2. Dynamics of DC Machine

1. EMF

We know that from Faraday's Law, we have:

\[ e = -BS\omega \sin(\omega t) \]

This is an sinusoidal wave within the part \(e < 0\), to have a DC voltage, we need to make EMF positive. We can use a brush to realize this, it will change the direction for the rotors in every \(180^\circ\). And we have:

\[ e = \left\{\begin{aligned} &-BS\omega \sin(\omega t), &0 < \omega t - 2n \pi \leq \pi \\ &BS\omega \sin(\omega t), &\pi < \omega t - 2n \pi \leq 2\pi \end{aligned}\right. \]

And this can be further derived into: \(e = |-BS\omega \sin(\omega t)|\). We can draw the figure of EMF:

If we add one more winding, each winding have \(90^\circ\) angles that crossover each other, and the EMF is:

\[ e = |-BS\omega \sin(\omega t)| + |BS\omega \sin(\omega t + \frac{\pi}{2})| \]

We can draw the figure:

2. Machine Structure

The DC machine have the structure:

The flux and EMF passing through the rotor have the following relationship with the angle of the rotor:

To calculate the EMF,

\[ d\psi = \vec B N_{coil} l dx \]

\(l\) is the length of the armature
\(N_{coil}\) is the number of the coils
\(dx = Rd\theta\), \(R\) is the radius of the armature

\[ e = N_{coil} e_{coil} = N_{coil} \frac{d\phi}{dt} = N_{coil} \vec B N_{turns} R l \frac{d\theta}{dt} \]

\(N_{turns}\) is the number of turns in each coil

We can get the final equation of the EMF:

\[ e = N_{coils} \underbrace{N_{turns} \vec B R l}_{\psi_{ae}} \omega = \underbrace{K \psi_{ae}}_{K_e} \omega \]

\(\omega\) is the angular speed of the rotor
\(\psi_{ae}\) is the flux between the rotor and stator
\(K_{e}\) is a constant of the DC machine

3. Equivalent Circuit of DC Machine

We first assume an ideal DC Machine, which only have the EMF, the equivalent circuit of this machine is:

And we know that windings have the resistance and inductance, we can add these components into the circuit:

And this is the simplified circuit for DC machine armature windings.

4. Mechanical Characteristics of DC Machine

We look at the armature structures:

We can find that the current have opposite directions, if we short circuit the armature windings, we will get \(\sum e = 0\). By using the Lorentz Law, we can define the force generated by the DC machine,

The torque and the force have the following relationships:

\[ T = 2FR \]

This converts the electrical power into mechanical power,

Electrical Power: \(P_{elec} = vi\)
Mechanical Power: \(P_{mech} = T \omega = F v = \frac{W_m}{t}\)

If there have no power loss between electrical and mechanical conversion, there have:

\[ \begin{aligned} vi &= T\omega \\ ei &= T\omega \\ K\psi \omega i &= T\omega \\ K\psi i &= T \end{aligned} \]

And from the same equation, we can also get \(e = K\psi \omega\).

Now considering other components in the electric circuits,

\(P_{R} = i^2R\), \(R = \frac{\rho l}{A}\)
\(W_L = \frac12 Li^2\), \(L = \frac{N^2}{\Theta}\), where \(\Theta = \frac{l}{\mu A}\)

And the time constant is: \(\tau = \frac{L}{R}\).

The voltage of the armature winding is:

\[ v_a = R_a i_a + L_a \frac{di_a}{dt} + e \]

And we look at the stator part, we can also use a circuit to generate the magnetic fields,

Thus we have,

\[ v_e = R_e i_e + L_e \frac{di_e}{dt} \]

\(L_e = \frac{\psi_e}{i_e}\)

5. Dynamic Equations of DC Machine

The dynamic equations of DC machine are:

\[ \left\{\begin{aligned} &v_a = R_a i_a + L_a \frac{di_a}{dt} + e \\ &v_e = R_e i_e + L_e \frac{di_e}{dt} \\ &e = k \psi \omega\\ &T = k \psi i_a \end{aligned}\right. \]

And \(L_e >> L_a\) because the flux pass through more path on the iron, and the permeability of iron is much larger than the air.

For the mechanical part:

\[ T - T_{RES} = J \frac{d\omega}{dt} (+ G\omega) \]

\(T_{RES}\) is the external torque
\(T\) is the torque generated by the electrical machine
\(J\) is the moment of inertia
\(G\) is the damping effect coefficient

6. Transfer Function Model

Within the dynamic equations, we can write them into transfer function,

\[ \left\{\begin{aligned} &v_a = R_a i_a + L_a si_a + e \\ &v_e = R_e i_e + L_e si_e \\ &e = k \psi \omega\\ &T = k \psi i_a \\ &T - T_{RES} = (J s + G)\omega \end{aligned}\right. \]

7. System Schematics of the DC Machine

By obtaining the TF of the system, the schematics can be given:

And for the DC machine that using the excitation circuit, the schematics is: