DC Motor Math Model

1 Nomenclature

nom

2 Diagram 

dcmotor

3 System equations

3.1 Electrical

Back emf is directly proportional to rotational speed of shaft:

e \propto \dot{\theta} \rightarrow e = K_{e} \dot{\theta}

Where:

e = back emf

\dot{\theta} = rotational speed of stator

K_{e} = constant

From Kirchhoff’s voltage law:

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

\sum_{k=1}^{n} V_{k} = 0

Therefore:

L \frac{di}{dt} + Ri = V - K_{e} \dot{\theta}

3.2 Mechanical

Torque is directly proportional to current and electric field strength:

Assuming armature controlled motor therefore electric field strength is constant

T \propto i \rightarrow  T = K_{t}i

Where:

T = torque

i = current

K_{t} = 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

\sum_{k=1}^{n} T_{k} = I \Ddot{\theta}

Therefore:

J \Ddot{\theta} + b\dot{\theta} = K_{t}i

4 Transfer function

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

Laplace transform equation:

F(s) = \mathcal{L} \left[ f(t) \right] = \int_{0}^{\infty} f(t)e^{-st}dt

General Laplace transforms:

\mathcal{L} [\dot y(t)] = sY(s) - y(0)

\mathcal{L} {\Ddot y(t)} = s^2Y(s) - sy(0) - y(0)

4.1 Electrical

\mathcal{L} \left[L \frac{di}{dt} + Ri\right] = \mathcal{L} \left[V - K_{e}\right]

Assuming i(0) = 0

sI(s)L + RI(s) = V(s) - K_{e}s \theta(s)

I(s)(Ls + R) = V(s) - K_{e}s \theta(s)

4.2 Mechanical

\mathcal{L} \left[J \Ddot{\theta} + b\dot{\theta}\right] = \mathcal{L} \left[K_{t}i\right]

Assuming

\theta(0) = 0

s(Js+b) \ominus(s) = K_{t}I(s)

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

G(s) = \frac{\dot\ominus(s)}{V(s)}

Voltage

From Equation I(s)(Ls + R) = V(s) - K_{e}s \theta(s)

V(s) = I(s)(Ls+R)+K_{e}s\ominus(s)

Rotational speed:

\dot\ominus(s) = \ominus(s)s

From Equation s(Js+b) \ominus(s) = K_{t}I(s) & Equation \dot\ominus(s)=\ominus(s)s

\dot\ominus(s) = \frac{K_{e}I(s)}{Js+b}

Open loop transfer function for speed control:

From Equation V(s)=I(s)(Ls+R)+K_{e}s\ominus(s) & Equation \dot\ominus(s)=\frac{K_{e}I(s)}{Js+b}:

G(s) = \frac {I(s)K_{e}} {I(s)((Js+b)[(Ls+R)+K_{t} \dot \ominus(s)])}

From Equation s(Js+b) \ominus(s) = K_{t}I(s)

K_{t} = \frac{\dot \ominus(s)(Js+b)} {I(s)}

From Equation K_{t} = \frac{\dot \ominus(s)(Js+b)} {I(s)}  & Equation \dot\ominus(s)=\ominus(s)s

G(s) = \frac {K_{e}} {(Js+b)(Ls+R)+K_{t}^2}

Open loop transfer function for position control:

From Equation \dot\ominus(s)=\ominus(s)s   & Equation G(s)=\frac{K_{e}}{(Js+b)(Ls+R)+K_{t}^2}

G(s) = \frac {sK_{e}} {(Js+b)(Ls+R)+K_{t}^2}

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