Calculation of Differential‑Mode Inductance for Toroidal Common‑Mode Choke
Before we start, let’s introduce integrated common‑mode & differential‑mode inductors. This is a magnetically integrated component combining the functions of a common‑mode inductor and a differential‑mode inductor within a single magnetic part. One component can suppress the two primary electromagnetic interferences in circuits: differential‑mode noise and common‑mode noise. Compared with discrete conventional solutions, the integrated inductor occupies the footprint of merely one part, substantially saving PCB space.
We have encountered numerous real‑world cases in practical engineering. For a toroidal common‑mode choke, customers specify a minimum differential‑mode inductance requirement. Many designers are confused. They know the design rules for common‑mode inductance, yet they cannot predict whether differential‑mode inductance will satisfy customer specifications. Consequently, prototypes have to be manufactured and measured. If results are unsatisfactory, magnetic materials and schemes will be revised repeatedly. This extends development cycles and brings heavy workload. Today we will resolve this challenge.
Here is the formula, along with definitions for each parameter:
Le: magnetic path length of toroidal core, in m
d: diameter of a circle with cross‑sectional area equivalent to core Ae, in m
N: turns of a single winding
μ₀: permeability of free space, μ₀ = 4π×10⁻⁷ H/m
θ: subtended arc angle of winding wrapped around the toroid, in rad (radians)
The above calculated value applies to the differential‑mode inductance when a magnetic sheet is inserted in the middle of the toroidal core. Standard separators are epoxy plates. To construct an integrated common‑mode & differential‑mode choke and boost differential‑mode inductance, the epoxy separator will be substituted by a magnetic sheet. Manganese‑zinc power‑grade ferrites (grade 44, 95) or low‑permeability metal powder cores are typical options. High‑permeability materials or nanocrystalline ribbons are rarely adopted for the inserted sheets.
Divide the computed inductance value by two, and you can obtain differential‑mode inductance without any inserted magnetic material.
Calculation Example
Design specifications:
Toroid T25×15×10C, 7 000‑perm manganese‑zinc high‑permeability ferrite. Ae = 4.893×10⁻⁵ m².
Minimum common‑mode inductance: 250 μH.
Minimum differential‑mode inductance: 10 μH.
Magnetic insert is allowed. Rated current: 2 A. Wire diameter: 0.7 mm.
Step 1. Complete common‑mode design. The required number of turns is 24.
Step 2. Simulate winding layout in CAD according to wire diameter, and acquire the wrapped arc angle for one winding of the two.
Step 3. Convert mechanical angle into radians.
Radian = Angle × π ÷ 180 = 160 × π ÷ 180 = 2.79 rad
Step 4. Calculate equivalent diameter for cross‑section Ae.
d = SQRT(4.893×10⁻⁵ ÷ π) × 2 = 0.00789 m
Step 5. Substitute parameters into the formula. The differential‑mode inductance with inserted magnetic sheet equals 49.3 μH.
Differential‑mode inductance without magnetic sheet = 49.3 ÷ 2 = 24.6 μH.
The value exceeds the 10 μH minimum requirement. Sufficient design margin is secured, and mass production is feasible.
What if the calculated inductance fails specifications? Adjust parameters based on the formula.
The most straightforward approach is adding turns under available winding space, while keeping the wrapped arc unchanged. Another alternative is selecting a larger toroidal core with greater Ae. In this instance, there is barely any spare space to expand the wrapped arc of windings, so increasing turn‑count is the primary option.
Differential‑Mode Inductance Measurement Procedure
Many engineers misunderstand the test setup. It is not identical to leakage‑inductance measurement of transformers.
We also include common‑mode impedance measurement for comparison.
Differential‑mode inductance measurement (globally‑recognized standard):
Short the dotted terminals of one winding pair. Connect the probe clips of measuring equipment to the dotted terminals of the second winding pair.
Transformer leakage‑inductance measurement (short one winding, test the other) is not accepted by international standards and reputable domestic manufacturers for common‑mode chokes.
Common‑mode impedance measurement:
Short dotted terminals on both windings. Connect measuring probes to the two shorted groups.
A large number of technicians mistakenly test the impedance of a single winding. The definition of common‑mode noise requires both windings to be measured simultaneously. A single‑winding impedance test cannot characterize common‑mode performance.
Notes on the formula
This equation is applicable to circular toroidal cores, including nanocrystalline and high‑permeability ferrite rings. The calculation barely depends on core permeability. It is determined mainly by geometric dimensions.
Prediction error ranges approximately ±5% ~ ±20%. Deviation originates primarily from the difference between CAD‑simulated winding arc and real‑world winding coverage.
The formula delivers reference values for preliminary scheme evaluation. Ample calculated margin indicates a qualified design.
Rectangular cores and racetrack‑shaped cores follow separate calculation formulas. We will publish guidelines for rectangular cores in the next chapter. Racetrack cores have not been fully investigated. They are widely used within three‑phase common‑mode filters, where integrated common‑mode plus differential‑mode structures are seldom required.
Above is the complete tutorial for differential‑mode inductance calculation on toroidal common‑mode inductors.