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9.11 Torque Study

This section reviews the torque that an asynchronous machine develops. Recalling the definitions of:

Work = τ·Δθ (Work; τ: torque; Δθ: change in angular displacement)

K = ½·I·w² (K = Kinetic energy; I: moment of inertia; w: angular velocity)

P = τ·w (P: power; τ: torque; w: angular speed)

Remember that power (P): .

Now on we look on the torque calculation (Cm).

If we express n₁ and n₂ in revolutions per second (rps) we have that:

Cm = Pmi/(2π·n₂) = Pmi/(2π·n₁·(1-s)) (9.19)

Recalling that:

s = (n₁ - n₂)/n₁ → n₂ = n₁(1-s) (9.20)

We can also define torque through the power that the stator transfers to the rotor when they are rotating in sync. Substituting (Eq. 9.15) we have:

Cm = Pmi/(2π·n₂) = Pmi/(2π·n₁·(1-s)) = P₁₂/(2π·n₁) (9.21)

This statement of torque is true, as the denominator comprises the speed of the rotating field, which remains constant in the motor as long as the power supply frequency remains unchanged.

For this reason, the power that the stator transmits to the rotor (P12) refers to as the motor torque in synchronous watts.

练习题

Which equation correctly expresses the relationship between power (P), torque (τ), and angular speed (ω)?

P = τ / ω

P = τ \cdot ω

P = τ^{2} \cdot ω

P = τ / ω^{2}

An asynchronous motor has a synchronous speed $n_{1} = 50$ rps and operates with a slip $s = 0.04$ . If the mechanical internal power $P_{mi} = 10$ kW, what is the torque $C_{m}$ developed by the machine?

C_{m} = \frac{10000}{2 π \cdot 50} \approx 31.8

N·m

C_{m} = \frac{10000}{2 π \cdot 50 \cdot 0.96} \approx 33.2

N·m

C_{m} = \frac{10000}{2 π \cdot 2} \approx 795.8

N·m

C_{m} = \frac{10000}{2 π \cdot 50 \cdot 0.04} \approx 795.8

N·m

The power that the stator transmits to the rotor ( $P_{12}$ ) is referred to as the motor torque in:

A. Mechanical watts

B. Rotor watts

C. Synchronous watts

D. Stator watts

Which of the following expressions correctly represent the torque $C_{m}$ of an asynchronous machine? (Select all that apply)

C_{m} = \frac{P _{mi}}{2 π \cdot n _{2}}

C_{m} = \frac{P _{mi}}{2 π \cdot n _{1} \cdot ( 1 - s )}

C_{m} = \frac{P _{12}}{2 π \cdot n _{1}}

C_{m} = \frac{P _{mi}}{2 π \cdot n _{1} \cdot s}

The work done by a torque is given by the equation: Work = $τ \cdot Δ θ$ , where $τ$ is torque and $Δ θ$ is the change in angular displacement.

In an asynchronous motor, the rotor speed $n_{2}$ can equal the synchronous speed $n_{1}$ during normal loaded operation.

The rotating field speed in an asynchronous motor remains constant as long as the power supply frequency remains unchanged.

The kinetic energy of a rotating body is given by $K = \frac{1}{2} I ω^{2}$ , where $I$ represents the ___ and $ω$ represents the angular velocity.

When the rotor speed is expressed in terms of synchronous speed and slip, the relationship is $n_{2} = n_{1} (1 - s)$ , where $s$ is the slip defined as $s = \frac{n_1 - n_2}{\text{___}}$ .

Explain why the expression $C_{m} = P_{12} / (2 π \cdot n_{1})$ provides a valid representation of motor torque, where $P_{12}$ is the power transferred from stator to rotor and $n_{1}$ is the synchronous speed.

A three-phase asynchronous motor has a synchronous speed $n_{1} = 1500$ rpm ( $25$ rps). When operating with a slip of $s = 0.04$ and developing a mechanical internal power $P_{mi} = 8$ kW, what is the motor torque $C_{m}$ ?

50.9

53.1

55.4

57.8

The power $P_{12}$ that the stator transmits to the rotor is called "motor torque in synchronous watts" because dividing it by $2 π \cdot n_{1}$ yields the torque, and the synchronous speed $n_{1}$ remains constant as long as the supply frequency is unchanged.

Which of the following statements correctly describe the relationships between slip, rotor speed, and torque in an asynchronous motor?

A. The rotor speed

n_{2}

never equals the synchronous speed

n_{1}

due to Faraday's law requiring relative motion between field and conductor

B. The torque can be calculated as

C_{m} = P_{mi} / (2 π \cdot n_{2})

where

n_{2}

is the actual rotor speed

C. Using the slip relationship

n_{2} = n_{1} (1 - s)

, the torque formula can be rewritten as

C_{m} = P_{mi} / (2 π \cdot n_{1} \cdot (1 - s))

D. The rotating field speed

n_{1}

remains constant regardless of motor load as long as supply frequency is unchanged

At motor start-up, the rotor is stationary ( $n_{2} = 0$ ), the slip equals ___, and the rotor EMF equals the standstill value $E_{2}$ .

Explain why the torque formula $C_{m} = P_{12} / (2 π \cdot n_{1})$ using "synchronous watts" is advantageous for motor analysis compared to $C_{m} = P_{mi} / (2 π \cdot n_{2})$ .

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9.11 Torque Study

9.11 Torque Study

练习题

Which equation correctly expresses the relationship between power (P), torque (τ), and angular speed (ω)?

An asynchronous motor has a synchronous speed n1​=50 rps and operates with a slip s=0.04. If the mechanical internal power Pmi​=10 kW, what is the torque Cm​ developed by the machine?

The power that the stator transmits to the rotor (P12​) is referred to as the motor torque in:

Which of the following expressions correctly represent the torque Cm​ of an asynchronous machine? (Select all that apply)

The work done by a torque is given by the equation: Work = τ⋅Δθ, where τ is torque and Δθ is the change in angular displacement.

In an asynchronous motor, the rotor speed n2​ can equal the synchronous speed n1​ during normal loaded operation.

The rotating field speed in an asynchronous motor remains constant as long as the power supply frequency remains unchanged.

The kinetic energy of a rotating body is given by K=21​Iω2, where I represents the ___ and ω represents the angular velocity.

When the rotor speed is expressed in terms of synchronous speed and slip, the relationship is n2​=n1​(1−s), where s is the slip defined as s = \frac{n_1 - n_2}{\text{___}}.

Explain why the expression Cm​=P12​/(2π⋅n1​) provides a valid representation of motor torque, where P12​ is the power transferred from stator to rotor and n1​ is the synchronous speed.

A three-phase asynchronous motor has a synchronous speed n1​=1500 rpm (25 rps). When operating with a slip of s=0.04 and developing a mechanical internal power Pmi​=8 kW, what is the motor torque Cm​?

The power P12​ that the stator transmits to the rotor is called "motor torque in synchronous watts" because dividing it by 2π⋅n1​ yields the torque, and the synchronous speed n1​ remains constant as long as the supply frequency is unchanged.

Which of the following statements correctly describe the relationships between slip, rotor speed, and torque in an asynchronous motor?

At motor start-up, the rotor is stationary (n2​=0), the slip equals ___, and the rotor EMF equals the standstill value E2​.

Explain why the torque formula Cm​=P12​/(2π⋅n1​) using "synchronous watts" is advantageous for motor analysis compared to Cm​=Pmi​/(2π⋅n2​).

An asynchronous motor has a synchronous speed $n_{1} = 50$ rps and operates with a slip $s = 0.04$ . If the mechanical internal power $P_{mi} = 10$ kW, what is the torque $C_{m}$ developed by the machine?

The power that the stator transmits to the rotor ( $P_{12}$ ) is referred to as the motor torque in:

Which of the following expressions correctly represent the torque $C_{m}$ of an asynchronous machine? (Select all that apply)

The work done by a torque is given by the equation: Work = $τ \cdot Δ θ$ , where $τ$ is torque and $Δ θ$ is the change in angular displacement.

In an asynchronous motor, the rotor speed $n_{2}$ can equal the synchronous speed $n_{1}$ during normal loaded operation.

The kinetic energy of a rotating body is given by $K = \frac{1}{2} I ω^{2}$ , where $I$ represents the ___ and $ω$ represents the angular velocity.

When the rotor speed is expressed in terms of synchronous speed and slip, the relationship is $n_{2} = n_{1} (1 - s)$ , where $s$ is the slip defined as $s = \frac{n_1 - n_2}{\text{___}}$ .

Explain why the expression $C_{m} = P_{12} / (2 π \cdot n_{1})$ provides a valid representation of motor torque, where $P_{12}$ is the power transferred from stator to rotor and $n_{1}$ is the synchronous speed.

A three-phase asynchronous motor has a synchronous speed $n_{1} = 1500$ rpm ( $25$ rps). When operating with a slip of $s = 0.04$ and developing a mechanical internal power $P_{mi} = 8$ kW, what is the motor torque $C_{m}$ ?

The power $P_{12}$ that the stator transmits to the rotor is called "motor torque in synchronous watts" because dividing it by $2 π \cdot n_{1}$ yields the torque, and the synchronous speed $n_{1}$ remains constant as long as the supply frequency is unchanged.

At motor start-up, the rotor is stationary ( $n_{2} = 0$ ), the slip equals ___, and the rotor EMF equals the standstill value $E_{2}$ .

Explain why the torque formula $C_{m} = P_{12} / (2 π \cdot n_{1})$ using "synchronous watts" is advantageous for motor analysis compared to $C_{m} = P_{mi} / (2 π \cdot n_{2})$ .