What is the relationship between inverter and the encoder connect to it?

The rotary encoder can be used as the electric motor speed feedback or as a position feedback signal to the variable frequency drive, for speed closed-loop control, or position closed-loop control

In vector control, generally it needs closed-loop control, of course it can be open-loop vector control (but not good as closed loop)

First, we should understand the variable frequency drive function, the VFD is convert 50hz AC into controllable 0 - 200hz AC power. The encoder is sent the motor speed analog signal to the controller, by the input rotational speed of the digital subtracting the real-time speed of the photoelectric encoder of the converted digital quantity, then to get a vector control. The control amount is converted by the controller to corresponding AC power frequency, to achieve motor closed loop speed control. So the encoder and variable frequency drive is complementary each other, if the encoder failure, the controller will cause alarm due to the analog output failed. Means the real-time speed is 0 for the motor, the electric motor will accelerate all the time, until to the motor's critical speed value.

Encoder usually be treated as the induction motor speed or the position measuring device, is used to feedback the speed or position signal to the controller, and then to control motor speed.
Vector variable frequency drive for motor speed closed-loop control, often requires install motor encoder to get real-time speed feedback signal to the VFD, to complete the requirements of the closed-loop control.

Detect speed in vector control or detect position in positioning system. Just in those two conditions.

First of all , we should be known about encoder and vfd also.

encoder gives the fixed number of pulses in one revolution to the VFD . So that VFD can varify the speed and position of motor . And based on that we know about the motor position .