Learning 1:

A good general structure to present thought process of creating a math model of a system (especially physical) is:

1. Nomenclature

2. Diagram(s)

3. System equation(s)

# Category: Math model

# DC Motor Math Model

**1 Nomenclature**

**2 Diagram **

**3 System equations**

**3.1 Electrical**

**Back emf is directly proportional to rotational speed of shaft:**

Where:

e = back emf

= rotational speed of stator

= constant

**From Kirchhoff’s voltage law:**

The directed sum of the potential differences (voltages) around any closed loopis zero.

Therefore:

**3.2 Mechanical**

**Torque is directly proportional to current and electric field strength:**

Assuming armature controlled motor therefore electric field strength is constant

Where:

T = torque

i = current

= constant

**From Newton’s Second Law for Rotation:**

If more than one torque acts on a rigid body about a fixed axis, then the sum of the torques equals the moment of inertia times the angular acceleration

**Therefore:**

**4 Transfer function**

In SI units, the motor torque and back emf constants are equal

**Laplace transform equation:**

**General Laplace transforms:**

**4.1 Electrical**

Assuming i(0) = 0

**4.2 Mechanical**

Assuming

**Open loop transfer function given input is voltage and output is rotational speed:**

**Voltage**

From Equation

**Rotational speed:**

From Equation & Equation

**Open loop transfer function for speed control:**

From Equation & Equation :

From Equation

From Equation & Equation

**Open loop transfer function for position control:**

From Equation & Equation