## Introduction

Calculating motor torque from available drive current can be confusing due to many different drive types and the multiple ways current is specified (I_{pk–sine}, I_{DC}, I_{RMS}). This paper will provide the key formulas for torque constant and motor current, from fundamental principles of three-phase motor theory. It will also walk through the many ways torque can be calculated, using a per-phase torque, phase-to-phase torque constant, and sinusoidal torque constant, while applying the appropriate currents based on drive type (trapezoidal or sine).

## Three Phase Motor Theory

The motor theory described in this paper is based on a brushless motor (BLAC or permanent magnet synchronous machines) configuration that has a sinusoidal back EMF waveform for each phase-toneutral winding. Assumptions are as follows: torque angle curves and current waveforms are assumed to be pure sinusoids of the same amplitude. For equations using the per-phase torque constant, this assumes that input current is supplied to the neutral (center tap) connection.

When applying current to a singular motor phase of a BLAC (phase-to-neutral), these types of motors generate sinusoidal torque waveforms versus electrical angle (as the shaft rotates mechanically). The electrical angle is N/2 multiplied by (*) mechanical angle, where N is the number of poles of the motor.

Figure 1 below shows a graphical drawing for the phasor representation of torque angle curve, with a three-phase brushless motor. The three vectors created by are shown as dotted lines, meant to represent that these vectors scale up and down in magnitude, as they rotate around the phase diagram, always staying 120 degrees apart. The magnitude of each vector varies based on current 𝑖_{𝑥} (constant for trapezoidal commutation, and sinusoidal for sine commutation) and the sinusoidal variation of the perphase torque constant (𝐾_{𝑡}_{∅𝑥}) with rotor position.

In this derivation, we are making the assumption that the 𝐾_{𝑡}_{∅ }values are equal in magnitude (actual variations are typically small, but for the purposes of these derivations, we are stating there are no variations), and separated by 120 electrical degrees.

*Figure 1 – Spatial Variation of Torque in the Re/Im plane*

## Trapezoidal Drive

Trapezoidal commutation allows current to only flow through two of the three motor phases at any given time, and the current flowing is considered to be DC (see Figure 2). This results in

𝑖_{𝑎}(𝑡) = −𝑖_{𝑐}(𝑡) & 𝑖_{𝑎}(𝑡) = *I*_{DC}. For the purposes of the derivation, we are ignoring the ramp up and ramp down of the current based on the time constant (RL) of the motor phases. With this configuration, 𝑖_{𝑥} values within the phasor diagram of Figure 1 are constant in time and have no variation (unlike sinusoidal drive, which will be discussed below).

*Figure 2 – Trapezoidal Current in Three-Phase Motor*

The general equation for motor torque is a function of the single-phase motor torque and the current flowing through each of the three phases, represented as follows:

### Equation (1)

As stated above for trapezoidal drives, current is only flowing in two of three phases at any one time, so the math reduces significantly, and the motor torque (𝑇_{𝑚𝑡rap}) can be solved algebraically or by using vector math as shown in Figure 3.

*Figure 3 – Vector Math for Trapezoidal Commutation*

Solving Equation (1) algebraically requires angle 𝐴∅ to be known. From the vector diagram of Figure 3, we can see that when current is only flowing in phase A and out phase C, the value is 30 degrees, and we arrive at Equation (2) below.

### Equation (2)

Since current 𝐼_{𝐷C} is only ever flowing in two phases at a time, an equivalent two-phase torque constant is then derived:

Equation (4) yields the result of the two-phase motor torque constant, as a function of the single-phase.

## Sinusoidal Drive

Sine drives use a commutation scheme that flows current in all three phases at all times, with the sum always equaling zero (𝐼_{𝐴}+𝐼_{𝐵}+𝐼_{𝐶} = 0). Different than a trapezoidal drive, the currents 𝑖_{𝑥} are now modulating sinusoidally as follows:

Where 𝜔t rotates with 𝐴𝜃 or 𝐴𝜃 ̇𝑡 and 𝜔=𝐴𝜃 ̇

Substituting Equations (5), (6), and (7) into Equation (1) yields:

### Equation (8)

Solving Equation (8) for where 𝜔t =𝐴∅ and 𝐴∅ = 0° yields:

Like Equation (3) above, which shows torque developed as a function of the single-phase torque constant and the current flowing in the motor, Equation (9) can be re-written to represent a sinusoidal torque constant. Equation (10) yields the equivalent torque constant for current flowing when using a sine drive.

### Back EMF Constant and Torque Constant

In Wye connected motors, the center tap (neutral) is rarely available, making phase-to-neutral measurements impossible. The best thing to do is work within the constraints and make phase-to-phase measurements, then work backwards through the equations to arrive at per-phase, phase-to-phase, and sine torque constants.

When working in SI units, the per-phase 𝐾_{𝑒∅} constant is equal to the single-phase torque constant 𝐾_{𝑡∅}. In a three-phase system, if we could measure a single-phase voltage, we would find that the relationship between single-phase and phase-to-phase voltage is:

This is because 𝐾_{𝑒} is directly correlated to 𝑉_{𝑏emf}

Given the relationship between 𝐾_{𝑒} and voltage shown in Equation (12), the voltage output can be substituted for 𝐾_{𝑒} in Equation (11), yielding Equation (13).

In Equation (4) above, we show how 𝐾_{𝑡∅∅𝑡rap} = √3 ∗ 𝐾_{𝑡∅} and when combined with 𝐾_{𝑒∅} = 𝐾_{𝑡∅} we get:

The two-phase representation 𝐾_{𝑡∅∅} is then equivalent to 𝐾_{𝑡∅∅𝑡rap}.

Therefore, Equation (13) is also equal to Equation (14).

### Comparing Trapezoidal with Sine Commutation

Since measurement of 𝐾_{e∅∅} (and subsequent conversion to 𝐾_{𝑡∅∅𝑡rap}) involves two phases at a time, it is convenient to find a conversion from 𝐾_{𝑡∅∅𝑡rap} to 𝐾_{𝑡sine.}

We know that 𝐾_{msine = 𝐾m𝑡rap therefore using Equations (3) and (9), solving for 𝐼𝐷C we see that:}

Substituting Equation (15) above into Equation (16), yields Equation (17).

### Equation (17)

Simplifying the equation then yields Equation (18).

### Equation (18)

It should be noted that referring to 𝐾_{tsine} as a line-to-line or phase-to-phase, torque constant, is generally not good pratice because it implies only two phases are working at once – which is only true for very brief moments when the phase-to-phase current of a single-phase pair passes through zero. It is useful, however, to scale between a two-phase (𝐾_{𝑡∅∅𝑡rap}) and a sinusoidal (𝐾_{𝑡sine}) torque constant, as many drives and motor companies spec current (I_{DC} or I_{pk-sine}) and torque constant (𝐾_{𝑡∅∅𝑡rap} or 𝐾_{𝑡sine}) differently.

### Static Kt Test Versus Ke to Kt Calculation

1. Back drive motor and calculate

2. Convert 𝐾_{e∅} to 𝐾_{tsine} and 𝐾_{t∅∅trap}

3. Apply various current outputs (0.1, 0.2, 0.2, and 0.3 𝐴_{𝑝k-sine}) from the drive, using the drive’s software (MotionLab 3)

4. Hold motor shaft in place long enough to get full reading on oscilloscope that is reading torque

5. Record oscilloscope output in volts and convert to oz-in

6. Calculate 4 values of 𝐾_{𝑡∅} (in units of 𝑁m/𝐴_{𝑝k-sine}) using Equation (10).

7. Use Average to calculate 𝐾_{tsine} .

8. Calculate Percent Error

9. Solve for 𝐾_{t∅∅trap}

10. Calculate Percent Error

### Calculations

Calculating 𝐾_{t∅} for measurements 1 and 2:

1.Current = 0.2 𝐴_{𝑝k-sine}

T_{m} = 0/94 oz-in = 0.00664

K_{t∅} = 0.0221 𝑁m/𝐴_{𝑝k-sine}

2.Current = 0.2 𝐴_{𝑝k-sine}

T_{m} = 0.92 oz-in = 0.00650

K_{t∅} = 0.0217 𝑁m/𝐴_{𝑝k-sine}

Then calculate the average:

Average K_{t∅} = 0.0219 Nm/𝐴_{𝑝k-sine}

- Use Average K
_{t∅} to calculate K_{tsine}

K_{tsine} = 1.5 * K_{t∅} = 1.5 * 0.0219 𝑁m/𝐴_{𝑝k-sine} = 0.0329 𝑁m/𝐴_{𝑝k-sine}

- Calculate percentage error

Expected K_{tsine} from datasheet = 0.034 𝑁m/𝐴_{𝑝k-sine}

Expected K_{tsine} from measured = 0.033 𝑁m/𝐴_{𝑝k-sine}

- Solve for K
_{t∅∅trap}

Average 𝐾_{t∅ = 0.0219 𝑁m/𝐴𝑝k-sine 𝐾t∅∅𝑡rap = √3 * 0.0219 = 0.038 𝑁m/𝐴DC}

Expected K_{t∅∅trap} from datasheet = 0.039 𝑁m/𝐴_{DC}

Expected K_{t∅∅trap} from measured = 0.038 𝑁m/𝐴_{DC}

## Summary of Test Data

For percent error calculations, the measured 𝐾_{e∅∅} was used as the ‘expected’ value, instead of the motor spec sheet value, due to variations of 10% allowed for magnet strength differences. The calculated 𝐾_{𝑡sine} with <1% error and the calculated 𝐾_{𝑡∅∅trap of ~3% error are both within the acceptable range given the error of the measurements. The calculations above, conducted with a slotless motor, also eliminate any cogging torque from altering measured data. Cogging torque can have a huge impact on static torque testing and should always be considered. For this reason, back driving motors and using 𝐾=e∅∅trap to arrive at 𝐾𝑡∅∅trap is the preferred approach. The static testing above is also a common method, but can quickly be made irrelevant when cogging is introduced.}

## Conclusion

The torque constant K_{t} is a critical motor parameter that determines the amount of output torque per input current. Because of the three-phase nature of brushless DC/AC/PMSM, and how they are driven, confusion quickly arises as to which Kt value to use when calculating output torque for a given input current. The two most common ways to state K_{t} are K_{t∅∅trap} with units of 𝑁m / 𝐴_{𝐷C} ⁄ and 𝐾_{𝑡sine} units of 𝑁m / 𝐴_{pk-sine} . The derivation of both comes from per-phase motor theory and the conversion between them has been shown. Care must be taken to understand how the motor will be driven in terms of the drive type, and how the motor manufacturer is stating K_{t}. When testing to confirm stated K_{t} values, be cautious when cogging is present. For this reason, it is recommended to back drive motors to measure 𝐾_{𝑒} and convert to 𝐾_{𝑡∅∅𝑡rap} and 𝐾_{𝑡sine} as shown in this paper.

## Acknowledgments

Per-phase torque derivations described in this paper were made possible through a partnership with Dr. Steve Prina, Co-Founder of Applimotion, now part of Celera Motion a Novanta Company