The transfer function voltage-velocity, G_{1}\left ( s \right ), is given by:
G_{1}\left ( s \right )=\frac{\Omega\left ( s \right ) }{V\left ( s \right )}=\frac{K_{t}}{\left ( s\cdot L_{a} +R_{a}\right )\left ( s\cdot J+B \right )+K_{e}\cdot K_{t}}
Multiplying G_{1}\left ( s \right ) by \frac{1}{s}, the transfer function voltage-position, H_{1}\left ( s \right ) , is obtained:
H_{1}\left ( s \right )=\frac{\Omega\left ( s \right ) }{V\left ( s \right )}\cdot \frac{1}{s}=\frac{K_{t}}{s\cdot \left ( s\cdot L_{a} +R_{a}\right )\left ( s\cdot J+B \right )+s\cdot K_{e}\cdot K_{t}}
The transfer function load torque-velocity, G_{2}\left ( s \right ) , is given by:
G_{2}\left ( s \right )=\frac{\Omega \left ( s \right )}{T_{f}\left ( s \right )+T_{L}\left ( s \right )}=\frac{-\frac{1}{s\cdot J+B}}{1+\frac{K_{e}\cdot K_{t}}{\left ( s\cdot L_{a} +R_{a}\right )\left ( s\cdot J+B \right )}}
Multiplying G_{2}\left ( s \right ) by \frac{1}{s} , the transfer function voltage-position, H_{2}\left ( s \right ) , is obtained :
H_{2}\left ( s \right )=\frac{\Omega \left ( s \right )}{T_{f}\left ( s \right )+T_{L}\left ( s \right )}\cdot \frac{1}{s}=\frac{-\frac{1}{s\cdot J+B}}{s+\frac{s\cdot K_{e}\cdot K_{t}}{\left ( s\cdot L_{a} +R_{a}\right )\left ( s\cdot J+B \right )}}
The system can be also represented using the state space model. The state variables are the angular velocity and the armature current. The input is the armature voltage and the output is the angular velocity.
\frac{d}{dt}\begin{bmatrix}
\dot{\theta }\left ( t \right )\\
i\left ( t \right )
\end{bmatrix} = \begin{bmatrix}
-\frac{B}{J} & \frac{K_{t}}{J}\\
-\frac{K_{e}}{L_{a}} & \frac{R_{a}}{L_{a}}
\end{bmatrix}\cdot \begin{bmatrix}
\dot{\theta }\left ( t \right )\\
i\left ( t \right )
\end{bmatrix}+\begin{bmatrix}
0\\
\frac{1}{L_{a}}
\end{bmatrix}\cdot v\left ( t \right )
y\left ( t \right ) = \begin{bmatrix}
1 & 0
\end{bmatrix}\cdot \begin{bmatrix}
\dot{\theta }\left ( t \right )\\
i\left ( t \right )
\end{bmatrix}
Enter the motor parameter values to determine the two models.
