State three merits of operating two three-phase induction motors in cascade.

Step 1: Calculate the initial back EMF ($E_{b1}$). The back EMF of a DC shunt motor is given by the formula $E_b = V - I_a R_a$. Given: $V = 300\, \text{V}$ $I_{a1} = 30\, \text{A}$ $R_a = 0.5\, \Omega$

$E_{b1} = V - I_{a1} R_a$ $E_{b1} = 300\, V - (30\, A)(0.5\, \Omega)$ $E_{b1} = 300\, V - 15\, \text{V}$ $E_{b1} = 285\, \text{V}$

Step 2: Determine the armature current for the new speed. For a DC shunt motor, torque $T$ is proportional to the product of flux $\Phi$ and armature current $I_a$, i.e., $T \propto \Phi I_a$. The problem states that the flux $\Phi$ remains constant and the load torque $T_L$ is kept constant. Therefore, the armature current must also remain constant. $I_{a2} = I_{a1} = 30\, \text{A}$

Step 3: Calculate the back EMF ($E_{b2}$) at the new speed. The back EMF is also proportional to the product of flux $\Phi$ and speed $N$, i.e., $E_b \propto \Phi N$. Since the flux $\Phi$ is constant, $E_b \propto N$. We can write the ratio: $\frac{E_{b2}}{E_{b1}} = \frac{N_2}{N_1}$ Given: $N_1 = 1500\, \text{r.p.m}$ $N_2 = 800\, \text{r.p.m}$

$E_{b2} = E_{b1} \left(\frac{N_2}{N_1}\right)$ $E_{b2} = 285\, V \left(\frac{800\, r.p.m}{1500\, r.p.m}\right)$ $E_{b2} = 285\, V \left(\frac{8}{15}\right)$ $E_{b2} = 19\, V \times 8$ $E_{b2} = 152\, \text{V}$

Step 4: Determine the total armature resistance required. Let $R_{ext}$ be the external resistance added in series with the armature. The total armature circuit resistance will be $R_a + R_{ext}$. Using the back EMF formula for the new condition: $E_{b2} = V - I_{a2} (R_a + R_{ext})$ $152\, V = 300\, V - 30\, A (0.5\, \Omega + R_{ext})$

Step 5: Solve for the external resistance $R_{ext}$. Rearrange the equation to solve for $R_{ext}$: $30\, A (0.5\, \Omega + R_{ext}) = 300\, V - 152\, \text{V}$ $30\, A (0.5\, \Omega + R_{ext}) = 148\, \text{V}$ $0.5\, \Omega + R_{ext} = \frac{148\, V}{30\, A}$ $0.5\, \Omega + R_{ext} = 4.9333... \, \Omega$ $R_{ext} = 4.9333... \, \Omega - 0.5\, \Omega$ $R_{ext} = 4.4333... \, \Omega$

Rounding to two decimal places: $R_{ext} \approx 4.43\, \Omega$

The value of resistance to be added in series with the armature is $\boxed{4.43 \Omega}$.