Toroidal Power Inductors

2.3 Power Inductors and Calculations

Switched-mode power supplies are becoming ever more widespread. The semiconductor manufacturers have made their contribution, offering a wide range these integrated circuits with simplified circuit design. Care must be taken in the selection of the appropriate storage choke to fully utilize the advantages of switching regulators.

The selection of cores and windings of storage chokes are optimized for use in switching converters and DC-DC converters.

Leading manufacturers of storage chokes following recommendations from various switching converter IC manufacturers, e.g. National Semiconductor, Linear Technology, STMicroelectronics, Texas Instruments, Exar, Diodes, MPS, ON Semiconductor, Semtech, Maxim and a special customized solutions can be found in their reference design guidelines.

The energy stored in the core in this process is:

To enable high energy storage and to minimize the resulting core losses, the toroidal core volume is divided into many electrically isolated regions. The iron powder used in our storage chokes therefore has three-dimensional, uniformly distributed, microscopic air gaps, which prevent eddy-current losses.

The disadvantage of reduced permeability is balanced by greater maximum energy storage and lower losses. Furthermore, these cores are extremely well suited for use in applications with high DC premagnetization.

Data book specifications

Open-circuit inductance L₀:

If the inductor is operated without DC premagnetization or with only a small AC current, the open-circuit inductance L₀ results.This value may be measured with sufficiently sensitive inductance measuring equipment for small AC voltages e.g. 0.1–0.5 V and a fixed measuring frequency between 1 kHz and 100 kHz, depending on the inductance value.

Inductance rating L_N:

In addition to the small AC voltage amplitude, the specified DC current is superimposed and the resulting inductance measured.

Current rating I_N:

The DC current, for which the inductance and wire thickness are specified and whose specifications are optimized. As shown in the graph below, inductance only saturates with a much larger current.

Fig. 2.43: Inductance with DC premagnetization; IN = 1 A; LN = 100 µH

DC resistance DCR:

The windings resistance value is measured with an ohmmeter at an ambient temperature of +25 °C.The test current for resistance measurement is a small DC current, which does not lead to a significant temperature increase in the wire. As values in the milliohm range are measured here, a 4-wire measurement must be made to minimize measurement errors.

Magnetic field energy E:

The energy, for which the core data and windings of the coil is optimized. This is specified in microjoules.The following simple and practically proven formulae can be used for dimensioning a storage choke. A brief extract from the extensive core material program and the following table should provide an overview of the choke dimensioning process. Depending on the application, further specifications from the core material data spectrum may be necessary.

Iron core material data:

The following table shows an overview of the most commonly used materials and their applications.

Tab. 2.22: Materials and their applications (source: Würth Elektronik)

Operating temperature:

The operating temperature of the iron powder core may be from –55 °C to +125 °C. Prolonged core operation above +75 °C however results in increased losses.

Insulation voltage:

The protective coating of the toroidal core uniquely identifies the core material and serves to protect against environmental effects and provides electrical isolation from the windings. Epoxy resin coatings are used and an insulation dielectric strength of 500 VDC is achieved as standard. Higher insulation voltages can also be offered.

A_L value: For every size of core an A_L value is specified to simply calculate the winding turns for the required choke; the tolerance is ±10%.
The standard means of measuring the A_L value is at B = 1 mT and f = 10 kHz.

Storage choke calculation:

The following demonstrates how a storage choke can be calculated for a switching converter application:
Example: switching converter (step-down controller – storage choke)

Requirements:

Inductance rating L_N = 100 µH Current rating (DC) I_N = 1 A Peak current through the inductance I_max = 1.5 A Ripple current = 20% of I_max = 0.3 A (see Chapter III/Applications) Switching frequency f = 52 kHz

A maximum AC flux density B_AC = 0.05 T is recommended for iron powder cores (to ensure low core losses). Also the inductance should be selected so the ripple current does not exceed 20%–30% of the maximum current.

Step 1: Choice of the core material and the necessary core volume (V) As the switching frequency is just 50 kHz, we firstly select the material 3W7538 with µ_r = 75.

Selected core:

3W7538, as switching frequency < 70 kHz; core no. 13 d_a = 12.7 mm; d_i = 7.7 mm; h = 4.83 mm

Magnetic data:

l = 3.19 cm; A = 0.112 cm2; V = 0.358 cm3 A_L value: 33 nH/N²

Step 2: Required winding turns

L in nH

A_L value in nH/N²

The final number of winding turns must be increased as a result of current dependent permeability. The correction factor for the A_L value is determined from the “effective permeability against DC premagnetization” graph (see Figure 2.44).

At H = 1724 A/m on the graph in Figure 2.44

→ Effective permeability with DC premagnetization = 80% of the initial permeability

To be certain that the full inductance rating of 100 µH exists with a DC current of 1 A, the final number of winding turns is calculated as:

Step 3: Determination of the DC resistance

The wire diameter can be ascertained from the relevant wire tables for the required current of 1 A, e.g. AWG 22 (d = 0.6 mm). This limits the self-heating of the wire to less than +10 °C.

The DC resistance of the windings is given by:

Step 4: Check for max. AC field flux density

with:

• Inductance rating L in H
• Ripple current ΔI in A
• Core cross-sectional area A in cm²
• Winding turns N
• Peak voltage of the choke U_s in V (during “t”)
• Duration of peak voltage t in s

Step 5: Calculation of core losses
The losses in the core material may be calculated from the following formula:

with

• Frequency f in Hz
• AC field flux density B in mT
• Core losses P_C in mW/cm³

For our examples this leads to:

The total core losses of the selected core are:

The losses in the windings equal:

The total losses of the storage choke are low at around 370 mW and the choke calculated is well suited for the application.

Source: EPCI ABC of CLR article