AC Generator Waveforms

Overview

In an AC generator, a rotating coil in a magnetic field produces:

• a sinusoidal magnetic flux linkage
• a sinusoidal induced emf

The emf arises because the flux linkage changes with time.

This page focuses on graph interpretation, phase relationships, and quick exam reasoning.

Main topic: Alternating Current Generators

Foundation topic: Electromagnetic Induction

Definition

AC-generator waveforms are the time-varying graphs of flux linkage and induced emf produced by a rotating coil.

Why It Matters

Waveform questions are among the most common places students lose marks because:

• flux linkage and emf are not the same graph
• maximum value is often confused with maximum gradient
• quarter-turn positions are easy to misread

Key Representations

Core Equations

Flux Linkage

$$N\Phi =NBA\cos (\omega t)$$

Induced emf

$$\mathcal{E}=NBA\omega \sin (\omega t)$$

Hence:

• both are sinusoidal
• emf is the time derivative of flux linkage
• the two graphs are phase shifted

Flux Linkage vs Time Graph

Flux linkage varies between:

$$+NBA\text{and}-NBA$$

Key Positions

At $t=0$

If the coil starts with maximum flux linkage:

$$N\Phi =+NBA$$

At Quarter Period

$$t=\frac{T}{4}$$

Then:

$$N\Phi =0$$

At Half Period

$$t=\frac{T}{2}$$

Then:

$$N\Phi =-NBA$$

At Full Period

$$t=T$$

Returns to the starting value.

emf vs Time Graph

Induced emf varies between:

$$+{\mathcal{E}}_0\text{and}-{\mathcal{E}}_0$$

where:

$${\mathcal{E}}_0=NBA\omega$$

Key Positions

At $t=0$

Flux linkage is maximum, so gradient is zero:

$$\mathcal{E}=0$$

At Quarter Period

Flux linkage crosses zero most steeply:

$$\mathcal{E}=+{\mathcal{E}}_0$$

At Half Period

Flux linkage is minimum, so gradient is zero:

$$\mathcal{E}=0$$

At Three-Quarter Period

$$\mathcal{E}=-{\mathcal{E}}_0$$

At Full Period

$$\mathcal{E}=0$$

90° Phase Relationship

Since:

$$\mathcal{E}=-\frac{d(N\Phi )}{dt}$$

emf reaches maximum one quarter cycle after flux linkage maximum.

So:

• flux linkage leads emf by $90^{\circ }$ with the chosen equations
• equivalently, emf is quarter-cycle shifted relative to flux linkage

In exam questions, focus on graph shape and zero/max positions.

Quarter-Turn Reasoning

Each quarter turn of the coil changes the orientation significantly.

0° Position

Coil face perpendicular to the field.

• maximum flux linkage
• zero emf

90° Position

Coil plane parallel to the field.

• zero flux linkage
• maximum emf

180° Position

• maximum negative flux linkage
• zero emf

270° Position

• zero flux linkage
• maximum negative emf

360° Position

Returns to the start.

Why emf Is Zero at Maximum Flux

Many students confuse this.

At maximum flux linkage, the graph has zero slope.

Thus:

$$\frac{d(N\Phi )}{dt}=0\Rightarrow \mathcal{E}=0$$

A large value does not mean a large rate of change.

Sign Conventions

Positive or negative emf depends on:

• chosen terminal labels
• chosen current direction
• chosen starting orientation

Graph shape matters more than arbitrary sign.

If the exam gives orientation, use the stated convention.

Quick Graph-Sketch Method

Step 1: Draw Flux Linkage First

Use a cosine shape starting at maximum if stated.

Step 2: Differentiate Mentally

• maximum value gives zero emf
• zero crossing with steepest slope gives maximum emf magnitude

Step 3: Apply Sign

Check slope:

• decreasing flux linkage gives one sign
• increasing flux linkage gives the opposite sign

Worked Graph Examples

Example 1

Flux linkage starts at maximum positive value.

Then the emf graph starts at:

$$0$$

and rises to positive maximum at $T/4$.

Example 2

Flux linkage graph crosses zero upward.

Then the gradient is positive, so emf has the corresponding sign based on the chosen equation.

Check the formula given.

Example 3

Rotational speed doubles.

Then:

• frequency doubles
• period halves
• peak emf doubles, since ${\mathcal{E}}_0=NBA\omega$

Common Exam Traps

• maximum flux mistaken as maximum emf
• forgetting phase shift
• mixing period and quarter period
• getting sign wrong from slope
• changing amplitude when only frequency changes

See Alternating Current Generators Common Exam Traps.

Summary Table

Coil Position	Flux Linkage	emf
0°	maximum +	0
90°	0	maximum +
180°	maximum -	0
270°	0	maximum -
360°	maximum +	0

Summary

For AC generators:

• flux linkage follows cosine-type variation
• emf follows sine-type variation
• emf depends on rate of change of flux linkage
• quarter-turn reasoning quickly predicts graph values

Links

• Alternating Current Generators
• Alternating Current Generators Common Exam Traps
• Electromagnetic Induction
• Alternating Current
• RMS and AC Power