Number of turns

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

The number of turns (German: Windungszahl) describes how many turns an inductor or winding has. It is one of the most important design parameters for inductive components and largely determines their electrical and magnetic properties.

1 Quantity symbol and unit

The quantity symbol for the number of turns is $N$.
The number of turns is a dimensionless quantity with the unit ‘1’ (one):

$$[N]=1$$

2 Practical and worked examples

2.1 Inductance of an inductor

The inductance of an inductor depends on the square of the number of turns:

$$L={\mu }_0\cdot {\mu }_r\cdot \frac{N^2\cdot A}{l}$$

Example: doubling the number of turns from $N_1=200$ to $N_2=400$ with all other parameters unchanged:

$$\frac{L_2}{L_1}={\left(\frac{N_2}{N_1}\right)}^2={\left(\frac{400}{200}\right)}^2=\underline{4}$$

The inductance quadruples!

2.2 Magnetomotive force

The magnetomotive force is proportional to the number of turns:

$$\Theta =N\cdot I$$

Example: inductor with $N=500$ turns at $I=2.0\mathrm{A}$:

$$\Theta =500\cdot 2.0\mathrm{A}=\underline{1000\mathrm{A}}$$

2.3 Transformer turns ratio

The voltage ratio of a transformer is determined by the turns ratio:

$$\frac{U_2}{U_1}=\frac{N_2}{N_1}$$

Example: primary side $N_1=2300$ turns for $U_1=230\mathrm{V}$, secondary side for $U_2=12\mathrm{V}$:

$$N_2=N_1\cdot \frac{U_2}{U_1}=2300\cdot \frac{12\mathrm{V}}{230\mathrm{V}}=\underline{120\text{ turns}}$$

2.4 Inductance with and without an iron core

Air-core inductor: $N=100$, $A=2.0\cdot 10^{-4}{\mathrm{m}}^2$, $l=0.05\mathrm{m}$, ${\mu }_r=1$

$$L_{air}={\mu }_0\cdot \frac{N^2\cdot A}{l}=4\pi \cdot 10^{-7}H/m\cdot \frac{100^2\cdot 2.0\cdot 10^{-4}{\mathrm{m}}^2}{0.05\mathrm{m}}=\underline{50.3\mu \mathrm{H}}$$

With an iron core (${\mu }_r=1000$):

$$L_{iron}={\mu }_r\cdot L_{air}=1000\cdot 50.3\mu \mathrm{H}=\underline{50.3\mathrm{mH}}$$

3 Typical orders of magnitude

• Small RF inductors: $N=5$ to $50$ turns
• Interference suppression inductors: $N=20$ to $200$ turns
• Mains filter inductors: $N=100$ to $1000$ turns
• Transformers (extra-low voltage): $N=50$ to $500$ per winding
• Transformers (mains voltage): $N=1000$ to $5000$ per winding
• Relays/contactors: $N=500$ to $3000$ turns
• Electric motors: $N=10$ to $100$ per slot
• High-voltage transformers: $N>10000$ turns

4 Example phrasings

• ‘The primary winding has $N_1=2300$ turns.’
• ‘If the number of turns is doubled, the inductance quadruples.’
• ‘The turns ratio is $N_1:N_2=10:1$.’
• ‘At least 800 turns are required for the desired magnetomotive force.’
• ‘The inductor is wound with 0.5 mm enamelled copper wire in 3 layers of 120 turns each.’

5 Related quantities

• Inductance:

$$L\propto N^2$$

• Magnetomotive force:

$$\Theta =N\cdot I$$

• Transformer ratio:

$$\frac{U_2}{U_1}=\frac{N_2}{N_1}$$

• Magnetic flux and flux linkage:

$$\Lambda =N\cdot \Phi$$

• Induced voltage:

$$u(t)=-N\cdot \frac{d\Phi }{dt}$$

6 Related components

• Inductor: the number of turns determines the inductance and the magnetic properties
• Transformer: the turns ratio determines the voltage transformation
• Relay and contactor: the number of turns influences the pick-up current and the pull-in force
• Electric motor: the number of turns per pole determines the torque and the rotational frequency