﻿ Helmholtz Coil Calculator

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# Advance Helmholtz Coil Design Calculator

Figure 1. Cross-section view of one coil showing diameter, coil length and width.

This calculator uses advance techniques to calculate magnetic field Helmholtz coils. It is the most comprehensive and accurate calculator available. In addition to magnetic field, it also calculates AC resistance, inductance, power dissipation, resonance capacitance, and much more. It support both solid and Litz wire.

Coil Inner Diameter: Coil copper inner diameter, excluding plastic bobbin.  Note the copper inner diameter is not same as the bobbin inner diameter. Copper inner diameter must be larger than bobbin inner diameter. See Figure 1 for details.

Coil Length: Enter the length of the solenoid or coil. See Figure 1 for details.

Number of Turns: Enter the number of turns in the coil or solenoid.

Coil Current: Enter the coil current. For DC-related calculations, this is the DC current. For AC-related calculations, this is the current amplitude which is defined as zero-to-peak. For example, 5A means the current is a sinewave from -5A to +5A.

Plastic-to-Plastic Separation: This is the distance between two coils, include the plastic bobbin thickness

Window Calculating Field Variation: This is the distance along the z-axis you want to calculate the magnetic variation. For example, the coil separation is 100mm and you want to know how uniform the magnetic field is over a distance of 50mm (or +/-25mm from the center). Enter 50. It will calculate the magnetic field from 0 to 25mm. 0mm is the middle between two coils. The magnetic field symmetrical.

Wire Outer Diameter with Insulation: Enter the overall wire diameter, include insulation.

Strand Diameter: For solid wire, enter copper wire diameter WITHOUT coated insulation. For Litz wire enter the single strand diameter WITHOUT coated insulation. This is the bare copper diameter. This diameter is used to calculate the resistance more accurately.

Number of Strands: For solid wire enter 1. For Litz wire enter the number of strands.

Frequency: Enter the operation frequency (sinewave). For DC, just leave the default 1kHz and use the DC parameters calculated below.

Plastic Thickness: Bobbin side thickness. You may enter 0 if you want to exclude bobbin in the calculation.

Capacitor Dissipation Factor: Enter resonance capacitor dissipation factor or loss tangent. For example, Polypropylene film capacitor have a dissipation factor of 5x10^-4.

Mutual Inductance Factor: Enter the mutual coupling factor for the two coils. Most Helmholtz coil coupling factor is 11%. Enter 1.11. The total inductance is calculated by the sum of each coil coil inductance and then multiply by 1.11.

Winding Porosity-Factor: This factor is used for calculating the proximity effect for AC resistance. Typical value is 0.85.

Winding Compact-Factor: Enter the coil winding density compact factor. If the coil is wound such that the distance between two copper layers is equal to the diameter (Figure 3), the compact factor is equal 1.0.  However, practical wounding is shown in Figure 4. This is the most tightly wound with no space wasted. The ideal compact factor is 0.866. A practical compact factor range is 0.88 to 1.0. Typical compact factor is 0.9.

Core Relative Permeability: Enter the core relative permeability constant, k. Enter 1 for air core.

Outputs:

Magnetic Field at Center: The calculated magnetic field at the center between two coils.

Minimum DC Driver Voltage Required: This is the voltage needed for DC magnetic field.

Coil DC Power Dissipation: This is how much power is dissipated for DC magnetic field.

Coil AC Resistance: This is the resistance at the frequency entered above. AC resistance is dominated by proximity effect and increase exponentially with frequency and winding layer. Reduce the number of winding layers, or reduce the wire diameter, or use Litz wire to reduce the resistance.

Coil AC-to-DC Resistance Ratio: Information only.  This is the ratio of AC resistance to DC resistance.

Minimum Coil+Cap AC Driver Voltage Required: This is the minimum voltage amplitude need to drive the coil using resonance technique. For example, 9.8V means you need a coil driver that can output sinewave from -9.8V to + 9.8V (or 19.6Vpp).

Minimum Coil Only AC Driver Voltage Required : Information only. Ignore it.

Coil AC Power Dissipation: This is the AC coil power dissipation at the current entered above.

Inductance: Coil inductance in micro-Henry (uH). Tip: For high frequency coils, keep the inductance as small as possible. It is better to reduce the number of turns, but increase the current by the same proposition.

Resonance Capacitor Capacitance: This is the capacitance needed to form an LC resonant tank. The coil impedance is reduced or canceled using this series resonant capacitor (Figure 4). Click here for more information about series resonant magnetic coil. It is calculated using the coil inductance and the user input frequency. Tip: Generally use Polypropylene film capacitor, especially at higher frequencies. Ceramic capacitor is acceptable if the ESR is low enough and can handle the power dissipation. Class-1 dielectric (i.e C0G/NPO) has lower loss than class-2 (i.e X7R).

Resonance Capacitor Voltage Rating: For resonance technique (Figure 4), this is the minimum capacitance voltage rating. This is the sinewave zero-to-peak amplitude voltage across the capacitor. This is also the voltage across the coil. Tip: capacitor voltage rating is degrades at higher frequencies, Figure 5. When design high frequency coil, this is the highest priority. Maximum practical voltage is about 15kV.

Capacitor ESR: Capacitor resistance at the frequency entered above. Tip: Use Polypropylene film capacitor which has the lowest loss tangent. NPO/C0G (Class-1 dielectric) ceramic capacitor maybe okay as long as the ESR is low enough and the capacitor can handle power dissipation. Class-2 dielectric (X7R) ceramic capacitor may be acceptable for low frequency or low current or both.

Capacitor Power Dissipation: This is how much power (heat) will be dissipated by the resonance capacitor. Make sure the capacitor can handle the power.

Penetration Ratio: Information only. Proximity effect penetration ratio.  Generally keep the ratio less than 0.5.

Skin Depth: Information only. Ignore.

Skin-Effect (Litz) Resistance: Information only. Ignore.

Total Impedance: This the magnitude of the coil impedance. It includes both the reactive and resistive components.

Number of Layers: Total number of layers of winding.

Coil Layer Height: Total number of layers adding up to the height or thickness.

Average Inter-Layer Max Voltage: This is the voltage between two adjacent copper layer. Make sure the voltage is less than the dielectric breakdown of the wire coating. Typical breakdown is 500-1000V at DC. Breakdown derates at higher frequency.

 Gauge Diameter 10 2.67716 11 2.39268 12 2.13868 13 1.91516 14 1.7145 15 1.53162 16 1.36652 18 1.09474 20 0.87884
 Gauge Diameter 10 2.58826 11 2.30378 12 2.05232 13 1.8288 14 1.628 15 1.45034 16 1.29032 18 1.02362 20 0.8128

Table 1. Wire gauge diameter with insulation in mm.

Table 2. Gauge diameter without insulation (bare copper) in mm.

Figure 2. Coil winding spacing factor of one.

Figure 3. Ideal compact factor. The distance between two layers is 0.866 of diameter.

Figure 4. High-current Waveform Amplifier drives coil and resonance capacitor.

Figure 5. Dielectric strength reduction over frequency (kHz). Source: UL.

### Coil Input Parameters

Coil Inner Diameter (mm):
Coil Lenghth (mm):
Number of Turns:
Coil Current (A):
Plastic-to-Plastic Separation (mm):
Window Calculating Field Variation (mm):
Litz Wire Outer Diameter with Insulation(mm):
Strand Diameter (without Insulation):(mm):
Number of Strands:
Frequency (kHz):
Copper Temperature (C):
Plastic Thickness (mm):
Capacitor Dissipation Factor:
Mutual Inductance Factor:
Winding Porosity-Factor:
Winding Compac-Factor:
Core Relative Permeability, k:

Magnetic Field at Center(mT):
Coil Outer Diameter(mm):
Copper Wire Lenghth, One Coil (ft):
DC Calculated Output Parameters
DC Resistance, Two Coils(Ohm):
Minimum DC Driver Voltge Requried, Two Coils (V):
Coil DC Power Dissipation,One Coil (W):
AC Calculated Output Parameters
Coil AC Resistance, Two Coils (Ohm):
Coil AC-to-DC Resistance Ratio:
Minimum Coil+Cap AC Driver Voltge Requried,Two Coils (V):
Minimum Coil Only AC Driver Voltge Requried, Two Coils (V):
Coil AC Power Dissipation, One Coil (Wrms):

Inductance, Two Coils (uH):
Resonance Capacitor Capacitance, Two Coils (nF):
Resonance Capacitor Voltage Rating, Two Coils (kV):
Capacitor ESR (Ohm):
Capacitor Power Dissipation (Wrms):

Uniformity Max(%):
Uniformity Min%):
Penetration Ratio:
Skin Depth (mm):
Skin-Effect (Litz) Resistance (Ohm):
Total Impedance, Two Coils (Ohm):
Number of Layers, One Coil:
Coil Layer Height(mm):
Average Inter-Layer Max Voltage:
Copper Wire Lenghth, One Coil (meter):

## High-Voltage Safety

- Caution: High voltage. Both the capacitor and the coil are high voltage at resonance.

- High voltage and high frequency can couple the electric field to your fingers and hand even without physically touching the coil or the capacitor.

- It is strongly recommended you use a plastic safety cage to enclose the capacitor and coil. At moderate high voltage and frequency (5kV and 10kHz), human hands must be least 6 inches away from any part of the coil and capacitor. At higher voltage and frequency (10kV and 100kHz), at least 12 inches.

- To start with, you may use a plastic container, flip it over and cover the capacitor and coil.

- It is recommended that you put the coil and capacitor on a plastic tray or platform and raise it above the table by at least 6 inches. This will prevent the electric field from coupling to the table frame which is made of metal.

- Use high-voltage wire for the high-voltage portion of the connection, namely the wire connecting the capacitor and the coil.

- Keep the ground return wire (or any low-voltage wires) at least 1 inch from any part of the coil and capacitor. If the low-voltage wire and the high-voltage wire are too close, dielectric breakdown may occur. For example, we used 40kV wire for a 15kV and 250kHz capacitor. When the ground return wire got close to the high-voltage wire, the strong electric field burned the 40kV wire insulation. At 250kHz, the dielectric strength have severely degraded to well less than 15kV.

- Avoid sharp or pointed metal such as solder joints or tiny wire sticking out. Sharp point will amplify the electric field and may cause corona or arc. Use a file to polish any sharp metal and make it round. You may use “Super Corona Dope” from MG Chemicals # 4226 to coat any high-voltage metal to avoid corona or arc.

## Coil Design Tips

Generally speaking, first priority is to reduce the resonance voltage. That means reduce the inductance. Keep the coil diameter as small possible. The inductance is proportional to the number of turns square. Reduce the number of turns by half will reduce the inductance by a factor of 4. Then increase the current by 2. The magnetic field remains about the same, but the resonance voltage is reduced by 2x.

AC resistance is dominated by proximity effect. Copper wire diameter either too large or too small will increase the resistance. Using the calculator to experiment with the different wire diameter for the lowest resistance. If the resistance is still too high, either use Litz wire or reduce the number of winding layers.

Pay attention to the “inter-layer max voltage”. This voltage must be within the dielectric breakdown strength of the copper wire enamel. Furthermore, the dielectric strength is degraded by high frequency AC electric field. The inter-layer max voltage should in the range of 500V.

Often time and at high frequency and high current, the coil design can only be one layer. More than one layer is not possible due to too high voltage between two layers. You also need to look at the voltage between two loops. This voltage should not be higher than 500V. The voltage between loops is the resonance voltage divided by the number of loops.

You may use “Super Corona Dope” from MG Chemicals # 4226 to enhance the dielectric strength.