## Step response of control system with PI controller

In this post, we will see the step response of control system with and without PI controller.

Step response without PI controller

A unity feedback closed loop system has a open loop transfer function $G(s)= \frac {2}{s^2+s+2}$  as shown below We can see that the steady state error for a unit step input  is 0.5 for the given system.

It is also seen that the response has settled at 0.5, so steady state error is 1-0.5=0.5 (Refer the response below) Step response with PI controller

We saw in the video lecture that PI controller is used to decrease the steady state error without affecting the stability.

Let us see, how the  PI controller denoted by $G_c(s)$ will improve the steady state error of the system G(s) described above. So after adding PI controller in the forward path , overall closed loop looks like the diagram below, where (R(s) is the input i.e unit step function, since we are interested in step response) Let $G_c(s)$ be $1+ \frac{1}{s}$

i.e $G_c(s) = \frac {s+1}{s}$

So the combined open loop transfer function (OLTF)  is $G(s) G_c(s)$ = $\frac{2s+2}{s^3+s^2+2s}$

After unit step input is applied as shown in the block diagram, the unit step response looks like below. So , we can see the difference in step response before and after adding Proportional Integral controller(PI). PI controller is used to improve steady state response in cases like these. Always remember that increasing the type of the system decreases the steady state error.

In this case, introducing PI controller in the forward path has helped us achieve  zero steady state error for step input instead of  0.5.  If you have any doubts regarding this topic, please use comments section.

## Control system questions

1. The breakaway point  of the root locus from the real axis of a  closed loop control system G(s)H(s) is (GATE 1995)

G(s)H(s) = $\frac {K (s+10)}{(s+2)(s+5)}$ lies

a) Between -2 and origin

b) Between -2 and -5

c) Between -10 and -infinity

d)  At infinity

2) A unity feedback system has open loop transfer function  G(s)H(s)= $\frac {K}{s(s+4)(s+16)}$ . Its root locus plot intersects the jw axis at

a) $\pm j2$

b) $\pm j4$

c) $\pm j8$

d) Does not intersect the jw axis

3) A unity feedback system has open loop transfer function  G(s)H(s)= $\frac {6K}{(s+1)(s+2)(s+3)}$ . What is the maximum value of K for which the given open loop TF is stable enough?

a) $\sqrt {11}$

b) 6

c) 10

d) 6 $\sqrt {11}$

4) The value of K for which the unity feedback system $G(s)$ = $\frac {K}{s(s+2)(s+4)}$ crosses the imaginary axis is

a) 8

b) 16

c) 48

d)64

…….

** BODE PLOT QUESTIONS***

5) The Bode plot for the gain magnitude of a minimum phase system G(s) is shown in the figure. The transfer function G(s) is a) $\frac {100}{(1+S/10)(1+S/250)}$

b) $\frac{40}{S(S+250)}$

c) $\frac {100}{(S+10)(S+250)}$

d) $\frac {100S}{(S+10)(S+250)}$

6) For a transfer function $D(s)=\frac {0.5s+1}{0.05s+1}$ . Maximum phase lead of the compensator is   (GATE EE-2000)

a) 52 deg at 4 rad/s

b) 52 deg at 10 rad/s

c) 55 deg at 12 rad/s

d) None of the above.

## GATE 2017 questions and answers

1. A bar of Gallium Arsenide(GaAs) is doped with silicon in such a way that silicon atoms occupy Gallium and Arsenic sites in GaAs crystal. Which of the following statements is true?

a) Silicon acts as p-type dopants in Arsenic sites and n-type dopants in Gallium sites

b) Silicon acts as n-type dopants in Arsenic sites and p-type dopants in Gallium sites

c) Silicon acts as p-type dopants in both Arsenic and Gallium sites

d) Silicon acts as n-type dopants in both Arsenic and Gallium sites

Correct Ans : a

Background: amphoteric dopant -> element which can act either as a donor or an acceptor in a given semiconductor; e.g. Si is an amphoteric dopant of GaAs where it acts as a donor on a Ga site or as an acceptor on an As site.

2) Hint : Apply Kirchoff’s law for the branch involving 10k, 5k and 3v source with vin =15v and for vin=-15v

3) A good transconductance amplifier should have

a) High input resistance and low output resistance

b) Low input resistance and high output resistance

c) High input and output resistance

d) Low input and output resistance

Correct Ans : c

4) Which one of the following is the general solution of this first order differential equation $\frac {dy}{dx}$= $(x+y-1)^2$ where x and y are real.

a) y=1+x + $tan^-1 (x+c)$

b) y=1+x+tan(x+c)

c) y=1-x+ $tan^-1 (x+c)$

d) y=1-x+tan(x+c)

where c is a constant

5) Correct Ans : 43.3 to 45.3

6) Assuming that transistors M1 and M2 are identical  and have a threshold voltage of 1V, the state of transistors M1 and M2  are respectively a) saturation, saturation

b) Linear, linear

c) saturation,linear

d) Linear, saturation

(more…)

## Control system | GATE Questions on Time domain analysis

1) For a second order system, damping ratio , 0< $\zeta$  <1 , the roots of the characteristic polynomial are

a) Real, but not equal

b) Real and equal

c) Complex conjugates

d) Imaginary

2) For a second order control system with the closed loop transfer function

T(s) = $\frac {9}{s^2+4s+9}$ , the settling time for  2 percent band is (In seconds)

a) 1.5

b)2

c) 3

d) 4

3) Consider a system with transfer function G(s)= $\frac {s+6}{Ks^2+s+6}$ . Its damping ratio will be 0.5 when the value of K is

a) 2/6

b) 3

c) 1/6

d) 6

4) A unity feedback control system has the open loop transfer function

G(s)= $\frac {4(1+2s)} {s^2(s+2)}$ . If the input to the system is a unit ramp, the steady state error will be

a) 0

b) 0.5

c) 2

d) Infinity

5) A causal system having the transfer function G(s) = $\frac {1}{s+2}$ is excited with 10 u(t). The time  at which the output reaches 99% of its steady state value is

a) 2.7 seconds

b) 2.5 seconds

c) 2.3 seconds

d) 2.1 seconds

6) A ramp input applied to an unity feedback system results in 5% steady state error. The type number and zero frequency gain of the system are

a) 1 and 20

b) 0 and 20

c) 0 and 1/20

d) 1 and 1/20

## Electromagnetic questions for GATE-2018

1) The dominant mode frequency  of a waveguide is $f_c$ with air  separating the parallel plates of waveguide. By introducing the dielectric with ( $\epsilon$ instead of air , the dominant frequency of the waveguide

a) Increases

b) Decreases

c) Remains same

d) Zero

2) Air filled waveguide has a dominant mode cut off frequency of 9 GHz. One of the dimensions of the waveguide is

a) 4.3 cm

b) 1.66 cm

c) 3.3 cm

d) 0.8 cm

3) A 50 ohm  characteristic impedance line is connected to load which has a reflection coefficient of 0.268. If vin=15v, net power delivered to load will be

a)0.139 W

b)1.39 W

c)0.278 W

d)2.78 W

4) A waveguide has a separation of 3 cm for the broader dimension and carries the dominant mode at an unknown frequency. If the wave impedance is 550 ohm, the unknown frequency f is

a) 7.66 GHz

b) 8.66  GHz

c) 6.66 GHz

d) 10.66 GHz

## Electromagnetic Waves GATE Questions and Answers

1.  A plane electromagnetic wave travelling along +z direction, it has its electric field given by $E_x$= 2 $cos(\omega t)$  and $E_y$= 2 $cos(\omega t+90)$

the wave is

a) Linearly polarized

b) Elliptically polarized

c) Left circularly polarized

d) Right circularly polarized

2) The electric field of an electromagnetic wave propagating in the positive z direction is $E$ = $i sin( \omega t$ $\beta z)$ $i sin( \omega t$ $\beta z +90)$ The wave is

a) Linearly polarized in the Z direction

b) Elliptically polarized

c) Left Hand circularly polarized

d) Right Hand circularly polarized

3) The electric field of a uniform plane electromagnetic wave in free space, along the positive x direction, is given by

E = $10(a_y +j a_z)$ $e^{-j25x}$

The frequency and polarization of the wave are

a) 1.2 GHz and left circular
b) 4 GHz and left circular
c) 1.2 GHz and right circular
d) 4 GHz and right circular

4) A plane wave propagating in the dielectric medium has an electric field given as
Ex  = E0Cos (2.6×1010t -100z). The phase velocity of plane wave is:

a) $10^6 m/s$

b) $2.6*10^8 m/s$

c) $3*10^8 m/s$

d) $2.6*10^{12} m/s$

## What is magnetic field intensity at centre of current carrying loop(Circle) 1) B= $\frac{\mu_oI}{4r}$

2) B= $\frac{\mu_oI}{2r}$

3) B= $\frac{\mu_oI}{2 \pi r}$

4) B= $\frac{\mu_oI}{4 \pi r}$