What is the order of the characteristic equation of an armature controlled DC motor? 

Explanation:

Characteristic Equation of Armature Controlled DC Motor

Definition: The characteristic equation of a system represents the mathematical relationship between input and output variables, governing the system's dynamic behavior. For an armature-controlled DC motor, the characteristic equation is derived based on the electrical and mechanical dynamics of the motor.

Working Principle: In an armature-controlled DC motor, the armature voltage is varied to control the motor's speed, while the field current is kept constant. The motor's behavior can be described by equations that relate the electrical and mechanical components, including the armature circuit, back EMF, torque, and rotational dynamics.

The system's characteristic equation is typically obtained from the transfer function, which is derived using Kirchhoff's Voltage Law (KVL) for the armature circuit and Newton's Second Law for rotational dynamics.

Mathematical Derivation:

For an armature-controlled DC motor:

• Armature Circuit: Applying KVL:
  \( V_a = I_aR_a + L_a\frac{dI_a}{dt} + E_b \)
  Where:
  • \( V_a \): Armature voltage
  • \( I_a \): Armature current
  • \( R_a \): Armature resistance
  • \( L_a \): Armature inductance
  • \( E_b \): Back EMF (\( E_b = K_e\omega \))
  • \( \omega \): Angular velocity
  • \( K_e \): Motor constant
• Mechanical Equation: Using Newton's Second Law:
  \( J\frac{d\omega}{dt} + B\omega = T \)
  Where:
  • \( J \): Moment of inertia
  • \( B \): Damping coefficient
  • \( \omega \): Angular velocity
  • \( T \): Developed torque (\( T = K_tI_a \))
  • \( K_t \): Torque constant

Combining these equations and eliminating intermediate variables results in a first-order differential equation relating the system's input (\( V_a \)) to its output (\( \omega \)). This equation represents the characteristic equation of the armature-controlled DC motor.

Correct Option Analysis:

The correct option is:

Option 1: First order equation

This option correctly represents the characteristic equation of an armature-controlled DC motor. The system dynamics involve the electrical and mechanical components, which combine to form a first-order differential equation. The inductance of the armature and the moment of inertia contribute to this first-order behavior.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: Second order equation

This option is incorrect because the characteristic equation of the armature-controlled DC motor is not second-order. A second-order system requires two energy storage elements (e.g., inductance and capacitance in an electrical system or inertia and elasticity in a mechanical system). In this case, the motor's dynamics involve only one energy storage element, resulting in a first-order system.

Option 3: Zero order equation

A zero-order equation implies that the output is directly proportional to the input without any dynamic behavior. This is not applicable to an armature-controlled DC motor, as the motor exhibits dynamic behavior governed by differential equations involving time-dependent terms.

Option 4: Third order equation

A third-order equation would arise in systems with three energy storage elements or complex dynamics involving higher-order derivatives. The armature-controlled DC motor does not exhibit such complexity, as its dynamics are adequately described by a first-order equation.

Conclusion:

The armature-controlled DC motor's characteristic equation is first-order, as derived from the combination of electrical and mechanical equations governing the motor's dynamics. Understanding these dynamics is crucial for designing control systems and analyzing motor performance. Other options are incorrect as they do not accurately represent the system's behavior.