a vector whose length is the RMS magnitude and whose angle is the phase shift relative to a chosen reference. The whole diagram rotates together at angular velocity ω = 2πf, but only the relative angles matter for steady-state analysis, so we draw the diagram frozen at one instant. The little wave panel in the corner is the time-domain projection of those rotating vectors — watch how the phase shifts in the diagram show up as horizontal shifts of the sinusoids.
Why Φ lags V₁ by 90°
From Faraday's law, V₁ ≈ −E₁ = +N₁·(dΦ/dt). If Φ = Φmax·sin(ωt), then dΦ/dt = ωΦmax·cos(ωt) — a cosine, which leads the sine by 90°. So V₁ leads Φ by 90°, equivalently Φ lags V₁ by 90°.
The no-load current Ie and its two components
With no load on the secondary, the primary still draws a small current Ie. It splits into:
- Im (magnetising component) — in phase with Φ, sets up the alternating flux. Carries reactive power only.
- Ic (core-loss component) — in phase with V₁, supplies the hysteresis and eddy-current losses in the iron. Carries real power.
Im dominates: typically Im is 8–10× larger than Ic, so Ie sits very close to the Φ direction (about 80–85° behind V₁). When the secondary is open and I₂ = 0, the primary current I₁ is just this small magnetising current Ie — nothing more.
The load component I₂′
When a load is connected on the secondary, the secondary current I₂ flows. The MMF balance equation N₁·i₁′ = N₂·i₂ means the primary draws an additional load component I₂′ with magnitude (N₂/N₁)·|I₂| and in the same phase as I₂ (with the standard "dotted-end-in" current convention). The total primary current is the vector sum:
If the secondary is opened (I₂ = 0), I₂′ collapses to zero and I₁ falls back to Ie.
Unity, lagging, and leading load — what changes?
- Unity PF (resistive load): I₂ in phase with V₂, so I₂′ along V₁ direction. I₁ lags V₁ by only a small angle — basically Ie dragging I₁ slightly downward.
- Lagging PF (inductive load): I₂ lags V₂ by an angle θ. I₂′ rotates downward by the same angle. The angle by which I₁ lags V₁ increases, because both Ie and I₂′ now point downward.
- Leading PF (capacitive load): I₂ leads V₂. I₂′ rotates upward. I₁ may end up slightly leading V₁ — but only slightly, because the magnetising current Ie always drags it down. The lead angle of I₁ is always less than the lead angle of I₂.
Keyboard shortcuts
- → next step · ← previous step
- R replay narration · M mute / unmute