{"id":3563,"date":"2021-03-02T19:31:25","date_gmt":"2021-03-02T19:31:25","guid":{"rendered":"https:\/\/novanta.com\/robotics-automation\/technical-paper\/what-motor-torque-constant-to-use-for-drive-type-theory-and-application\/"},"modified":"2026-04-23T18:11:23","modified_gmt":"2026-04-23T18:11:23","slug":"what-motor-torque-constant-to-use-for-drive-type-theory-and-application","status":"publish","type":"novanta_tech_paper","link":"https:\/\/novanta.com\/robotics-automation\/technical-paper\/what-motor-torque-constant-to-use-for-drive-type-theory-and-application\/","title":{"rendered":"What Motor Torque Constant to Use for Drive Type \u2013 Theory and Application"},"content":{"rendered":"\n
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Introduction<\/h2>\n\n\n\n

Calculating motor torque from available drive current can be confusing due to many different drive types and the multiple ways current is specified (Ipk\u2013sine<\/sub>, IDC<\/sub>, IRMS<\/sub>). 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).<\/p>\n\n\n\n

Three Phase Motor Theory<\/h2>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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\"Three<\/img>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 \ud835\udc56\ud835\udc65<\/sub> (constant for trapezoidal commutation, and sinusoidal for sine commutation) and the sinusoidal variation of the perphase torque constant (\ud835\udc3e\ud835\udc61\u2205\ud835\udc65<\/sub>) with rotor position.<\/p>\n\n\n\n

In this derivation, we are making the assumption that the \ud835\udc3e\ud835\udc61<\/sub>\u2205 <\/sub>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.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Figure 1 \u2013 Spatial Variation of Torque in the Re\/Im plane<\/em><\/p><\/p>\n\n\n\n

Trapezoidal Drive<\/h2>\n\n\n\n

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<\/p>\n\n\n\n

\ud835\udc56\ud835\udc4e<\/sub>(\ud835\udc61) = \u2212\ud835\udc56\ud835\udc50<\/sub>(\ud835\udc61) & \ud835\udc56\ud835\udc4e<\/sub>(\ud835\udc61) = IDC<\/sub><\/em>. 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, \ud835\udc56\ud835\udc65<\/sub> values within the phasor diagram of Figure 1 are constant in time and have no variation (unlike sinusoidal drive, which will be discussed below).<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Figure 2 \u2013 Trapezoidal Current in Three-Phase Motor<\/em><\/p><\/p>\n\n\n\n

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:<\/p>\n\n\n\n

Equation (1)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

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 (\ud835\udc47\ud835\udc5a\ud835\udc61rap<\/sub><\/sub>) can be solved algebraically or by using vector math as shown in Figure 3.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Figure 3 \u2013 Vector Math for Trapezoidal Commutation<\/em><\/p><\/p>\n\n\n\n

Solving Equation (1) algebraically requires angle \ud835\udc34\u2205 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.<\/p>\n\n\n\n

Equation (2)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solving for torque:<\/p>\n\n\n\n

Equation (3)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Since current \ud835\udc3c\ud835\udc37C<\/sub> is only ever flowing in two phases at a time, an equivalent two-phase torque constant is then derived:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Where:<\/p>\n\n\n\n

Equation (4)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Equation (4) yields the result of the two-phase motor torque constant, as a function of the single-phase.<\/p>\n\n\n\n

Sinusoidal Drive<\/h2>\n\n\n\n

Sine drives use a commutation scheme that flows current in all three phases at all times, with the sum always equaling zero (\ud835\udc3c\ud835\udc34<\/sub> \ud835\udc3c\ud835\udc35<\/sub> \ud835\udc3c\ud835\udc36<\/sub> = 0). Different than a trapezoidal drive, the currents \ud835\udc56\ud835\udc65<\/sub> are now modulating sinusoidally as follows:<\/p>\n\n\n\n

Equation (5)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Equation (6)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Equation (7)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Where \ud835\udf14t rotates with \ud835\udc34\ud835\udf03 or \ud835\udc34\ud835\udf03 \u0307\ud835\udc61 and \ud835\udf14=\ud835\udc34\ud835\udf03 \u0307<\/p><\/p>\n\n\n\n

Substituting Equations (5), (6), and (7) into Equation (1) yields:<\/p>\n\n\n\n

Equation (8)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Solving Equation (8) for where \ud835\udf14t =\ud835\udc34\u2205 and \ud835\udc34\u2205 = 0\u00b0 yields:<\/p>\n\n\n\n

Equation (9)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

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.<\/p>\n\n\n\n

Equation (10)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Back EMF Constant and Torque Constant<\/h3>\n\n\n\n

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.<\/p>\n\n\n\n

When working in SI units, the per-phase \ud835\udc3e\ud835\udc52\u2205<\/sub><\/sub> constant is equal to the single-phase torque constant \ud835\udc3e\ud835\udc61\u2205<\/sub><\/sub>. 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:<\/p>\n\n\n\n

Equation (11)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

This is because \ud835\udc3e\ud835\udc52<\/sub> is directly correlated to \ud835\udc49\ud835\udc4femf<\/sub><\/p>\n\n\n\n

Equation (12)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Given the relationship between \ud835\udc3e\ud835\udc52<\/sub> and voltage shown in Equation (12), the voltage output can be substituted for \ud835\udc3e\ud835\udc52<\/sub> in Equation (11), yielding Equation (13).<\/p>\n\n\n\n

Equation (13)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

In Equation (4) above, we show how \ud835\udc3e\ud835\udc61\u2205\u2205\ud835\udc61rap<\/sub><\/sub><\/sub> = \u221a3 \u2217 \ud835\udc3e\ud835\udc61\u2205<\/sub><\/sub> and when combined with \ud835\udc3e\ud835\udc52\u2205<\/sub><\/sub> = \ud835\udc3e\ud835\udc61\u2205<\/sub><\/sub> we get:<\/p>\n\n\n\n

Equation (14)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

The two-phase representation \ud835\udc3e\ud835\udc61\u2205\u2205<\/sub> is then equivalent to \ud835\udc3e\ud835\udc61\u2205\u2205\ud835\udc61rap<\/sub>.
\nTherefore, Equation (13) is also equal to Equation (14).<\/p>\n\n\n\n

Comparing Trapezoidal with Sine Commutation<\/h3>\n\n\n\n

Since measurement of \ud835\udc3ee\u2205\u2205<\/sub><\/sub> (and subsequent conversion to \ud835\udc3e\ud835\udc61\u2205\u2205\ud835\udc61rap<\/sub><\/sub><\/sub>) involves two phases at a time, it is convenient to find a conversion from \ud835\udc3e\ud835\udc61\u2205\u2205\ud835\udc61rap<\/sub><\/sub><\/sub> to \ud835\udc3e\ud835\udc61sine<\/sub>.<\/sub><\/p>\n\n\n\n

We know that \ud835\udc3emsine<\/sub> = \ud835\udc3em\ud835\udc61rap<\/sub> therefore using Equations (3) and (9), solving for \ud835\udc3c\ud835\udc37C<\/sub> we see that:<\/sub><\/sub><\/p>\n\n\n\n

Equation (15)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

and<\/p>\n\n\n\n

Equation (16)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Substituting Equation (15) above into Equation (16), yields Equation (17).<\/p>\n\n\n\n

Equation (17)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Simplifying the equation then yields Equation (18).<\/p>\n\n\n\n

Equation (18)<\/h3>\n\n\n\n
\"\"<\/figure>\n\n\n\n

It should be noted that referring to \ud835\udc3etsine<\/sub><\/sub> 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 \u2013 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 (\ud835\udc3e\ud835\udc61\u2205\u2205\ud835\udc61rap<\/sub><\/sub><\/sub>) and a sinusoidal (\ud835\udc3e\ud835\udc61sine<\/sub><\/sub>) torque constant, as many drives and motor companies spec current (IDC<\/sub> or Ipk-sine<\/sub>) and torque constant (\ud835\udc3e\ud835\udc61\u2205\u2205\ud835\udc61rap<\/sub><\/sub><\/sub> or \ud835\udc3e\ud835\udc61sine<\/sub><\/sub>) differently.<\/p>\n\n\n\n

Examples: Lab Test Data<\/h3>\n\n\n\n

Test Components:<\/p>\n\n\n\n