# Equivalent Circuit of a Double Cage Induction Motor

For a double cage induction motor, the suffixes "s" and "r" are used to denote stator and rotor respectively.

Let,

• 𝑅𝑠 = Stator resistance per phase
• 𝑋𝑠 = Stator reactance per phase
• 𝑅′𝑟𝑜 = Rotor resistance per phase of outer cage referred to stator
• 𝑋′𝑟𝑜 = Standstill leakage reactance per phase of outer cage referred to stator
• 𝑅′𝑟𝑖 = Rotor resistance per phase of inner cage referred to stator
• 𝑋′𝑟𝑖 = Standstill leakage reactance per phase of inner cage referred to stator
• 𝑠 = Fractional slip

If it is considered that the main flux completely links both the cages, then the impedances of the two cages can be considered in parallel. Figure-1 shows the equivalent circuit of the double cage induction motor at slip s. Here, the impedances of the rotor are given by,

$$\mathrm{Outer\:cage\: impedace,\: 𝑍′_{𝑟𝑜} =\frac{𝑅′_{𝑟0}}{𝑠}+ 𝑗𝑋′_{𝑟𝑜}}$$

$$\mathrm{Inner \:cage \:impedace,\: 𝑍\:′_{𝑟𝑖} =\frac{𝑅′_{𝑟𝑖}}{𝑠}+ 𝑗𝑋′_{𝑟𝑖}}$$

And, the impedance of the stator is given by,

$$\mathrm{Impedace\: of \:the\: stator, \:𝑍_𝑠 = 𝑅_𝑠 + 𝑗𝑋_𝑠}$$

In the equivalent circuit, if the parallel branches containing R0 and X0 are neglected, then the approximate equivalent circuit of the double cage induction motor is obtained as shown in the figure below. The equivalent impedance per phase of the motor referred to stator is given by,

$$\mathrm{𝑍_{𝑒𝑠} = 𝑍_𝑠 + (𝑍′_{𝑟𝑜} || 𝑍′_{𝑟𝑖})}$$

$$\mathrm{⇒ 𝑍_{𝑒𝑠} = (𝑅_𝑠 + 𝑗𝑋_𝑠) + (\frac{𝑍′_{𝑟𝑜} 𝑍′_{𝑟𝑖}}{𝑍′_{𝑟𝑜}+ 𝑍′_{𝑟𝑖}})}$$

The current through the outer cage of the rotor is given by,

$$\mathrm{𝐼′_{𝑟𝑜} =\frac{𝐸′_𝑟}{𝑍′_{𝑟𝑜}}}$$

And the current through the inner rotor is given by,

$$\mathrm{𝐼′_{𝑟𝑖} =\frac{𝐸′_𝑟}{𝑍′_{𝑟𝑖}}}$$

Therefore, the rotor current referred to the stator is equal to the phasor sum of the currents through the outer and inner cages, i.e.,

$$\mathrm{𝐼′_𝑟 = 𝐼′_{𝑟𝑜} + 𝐼′_{𝑟𝑖}}$$