Question 6.17: An Armature-Controlled DC Motor Consider the dynamic system ...

a. This electromechanical system includes an armature circuit

R_{\mathrm{a}} i_{\mathrm{a}}=v_{\mathrm{a}}-\mathrm{e}_{\mathrm{b}}

a rotational system

I_{\mathrm{m}} \dot{\omega}_{\mathrm{m}}+B_{\mathrm{m}} \omega_{\mathrm{m}}=\tau_{\mathrm{m}}

and couplings between the electrical and mechanical subsystems

\mathrm{e}_{\mathrm{b}}=K_{\mathrm{e}} \omega_{\mathrm{m}} \quad \tau_{\mathrm{m}}=K_{\mathrm{t}} i_{\mathrm{a}}

Following Figure 6.45, we can construct a Simulink block diagram (see Figure 6.67), which shows the major components of the DC motor and their interconnections. The dynamics of the mechanical rotational system is represented using a Transfer Fen block. The armature resistance, torque constant, and back emf constant are represented using Gain blocks.

b. Taking Laplace transform of the two equations given and denoting s \Theta_{\mathrm{m}}(s)=\Omega_{\mathrm{m}}(s) provides

\begin{gathered} R_{\mathrm{a}} I_{\mathrm{a}}(s)+K_{\mathrm{e}} \Omega(s)=V_{\mathrm{a}}(s), \\ I_{\mathrm{m}} s \Omega_{\mathrm{m}}(s)+B_{\mathrm{m}} \Omega_{\mathrm{m}}(s)-K_{\mathrm{t}} I_{\mathrm{a}}(s)=0 . \end{gathered}

Using Cramer’s rule to solve for the transfer functions I_{\mathrm{a}}(s) / V_{\mathrm{a}}(s) and \Omega_{\mathrm{m}}(s) / V_{\mathrm{a}}(s) yields

\begin{gathered} \frac{I_{\mathrm{a}}(s)}{V_{\mathrm{a}}(s)}=\frac{I_{\mathrm{m}} s+B_{\mathrm{m}}}{R_{\mathrm{a}} I_{\mathrm{m}} s+R_{\mathrm{a}} B_{\mathrm{m}}+K_{\mathrm{t}} K_{\mathrm{e}}}, \\ \frac{\Omega_{\mathrm{m}}(s)}{V_{\mathrm{a}}(s)}=\frac{K_{\mathrm{t}}}{R_{\mathrm{a}} I_{\mathrm{m}} s+R_{\mathrm{a}} B_{\mathrm{m}}+K_{\mathrm{t}} K_{\mathrm{e}}}, \end{gathered}

both of which can easily be represented using a Transfer Fcn block (see Figure 6.68).

c. Let x_{1}=\theta_{\mathrm{m}}, x_{2}=\dot{\theta}_{\mathrm{m}^{\prime}} and u=v_{\mathrm{a}} and the state-variable equations are

or in matrix form

\left\{\begin{array}{l} \dot{x}_{1} \\ \dot{x}_{2} \end{array}\right\}=\left[\begin{array}{cc} 0 & 1 \\ 0 & -\left(\frac{K_{\mathrm{t}} K_{\mathrm{e}}}{I_{\mathrm{m}} R_{\mathrm{a}}}+\frac{B_{\mathrm{m}}}{I_{\mathrm{m}}}\right) \end{array}\right]\left\{\begin{array}{l} x_{1} \\ x_{2} \end{array}\right\}+\left[\begin{array}{c} 0 \\ \frac{K_{\mathrm{t}}}{I_{\mathrm{m}} R_{\mathrm{a}}} \end{array}\right] u .

The output equations are

\begin{gathered} y_{1}=i_{\mathrm{a}}=\frac{v_{\mathrm{a}}-K_{\mathrm{e}} \dot{\theta}_{\mathrm{m}}}{R_{\mathrm{a}}}=-\frac{K_{\mathrm{e}}}{R_{\mathrm{a}}} x_{2}+\frac{1}{R_{\mathrm{a}}} u, \\ y_{2}=\dot{\theta}_{\mathrm{m}}=x_{2} \end{gathered}

or in matrix form

\left\{\begin{array}{l} y_{1} \\ y_{2} \end{array}\right\}=\left[\begin{array}{cc} 0 & -\frac{K_{\mathrm{e}}}{R_{\mathrm{a}}} \\ 0 & 1 \end{array}\right]\left\{\begin{array}{l} x_{1} \\ x_{2} \end{array}\right\}+\left[\begin{array}{c} \frac{1}{R_{\mathrm{a}}} \\ 0 \end{array}\right] u

The system can be represented using a State-Space block (see Figure 6.69) with A, B, C, D matrices defined above. A Demux block from the Simulink library of Signal Routing is used to split the output vector signal into two scalar signals i_{\mathrm{a}}(t) and \omega_{\mathrm{m}}(t).

d. The Simscape block diagram of the DC motor (see Figure 6.70) consists of elements from two domains, electrical and mechanical rotational. Note that each domain requires at least one reference block. As shown in Figure 6.70, both Electrical Reference and Mechanical Rotational Reference blocks are attached to the appropriate circuit.

Define all the parameters in MATLAB Workspace and run the Simulink or Simscape models. The result can be plotted as shown in Figures 6.71 and 6.72 .