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Qwestrum Engineering360 · Electrical & Electronics · Electrical Machines – II

Synchronous Machine

A synchronous machine runs exactly at N_s = 120f/P and its real power output follows P = (E_fV/X_s)sinδ; changing field excitation controls reactive power, giving the characteristic V-curves.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

  • Salient pole: direct and quadrature reactances X_d, X_q
  • V-curves: minimum armature current at unity pf
  • Synchronising: check phase, frequency, voltage before closing breaker

Topic details

Introduction

Synchronous generators (alternators) supply almost all grid power, and synchronous motors provide power-factor correction. The power-angle equation P = (E_fV/X_s)sinδ shows that steady-state power is limited to E_fV/X_s at δ = 90°.

Scope in B.Tech and GATE syllabus

Exam problems compute E_f from V, armature current and synchronous reactance for a given power factor, then find voltage regulation. The phasor addition E_f = V + I_a(R_a + jX_s) must respect the pf angle sign (lagging vs leading).

Key relations & formulas

Ef=4.44fNΦE_{f} = 4.44 f N \Phi
(no-load emf per phase)
V=Ef−Ia(Ra+jXs)V = E_{f} - I_{a} (R_{a} + j X_{s})
(armature reaction included)
P=(EfVXs)sinδP = (E_{f} \frac{V}{X_{s}}) sin \delta
(power angle δ)
Ns=120fPN_{s} = 120 \frac{f}{P}
(rpm)

Notation and sign conventions

Relation 1 —
Ef=4.44fNΦE_{f} = 4.44 f N \Phi
Ef=4.44fNΦE_{f} = 4.44 f N \Phi
(no-load emf per phase)
Write this relation with symbols exactly as in Electrical Machines — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
V=Ef−IaV = E_{f} - I_{a}
V=Ef−Ia(Ra+jXs)V = E_{f} - I_{a} (R_{a} + j X_{s})
(armature reaction included)
Write this relation with symbols exactly as in Electrical Machines — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
P=P =
P=(EfVXs)sinδP = (E_{f} \frac{V}{X_{s}}) sin \delta
(power angle δ)
Write this relation with symbols exactly as in Electrical Machines — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
Ns=120fPN_{s} = 120 \frac{f}{P}
Ns=120fPN_{s} = 120 \frac{f}{P}
(rpm)
Write this relation with symbols exactly as in Electrical Machines — Nagrath & Kothari before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

For a cylindrical-rotor machine at lagging pf, E_f = √[(V cosφ + I_aR_a)² + (V sinφ + I_aX_s)²]. For leading pf, the X_s term subtracts. Regulation = (E_f − V)/V, and can be negative at leading pf.

Governing relations in practice

Over-excitation makes the machine deliver reactive power (behaves capacitive as a motor); under-excitation absorbs it. Plotting armature current versus field current at constant load gives the V-curve, minimum at unity pf.

Design and analysis considerations

Before paralleling to the grid, three conditions must match: voltage magnitude, frequency, and phase sequence/angle — checked with a synchroscope or lamp method.

Assumptions and validity limits

State assumptions explicitly before using any relation for synchronous machine — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Electrical Machines II viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Electrical Machines II papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to synchronous machine.
4. Use equation 1:
Ef=4.44fNΦE_{f} = 4.44 f N \Phi
.
5. Use equation 2:
V=Ef−IaV = E_{f} - I_{a}
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Synchronous Machine appears in industrial motors and generators. In Indian electrical curricula this topic is tested because it connects theory to induction and synchronous machines.
GATE and semester exams often combine synchronous machine with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use synchronous machine?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Adding I_aX_s in phase with V instead of accounting for the pf angle
• Using the wrong sign of the reactance term for leading vs lagging pf
• Confusing salient-pole (two-reactance) with cylindrical-rotor analysis
• Forgetting synchronous speed is fixed by f and P (no slip)

Quick revision checklist

Before attempting synchronous machine problems, confirm you can:
1. Salient pole: direct and quadrature reactances X_d, X_q
2. V-curves: minimum armature current at unity pf
3. Synchronising: check phase, frequency, voltage before closing breaker
Revise the solved examples in Electrical Machines — Nagrath & Kothari and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

EMF and regulation of an alternator

Problem

A 3-phase star alternator has per-phase terminal voltage 231 V and delivers 15 A at 0.8 pf lagging. R_a = 0.5 Ω, X_s = 5 Ω. Find E_f per phase and the voltage regulation.

Solution

cosφ = 0.8, sinφ = 0.6.
E_f = √[(V cosφ + I_aR_a)² + (V sinφ + I_aX_s)²]
= √[(231×0.8 + 15×0.5)² + (231×0.6 + 15×5)²]
= √[(184.8 + 7.5)² + (138.6 + 75)²] = √[192.3² + 213.6²]
= √(36979 + 45625) = √82604 = 287.4 V.
Regulation = (287.4 − 231)/231 = 24.4%.

Conceptual check — Synchronous Machine

Problem

In a Electrical Machines II semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of synchronous machine." What should a complete answer include?

Exams & GATE

Nagrath & Kothari Ch. 8 — phasor diagram for lagging/leading pf.

📖 Standard books (India)

  • Electrical Machines — Nagrath & Kothari

    Read: Syllabus unit

    Transformers, DC machines, and induction motors

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