Answer MET Question 49

Question: A. Sketch an arrangement showing the principal of proportional plus integral (P+I) control loop.  

B. Compare the series and parallel resonance circuits. Find the frequency at which the following circuit resonates.

Answer:A.

A controller  attempts  to  minimize  the error by  adjusting  the  process  through  use  of  a manipulated variable.

Proportional Term
The proportional term produces an output value that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp, called the proportional gain constant.The proportional term is given by:
$\displaystyle \small \mathrm{P_{out} = K_{p}e(t)}$
A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable. In contrast, a small gain results in a small output response to a large input error, and a less responsive or less sensitive controller.  If  the proportional  gain  is  too  low,  the  control  action  may  be  too  small  when responding  to  system  disturbances.  Tuning  theory  and  industrial  practice  indicate  that  the proportional term should contribute the bulk of the output change.

Integral Term
The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The integral in a PID controller is the sum of the instantaneous error over time  and  gives  the  accumulated  offset  that  should  have  been  corrected  previously. The accumulated error is then multiplied by the integral gain $\displaystyle \small \mathrm{K_i}$  and added to the controller output. 
$\displaystyle \small \mathrm{I_{out}=K_i\int e(\tau)d\tau}$
The integral term accelerates the movement of the process towards set-point and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral  term  responds  to  accumulated  errors  from  the  past,  it  can  cause  the  present  value to over shoot the set-point value.

 

B. Resonance in the RLC circuit is the condition when reactances of capacitor and inductor coil are equal in magnitude.

Capacitors and Inductors are both components which can store energy: capacitors store it in an electric field and inductors in a magnetic field.

Ideal capacitors and inductors are assumed to have zero resistance and so have a purely imaginary impedance,

ZC=1jωC=−jωC and ZL=jωL

and their reactances to be, XC=−1/ωC and XL=ωL

Series Resonance Circuit	Parallel Resonance Circuit
A series resonance circuit has a capability to draw heavy currents and power from the mains. So it is regarded as an acceptor circuit.	A parallel resonance circuit has a capability to very small currents and power from the mains. So it is regarded as a rejector circuit.
Current at resonance is maximum and given by V/R	Current at resonance is minimum and given by VCR/L
Resonant frequency  = $\displaystyle \small \mathrm{\frac{1}{2\pi \sqrt{LC}}}$	Resonant frequency = $\displaystyle \small \mathrm{\frac{1}{2\pi \sqrt{\frac{1}{LC}-\left ( \frac{R}{L} \right )^2}}}$
Power factor is unity	Power factor is unity
Effective impedance is minimum and given by R	Effective impedance is maximum = $\displaystyle \small \mathrm{\frac{L}{CR}}$