Force and Torque Calculator using Co-energy

An engineering tool for the calculation of force and torque using the concept of co-energy in magnetically linear electromechanical systems.

Linear Force Calculator

Rotational Torque Calculator

What is the Calculation of Force and Torque Using Co-energy?

The calculation of force and torque using the concept of co-energy is a fundamental method in the analysis of electromechanical systems like motors, actuators, and solenoids. Co-energy is a non-physical quantity, expressed in Joules, that serves as a powerful mathematical tool. While field energy is the integral of current with respect to flux linkage, co-energy is the integral of flux linkage with respect to current. In magnetically linear systems (where inductance is constant with current), energy and co-energy are numerically equal.

The primary advantage of using co-energy is that it allows for the calculation of force or torque by taking the partial derivative of the co-energy function with respect to a mechanical variable (like position ‘x’ or angle ‘θ’) while holding the current constant. This is often more convenient than the energy method, which requires holding flux linkage constant. This calculator helps engineers and students perform this calculation of force and torque using the concept of co-energy for linear and rotational systems.

The Co-energy Formula and Explanation

For an electromechanical system where the inductance `L` is a function of a mechanical position variable (let’s say `z`), the co-energy `W’` for a single excitation coil is given by:

W'(i, z) = ½ × L(z) × i²

The mechanical force `F` (for linear motion) or torque `T` (for rotational motion) is the partial derivative of the co-energy with respect to the mechanical variable.

For Linear Force:

F(i, x) = ∂W'(i, x) / ∂x = ½ × i² × (dL/dx)

For Rotational Torque:

T(i, θ) = ∂W'(i, θ) / ∂θ = ½ × i² × (dL/dθ)

Variables Used in Co-energy Calculations

Variable	Meaning	Unit (SI)	Typical Range
F	Electromagnetic Force	Newtons (N)	mN to kN
T	Electromagnetic Torque	Newton-meters (Nm)	mNm to kNm
W’	Co-energy	Joules (J)	Depends on system
i	Coil Current	Amperes (A)	mA to kA
L	Inductance	Henries (H)	μH to H
dL/dx	Inductance Gradient (Linear)	Henries/meter (H/m)	0.1 to 10 H/m
dL/dθ	Inductance Gradient (Angular)	Henries/radian (H/rad)	0.01 to 1 H/rad

Practical Examples

Example 1: Calculating Force in a Solenoid

Consider a linear solenoid actuator where the inductance changes as the plunger moves. A measurement shows that the inductance gradient is constant over a small range of motion.

• Inputs:
  • Coil Current (i): 5 A
  • Inductance Gradient (dL/dx): 0.8 H/m
• Calculation:
  • Force (F) = ½ × (5 A)² × 0.8 H/m
  • Force (F) = ½ × 25 A² × 0.8 H/m
  • Result: Force (F) = 10 Newtons

Example 2: Calculating Torque in a Reluctance Motor

A simple switched reluctance motor has a stator and rotor. As the rotor turns, the inductance of the stator winding changes. We want to find the instantaneous torque at a specific rotor position.

• Inputs:
  • Coil Current (i): 20 A
  • Angular Inductance Gradient (dL/dθ): 0.15 H/rad
• Calculation:
  • Torque (T) = ½ × (20 A)² × 0.15 H/rad
  • Torque (T) = ½ × 400 A² × 0.15 H/rad
  • Result: Torque (T) = 30 Newton-meters

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How to Use This Co-energy Calculator

This tool simplifies the calculation of force and torque using the concept of co-energy. Follow these steps for an accurate result:

1. Select the Calculator: Choose between the “Linear Force Calculator” for devices like solenoids and linear actuators, or the “Rotational Torque Calculator” for motors and rotary actuators.
2. Enter Coil Current (i): Input the steady-state DC current flowing through the device’s coil in Amperes (A).
3. Enter the Inductance Gradient:
   • For the force calculator, enter the value of dL/dx in Henries per meter (H/m). This value represents how much the coil’s inductance changes for each meter of linear movement.
   • For the torque calculator, enter the value of dL/dθ in Henries per radian (H/rad). This value represents how much the coil’s inductance changes for each radian of rotation.
4. Interpret the Results: The calculator instantly displays the resulting force in Newtons (N) or torque in Newton-meters (Nm). Intermediate values like current squared and the co-energy gradient are also shown to provide insight into the calculation.

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Key Factors That Affect Force and Torque Calculation

Current Level (i)

As seen in the formula, force and torque are proportional to the square of the current. Doubling the current will quadruple the output force or torque, assuming the system remains magnetically linear.

Inductance Gradient (dL/dz)

This is the most critical factor. The entire principle relies on the fact that inductance changes with position. A larger change in inductance per unit of movement (a steeper gradient) results in a larger force/torque for the same current. The geometry of the magnetic circuit—especially the air gap—is the primary determinant of this value.

Magnetic Saturation

This calculator assumes a magnetically linear system, meaning the inductance (L) does not change with current (i). In real-world materials, as current increases, the core material can saturate, causing inductance to drop. In such cases, the co-energy formula is more complex, and this simplified calculator would overestimate the force/torque.

Air Gap Dimensions

The shape, length, and area of the air gap in a magnetic circuit have the most significant impact on the inductance and its gradient. Force and torque are produced in a way that tends to minimize the reluctance of the magnetic path, which often means minimizing the air gap.

Fringing Flux

The formulas assume ideal flux paths. In reality, “fringing” flux can leak out of the intended path, especially around air gaps. This can affect the actual inductance gradient and lead to discrepancies between calculated and measured force/torque.

Mechanical and Electrical Losses

This calculation provides the ideal electromagnetic force/torque. It does not account for real-world losses such as friction in a mechanical system or resistive heating (I²R losses) in the coil.

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Frequently Asked Questions (FAQ)

1. What is the difference between energy and co-energy?

Energy (Wf) is the area under the flux-linkage vs. current (λ-i) curve, integrated with respect to flux linkage. Co-energy (W’) is the area to the left of the λ-i curve, integrated with respect to current. They are complementary areas. For linear systems, their values are equal.

2. Why use co-energy to find force instead of energy?

The co-energy method allows you to calculate force by taking a derivative while holding current constant (F = ∂W’/∂x). The energy method requires holding flux linkage constant (F = -∂Wf/∂x). In many practical scenarios, it’s easier to analyze a system with a known, constant current source, making the co-energy approach more straightforward.

3. Is this calculator valid for systems with permanent magnets?

No. This calculator and the formulas used are for singly-excited systems where the magnetic field is produced solely by current in a coil. Systems with permanent magnets have a different energy and co-energy relationship that includes the energy stored in the magnet itself.

4. What happens if the system is magnetically non-linear (saturated)?

If the system is saturated, the relationship between flux linkage and current is no longer linear, and inductance `L` becomes a function of current `i`. The simple formula W’ = ½ * L * i² no longer applies. One must use the integral definition W’ = ∫ λ(i) di, and the calculation becomes significantly more complex. This calculator would not be accurate in that case.

5. How do I find the inductance gradient (dL/dx) for my device?

This value can be determined through several methods: analytical calculation for simple geometries, using Finite Element Analysis (FEA) software for complex designs, or empirical measurement by plotting the inductance at different physical positions and calculating the slope of the curve.

6. What units are used in this calculator?

The calculator uses the standard SI unit system. Current is in Amperes (A), linear inductance gradient in Henries per meter (H/m), angular inductance gradient in Henries per radian (H/rad), force in Newtons (N), and torque in Newton-meters (Nm).

7. If inductance is constant, can I still calculate a force?

No. If the inductance `L` does not change with position `x` (meaning dL/dx = 0), then the co-energy does not change with position, and the resulting force is zero. This is a key principle: electromechanical force is only produced when inductance varies with mechanical position.

8. What is a typical value for an inductance gradient?

It varies widely based on the device. Small actuators might have gradients in the range of 0.1 to 2 H/m. Large industrial machines or specialized devices like railguns can have much higher values.

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