Critical resistance, critical speed and shunt generator characteristics

Critical field resistance

The critical field resistance $R_c$ is the maximum resistance of the shunt field circuit at which a self-excited shunt generator will just build up voltage at a given speed. Graphically it is the slope of the tangent drawn from the origin to the initial (linear) part of the OCC.

  E ^      Rc line (tangent)
    |     /      Rf < Rc line
    |    /      /
    |   /     /   ____---- OCC (speed N)
    |  /    / .--'
    | /   /.-'   builds up here
    |/  ./
    | ./     Rf > Rc: no build-up
    |/
    +----------------------> If

• $R_f < R_c$: field line cuts the OCC at a high voltage, so the machine builds up.
• $R_f > R_c$: field line lies above the OCC; voltage stays near residual value.

Critical speed

The critical speed $N_c$ is the speed at which a given field resistance $R_f$ becomes the critical resistance, i.e. the minimum speed at which the generator just builds up. Since the OCC ordinates are proportional to speed:

$$\frac{N_c}{N} = \frac{R_f}{R_c} \quad\Rightarrow\quad N_c = N\,\frac{R_f}{R_c}$$

  E ^        Rf line
    |       /     OCC at N
    |      /   ___----
    |     /.--'
    |    //    OCC at Nc: tangent = Rf line
    |   //  __---
    |  //.-'
    | /.'     OCC below Nc: no build-up
    |/
    +----------------------> If

Example: $R_c = 220\ \Omega$ at 1500 rpm, $R_f = 176\ \Omega$: $N_c = 1500 \times 176/220 = 1200$ rpm.

Performance characteristics of DC shunt generator

1. Open-circuit characteristic (OCC, $E_0$ vs $I_f$): starts at the residual voltage, rises linearly, then bends due to saturation.

2. External characteristic ($V$ vs $I_L$):

  V ^
 E0 +----.___
    |        ''--.__        (a) drop: IaRa
    |               '--.    (b) drop: armature reaction
    |                   '.  (c) drop: lower If
    |                     \
    |              Ic   ,-'  turn-back
    |           <------'
    +------------------------> IL
    0     (short circuit current small)

• Terminal voltage falls as load increases, because of (a) armature resistance drop $I_aR_a$, (b) armature reaction weakening flux, and (c) reduced field current since $I_f = V/R_f$ falls with $V$.
• Beyond a critical load current $I_c$ (load resistance below critical), the curve turns back: further decrease in load resistance lowers both $V$ and $I_L$. On a dead short circuit, the current is small (only residual EMF acts), so a shunt generator protects itself.

3. Internal characteristic ($E$ vs $I_a$): lies above the external curve by $I_aR_a$, showing the effect of armature reaction and reduced field current only.

Comment: the shunt generator gives an almost constant voltage for normal loads (voltage drop about 5-15% from no load to full load), so it is used for battery charging, excitation of alternators and lighting where moderate regulation is acceptable.