Doubly Excited Magnetic System | Electrical Machines - Electrical Engineering (EE) PDF Download

Introduction

• A doubly-excited system is the type of magnetic system in which two independent coils are used to produce magnetic field. Examples of doubly-excited systems are synchronous machine, separately excited DC machines, loudspeakers, tachometers etc.
  
• Consider a doubly-excited system as shown in the figure, it consists of a stator wound with a coil having a resistance of R1 and a rotor wound with a coil of resistance R2. Both the coils are excited by independent voltage sources.
• Following assumptions are made to analyse a doubly excited system −
  • For any rotor position the relationship between flux-linkage (ψ) and current is linear.
  • Hysteresis and eddy current losses are neglected.
  • The coils have negligible leakage flux.
  • The electric fields are neglected and the magnetic fields are predominating.
• The expressions for the flux linkage of the two winding are given by,
  ψ₁ = L₁i₁ + Mi₂
  ψ₂ = L₂i₂ + Mi₁
• By applying KVL, the instantaneous voltage equations for the two coils being,
  
  
• Substituting the values of ψ₁ and ψ₂ in the corresponding voltage equations, we get,
  
  
• Since, the inductances are dependent on the position of the rotor angle θ_m which is a function of time and are independent of the currents. Similarly, the currents are the function of time and are independent of inductances. Therefore, the eq. (1) and (2) can be written as,
  
  
• Now, multiplying eqn. (3) by i₁ and eqn. (4) by i₂, we get
  
  
• The eqns.(5) and (6) are the power expressions for the two coils.
• Now, integrating eqns.(5) and (6) with respect to time and adding, we have,
  ...(7)
• The eqn.(7) is the energy equation for the doubly-excited magnetic system. It shows that the total electrical energy input to the system is equal to the sum of two parts, where the first part is the energy to electrical losses and the second is useful electrical energy, i.e
  [Total Electrical Energy Input W_e] = [Energy to Electrical Losses W_{electric.losses}] +[Useful Electrical Energy W_f + W_m]
• Where,
  ... (8)

Energy Stored in the Magnetic Field

• The instantaneous value of energy stored in the magnetic field depends upon the inductance and values of current at that instant. For any stationary position of the rotor, the mechanical output is zero, and all the useful electrical energy input is converted into field energy. As the inductance values are constant. Therefore, terms dL₁, dL₂ and dM become zero. Hence, from eqns.(8), we get,
  
  

Electromagnetic Torque

• When the rotor rotates, the rate of change of field energy with respect to time is given by differentiating the eqn.(9),
  ...(10)
• Integrating the eqn.(10) with respect to time, we get,
  
• The eqn. (11) is a general equation for a moving transducer in which L₁, L₂, M, i₁ and i₂ are all varying with position and time. Now, comparing eqn.(11) with eqn.(8), we obtain,
  
• Differentiating eqn.(12) with respect to rotor angle θ_m, we get,
  
• The first two terms of the equation (13) are the reluctance torque and the last term is known as the co-alignment torque, that is due to the two superimposed magnetic fields try to align.

Note: For the machines having uniform air gaps the reluctance torque is not produced.

Numerical Example

For a doubly excited system, the inductances are given as follows −
L₁ = 10 + 2cos2θ; L₂ = 5 + 2cos2θ;M = 20cosθ

The coils are excited by current i₁ = 0.5A and i₂ = 0.6A.

• Find the torque as a function of θ.
• Find the energy stored in the system as a function of θ.

The torque produced in a doubly-excited system is given by

here,

⇒ τ_e = 12 × (0.5)² × (−4sin2θ) + 1/2 × (0.6)² × (−6sin2θ) + (0.5) × (0.6) × −20sinθ

∴ τ = −1.58sin2θ − 6sinθNm

The energy stored in doubly-excited system is given by,

⇒ W_f = 1/2(10 + 2cos2θ)(0.5)² + 1/2(5 + 2cos3θ)(0.6)² + (20cosθ)(0.5)(0.6)
⇒ W_f = 2.15 + 0.79cos2θ + 6cosθ

Difference between Singly Excited and Doubly Excited System

• In electrical machines and instruments, an excitation system is the one which receives electrical energy to produce the working magnetic flux. The excitation system is the crucial component every electromechanical energy conversion device such as electric motor.
• Depending upon the number of input supplies, any electromechanical energy conversion system can be grouped into the following two categories −
  Singly Excited System and Doubly Excited System

What is a Singly Excited System?

• The electromechanical energy conversion system in which only one electrical input is given to the system to produce the working magnetic flux is known as a singly excited system. It is so called because there is only one excitation provided in the system. A schematic of a singly excited system is shown in the following figure.
  
• The singly excited system consists of a coil wound around a magnetic core and is connected to a source of power. In the singly excited system, the rotor made of a ferromagnetic material experiences a torque which rotates the rotor to position it to give minimum reluctance for the magnetic flux. The torque produced in the singly excited system is termed as reluctance torque or saliency torque.

What is a Doubly Excited System?

• A doubly excited system is the one in which two separate sources of excitation are provided to produce the magnetic field. One example of a doubly excited system is the synchronous motor, where one electrical excitation is provided to stator and the other to the rotor. A schematic of a doubly excited system is shown in the following figure.
  
• Here, the doubly excited system consists of a stator wound with a coil having a resistance of R1 and a rotor wound with a coil of resistance R2. Both the coils are excited by independent voltage sources.

The following table highlights the major differences between singly excited system and doubly excited system −

Conclusion

• The most significant difference between a singly excited system and a doubly excited system is that a singly excited system has only one coil to produce the desired excitation, whereas a doubly excited system consists of two independent coils to produce the desired excitation.