Unit - 3

Relationship between time and frequency response

Q1) Plot polar plot for T(S) = 1/S + 1

A1)

For polar plot substitute S=jw.

TF = 1/1 + jw

(2). Magnitude M = 1 + 0j / 1 + jw = 1/√1 + w2

(3). Phase φ = tan-1(0)/ tan-1w = - tan-1w

W M φ

0 1 00

1 0.707 -450

∞ 0 -900

The plot is shown in fig. 1

Fig 1 Polar Plot T(S) = 1/S + 1

Q2) Plot polar plot for T(S) = 1/(S+1) (S+2)

A2)

(1). S = jw

TF = 1/(1+jw) (2+jw)

(2). M = 1/(1+jw) (2+jw) = 1/-w2 + 3jw + 2

M = 1/√1 + w2√4 + w2

(3). Φ = - tan-1 w - tan-1(w/2)

W M Φ

0 0.5 00

1 0.316 -71.560

2 0.158 -108.430

∞ 0 -1800

The plot is shown in fig 2

Fig 2 Polar Plot for T(S) = 1/(S+1) (S+2)

Intersection of polar plot with imaginary axis will be when real part of Transfer function = 0

M = 1/ (jw + 1) (jw + 2)

= 1/-w2 + j3w + 2

Real part

Re(M) = 1/(2-w2) +j3w x (2-w2)-3jw/(2-w2)-3jw

Re(M) = (2-w2)/(2-w2+9w2) - 3jw/(2-w2) +9w2

Equating Real part = 0

(2-w2)/(2-w2) + 9w2 = 0

W = +-√2

For w=√2 real part on the polar plot becomes zero.

So, polar plot intersects imaginary axis at w=√2 at φ = -900

Q3) Plot polar plot for T(S) = 1/(S+1) (S+2) (S+3)

A3)

(1). Substitute S =jw

(2). M = 1/√1+w2 √4+w2 √9+w2

(3). Φ = -tan-1w – tan-1 w/2 – tan-1w/3

W M φ

0 0.16 0

1 0.1 -900

2 0.04 -142.10

∞ 0 -270

(4). Intersection of Polar plot with Real axis

M = 1/(S+1) (S+2) (S+3)

= 1/S3+6S2+11S+6

=1/(jw)2+6(jw)2+11jw+6

= 1/(6-6w2) +j(11w-w3)

Re(M) = 0 [Intersection with imaginary axis]

M = 6-6w2/(6-6w2)2+(11w-w2)2 - j(11w-w3)/(6-6w2)2+(11w-w2)2

Re(M) = 0

6-6w2 = 0

w=1

Im(M) = 0[Intersection with Real axis]

11w = w3

The plot is shown in fig 3

Fig 3 Polar Plot for T(S) = 1/(S+1) (S+2) (S+3)

Q4) For T(S) = 1/S(S+1) plot polar plot?

A4)

(1). M = 1/W√1+w2

(2). Φ = -900 - tan -1(W/T)

W M φ

0 ∞ -900

1 0.707 -1350

2 0.45 -153.40

∞ 0 -1800

The plot is shown in fig.4

Fig 4 Polar Plot for T(S) = 1/S(S+1)

Q5) For T(s) = 1/S2(S+1) plot polar plot?

A5)

(1). M = 1/w2√1+jw

(2). Φ = -1800 – tan-1W/T

The plot is shown in fig.5

Fig 5 Polar Plot for T(s) = 1/S2(S+1)

Q6) Sketch the bode plot for transfer function G(S) =

A6)

Replace S = j

G (j=

This is type 0 system. So initial slope is 0 dB decade. The starting point is given as

20 log10 K = 20 log10 1000

= 60 dB

Corner frequency 1 = = 10 rad/sec

2 = = 1000 rad/sec

Slope after 1 will be -20 dB/decade till second corner frequency i.e., 2 after 2 the slope will be -40 dB/decade (-20+(-20)) as there are poles

- For phase plot

= tan-1 0.1 - tan-1 0.001

For phase plot

100 -900

200 -9.450

300 -104.80

400 -110.360

500 -115.420

600 -120.00

700 -124.170

800 -127.940

900 -131.350

1000 -134.420

The plot is shown in figure.

Fig 6 Magnitude Plot for G(S) =

Q7) For the given transfer function determine G(S) =

Gain cross over frequency phase cross over frequency phase mergence and gain margin

A7)

Initial slope = 1

N = 1, (K)1/N = 2

K = 2

Corner frequency

1 = = 2 (slope -20 dB/decade

2 = = 20 (slope -40 dB/decade

2. phase

= tan-1 - tan-1 0.5 - tan-1 0.05

= 900- tan-1 0.5 - tan-1 0.05

1 -119.430

5 -172.230

10 -195.250

15 -209.270

20 -219.30

25 -226.760

30 -232.490

35 -236.980

40 -240.570

45 -243.490

50 -245.910

Finding gc (gain cross over frequency

M =

4 = 2 ( (

6 (6.25104) + 0.2524 +2 = 4

Let 2 = x

X3 (6.25104) + 0.2522 + x = 4

X1 = 2.46

X2 = -399.9

X3 = -6.50

For x1 = 2.46

gc = 3.99 rad/sec (from plot)

For phase margin

PM = 1800 -

= 900 – tan-1 (0.5×gc) – tan-1 (0.05 × gc)

= -164.50

PM = 1800 - 164.50

= 15.50

For phase cross over frequency (pc)

= 900 – tan-1 (0.5 ) – tan-1 (0.05 )

-1800 = -900 – tan-1 (0.5 pc) – tan-1 (0.05 pc)

-900 – tan-1 (0.5 pc) – tan-1 (0.05 pc)

Taking than on both sides

Tan 900 = tan-1

Let tan-1 0.5 pc = A, tan-1 0.05 pc = B

= 00

= 0

1 =0.5 pc 0.05pc

pc = 6.32 rad/sec

The plot is shown in figure.

Fig 7 Magnitude Plot for G(S) =

Q8) For the given transfer function G(S) = .Plot the rode plot. Find PM and GM

A8)

T1 = 0.5 1 = = 2 rad/sec

Zero so, slope (20 dB/decade)

T2 = 0.2 2 = = 5 rad/sec

Pole, so slope (-20 dB/decade)

T3 = 0.1 = T4 = 0.1

3 = 4 = 10 (2 pole) (-40 db/decade)

- Initial slope 0 dB/decade till 1 = 2 rad/sec
- From 1 to2 (i.e., 2 rad /sec to 5 rad/sec) slope will be 20 dB/decade
- From 2 to 3 the slope will be 0 dB/decade (20 + (-20))
- From 3,4 the slope will be -40 dB/decade (0-20-20)

Phase plot

= tan-1 0.5 - tan-1 0.2 - tan-1 0.1 - tan-1 0.1

500 -177.30

1000 -178.60

1500 -179.10

2000 -179.40

2500 -179.50

3000 -179.530

3500 -179.60

GM = 00

PM = 61.460

The plot is shown in figure.

Fig 8 Magnitude and Phase plot for G(S) =

Q9) For the given transfer function plot the bode plot (magnitude plot) G(S) =

A9)

Given transfer function

G(S) =

Converting above transfer function to standard from

G(S) =

=

- As type 1 system, so initial slope will be -20 dB/decade
- Final slope will be -60 dB/decade as order of system decides the final slope
- Corner frequency

T1 = , 11= 5 (zero)

T2 = 1, 2 = 1 (pole)

4. Initial slope will cut zero dB axis at

(K)1/N = 10

i.e., = 10

5. finding n and

T(S) =

T(S)=

Comparing with standard second order system equation

S2+2ns +n2

n = 11 rad/sec

n = 5

11 = 5

= = 0.27

5. Maximum error

M = -20 log 2

= +6.5 dB

6. As K = 10, so whole plot will shift by 20 log 10 10 = 20 dB

The plot is shown in figure.

Fig 9 Magnitude plot for G(S) =

Q10) For the given plot determine the transfer function

Fig 10 Magnitude plot for Q10

A10) From figure 11 we can conclude that

- Initial slope = -20 dB/decade so type -1
- Initial slope alls 0 dB axis at = 10 so

K1/N N = 1

(K)1/N = 10.

3. corner frequency

1 = = 0.2 rad/sec

2 = = 0.125 rad/sec

3. At = 5 the slope becomes -40 dB/decade, so there is a pole at = 5 as

Slope changes from -20 dB/decade to -40 dB/decade

4. At = 8 the slope changes from -40 dB/decade to -20 dB/decade hence

5. is a zero at = 8 (-40+(+20) =20)

6. Hence transfer function is

T(S) =

Q11) For T(S) = 1/S plot polar plot?

A11)

(1). S = jw

(2). M = 1/W

(3). Φ = -tan-1(W/O) = -900

W M φ

0 ∞ -900

1 1 -900

2 0.5 -900

∞ 0 -900

The plot is shown in fig.

Fig 11 Polar plot for T(S) = 1/S

Q12) For the transfer function below plot the Nyquist plot and also comment on stability? G(S) = 1/S+1

A12) N = Z – P (No pole of right half of S plane P = 0)

P = 0, N = Z

NYQUIST PATH:

P1 = W – (0 to - ∞)

P2 = ϴ (- π/2 to 0 to π/2)

P3 = W (+∞ to 0)

Fig 12

Substituting S = jw

G(jw) = 1/jw + 1

M = 1/√1+W2

Φ = -tan-1(W/I)

For P1: - W (0 to -∞)

W M φ

0 1 0

-1 1/√2 +450

-∞ 0 +900

Path P2:

W = Rejϴ R ∞ϴ -π/2 to 0 to π/2

G(jw) = 1/1+jw

= 1/1+j(Rejϴ) (neglecting 1 as R ∞)

M = 1/Rejϴ = 1/R e-jϴ

M = 0 e-jϴ = 0

Path P3:

W = -∞ to 0

M = 1/√1+W2, φ = -tan-1(W/I)

W M φ

∞ 0 -900

1 1/√2 -450

0 1 00

The Nyquist Plot is shown in fig.

Fig 13 Nyquist plot for G(S) = 1/S+1

From plot we can see that -1 is not encircled so, N = 0

But N = Z, Z = 0

So, system is stable.

Q13) For the transfer function below plot the Nyquist Plot and comment on stability G(S) = 1/ (S + 4) (S + 5)

A13) N = Z – P, P = 0, No pole on right half of S-plane

N = Z

NYQUIST PATH

P1 = W (0 to -∞)

P2 = ϴ (-π/2 to 0 to +π/2)

P3 = W (∞ to 0)

Path P1 W (0 to -∞)

M = 1/√42 + w2 √52 + w2

Φ = -tan-1(W/4) – tan-1(W/5)

W M Φ

0 1/20 00

-1 0.047 25.350

-∞ 0 +1800

Path P3 will be the mirror image across the real axis.

Path P2: ϴ (-π/2 to 0 to +π/2)

S = Rejϴ

G(S) = 1/ (Rejϴ + 4) ( Rejϴ + 5)

R∞

= 1/ R2e2jϴ = 0.e-j2ϴ = 0

The plot is shown in fig 14 From plot N=0, Z=0, system stable.

Fig 14 Nyquist plot for G(S) = 1/ (S + 4) (S + 5)

Q14) For the given transfer function, plot the Nyquist plot and comment on stability G(S) = k/S2(S + 10)?

A14) As the poles exists at origin. So, first time we do not include poles in Nyquist plot. Then check the stability for second case we include the poles at origin in Nyquist path. Then again check the stability.

PART – 1: Not including poles at origin in the Nyquist Path.

P1 W (∞ Ɛ) where Ɛ 0

P2 S = Ɛejϴ ϴ (+π/2 to 0 to -π/2)

P3 W = -Ɛ to -∞

P4 S = Rejϴ, R ∞, ϴ = -π/2 to 0 to +π/2

For P1

M = 1/w. w√102 + w2 = 1/w2√102 + w2

Φ = -1800 – tan-1(w/10)

W M Φ

∞ 0 -3 π/2

Ɛ ∞ -1800

Path P3 will be mirror image of P1 about Real axis.

G(Ɛ ejϴ) = 1/( Ɛ ejϴ)2(Ɛ ejϴ + 10)

Ɛ 0, ϴ = π/2 to 0 to -π/2

= 1/ Ɛ2 e2jϴ (Ɛ ejϴ + 10)

= ∞. e-j2ϴ [-2ϴ = -π to 0 to +π]

Path P2 will be formed by rotating through -π to 0 to +π

Path P4 S = Rejϴ R ∞ ϴ = -π/2 to 0 to +π/2

G(Rejϴ) = 1/ (Rejϴ)2(10 + Rejϴ)

= 0

N = Z – P

No poles on right half of S plane so, P = 0

N = Z – 0

Fig 15 Nyquist plot for G(S) = k/S2(S + 10)

But from plot shown in fig 15. It is clear that number of encirclements in Anticlockwise direction. So,

N = 2

N = Z – P

2 = Z – 0

Z = 2

Hence, system unstable.

PART 2 Including poles at origin in the Nyquist Path.

P1 W (∞ to Ɛ) Ɛ 0

P2 S = Ɛejϴ Ɛ 0 ϴ (+π/2 to +π to +3π/2)

P3 W (-Ɛ to -∞) Ɛ 0

P4 S = Rejϴ, R ∞, ϴ (3π/2 to 2π to +5π/2)

M = 1/W2√102 + W2, φ = - π – tan-1(W/10)

P1 W (∞ to Ɛ)

W M φ

∞ 0 -3 π/2

Ɛ ∞ -1800

P3(mirror image of P1)

P2 S = Ɛejϴ

G(Ɛejϴ) = 1/ Ɛ2e2jϴ (10 + Ɛejϴ)

Ɛ 0

G(Ɛejϴ) = 1/ Ɛ2e2jϴ (10)

= ∞. e-j2ϴϴ (π/2 to π to 3π/2)

-2ϴ = (-π to -2π to -3π)

P4 = 0

Fig 16 Nyquist Plot for G(S) = k/S2(S + 10)

The plot is shown in fig. From the plot it is clear that there is no encirclement of -1 in Nyquist path. (N = 0). But the two poles at origin lies to the right half of S-plane in Nyquist path (P = 2) [see path P2]

N = Z – P

0 = Z – 2

Z = 2

Hence, system is unstable.

Path P2 will be formed by rotating through -π to -2π to -3π

Q15) Explain relation between time and frequency response?

A15)

The calculation of system parameters for higher order systems is difficult in time domain analysis. This can be overcome by frequency domain analysis. As the frequency response of a linear time invariant system is independent of amplitude and phase of the input test signal. The effect of noise and parameter variation in frequency domain can be easily computed.

In frequency response can be determine by calculating the phase and amplitude oof the given system transfer function. If a sinusoidal signal XG(j)sint then its amplitude will be

C(t) = XG(j)sint

= X|G(j)|sint

Magnitude = |G(j)|

Phase =

In polar form = |G(j)|

The transfer function of second order system is shown as

C(S)/R(S) = W2n / S2 + 2ξWnS + W2n - - (1)

ξ = Ramping factor

Wn = Undamped natural frequency for frequency response let S = jw

C(jw) / R(jw) = W2n / (jw)2 + 2 ξWn(jw) + W2n

Let U = W/Wn above equation becomes

T(jw) = W2n / 1 – U2 + j2 ξU

So,

| T(jw) | = M = 1/√(1 – u2)2 + (2ξU)2 - - (2)

T(jw) = φ = -tan-1[ 2ξu/(1-u2)] - - (3)

For sinusoidal input the output response for the system is given by

C(t) = 1/√(1-u2)2 + (2ξu)2Sin[wt - tan-1 2ξu/1-u2] - - (4)

The frequency where M has the peak value is known as Resonant frequency Wn. This frequency is given as (from eqn (2)).

DM/du|u=ur = Wr = Wn√(1-2ξ2) - - (5)

From equation (2) the maximum value of magnitude is known as Resonant peak.

Mr = 1/2ξ√1-ξ2 - - (6)

The phase angle at resonant frequency is given as

Φr = - tan-1 [√1-2ξ2/ ξ] - - (7)

As we already know for step response of second order system the value of damped frequency and peak overshoot are given as

Wd = Wn√1-ξ2 - - (8)

Mp = e- πξ2|√1-ξ2 - - (9)

Fig 17 Frequency Domain Specification

The comparison of Mr and Mp is shown in figure. The two performance indices are correlated as both are functions of the damping factor ξ only. When subjected to step input the system with given value of Mr of its frequency response will exhibit a corresponding value of Mp.

Similarly the correlation of Wr and Wd is shown in figure for the given input step response [ from eqn(5) & eqn(8) ]

Wr/Wd = √(1- 2ξ2)/(1-ξ2)

Mp = Peak overshoot of step response

Mr = Resonant Peak of frequency response

Wr = Resonant frequency of Frequency response

Wd = Damping frequency of oscillation of step response.

From fig (17) it is clear that for ξ> 1/2, value of Mr does not exists.

Q16) For the following system shown below find the phase margin and gain margin if the value of k=500by using Nyquist plot.

A16)

G(s)H(s) = 500/s(s+6) (s+9)

The polar plot for above OLTF is shown in figure below.

Fig 18 Polar Plot for given system

From figure we see A=0.6,

The GM in dB = -20logA.

= 4.43dB

PM = 1800-1650

= 150

Q17) For the system shown in figure, find the GM and PM using Nyquist plot for k=10. Find also range of k for stability?

A17)

The transfer function is G(s) H(s) = 10(s+2(s+4)/s(s2 – 3s+10)

As the OLTF is unstable system so PM and GM concepts cannot be applied.

G(jω) H(jω) = 10(jω+2) (jω+4)/ jω(-ω2 -3 jω+10)

| G(jω) H(jω)|= 10(√4+ω2)( √16+ω2)/ ω√[(10-ω)2+9 ω2]

G(jω)H(jω) = -900 +tan-1(ω/2) + tan-1(ω/4) - tan-1 (-3ω/10- ω2)

The polar plot for above system is shown below.

Fig 19 Polar plot for the given system

From figure we have A=2.6 and ɸ =-1350

The GM in dB = -20logA.

= -8.3dB

As GM is negative it indicates that system is open loop unstable.

PM = 1800-1650 = 450

Q18) Write short note on Nicholas chart?

A18)

The Nichols plot is named after the American engineer N.B Nichols who formulated this plot. Compensators can be designed using Nichol’s plot. Nichols plot technique is however also used in designing of dc motor. This is used in signal processing and control design. Nyquist plot in complex plane shows how phase of transfer function and frequency variation of magnitude are related. • Angle of positive real axis determines the phase and distance from origin of complex plane determines the gain.

Advantages

- Gain and phase margin can be determined easily and also graphically.
- Closed loop frequency response is obtained from open loop frequency response.
- Gain of the system can be adjusted to suitable values.
- Nichol’s chart provides frequency domain specifications.

Disadvantage

Using Nichols plot small changes in gain cannot be encountered easily.

Q19) What are M circles? Derive its equation for forming circle?

A19)

Constant magnitude loci that are M-circles and constant phase angle loci that are N-circles are the fundamental components. The constant M and constant N circles in G (jω) plane can be used for the analysis and design of control systems. However, the constant M and constant N circles in gain phase plane are prepared for system design and analysis as these plots supply information with fewer manipulations. Gain phase plane is the graph having gain in decibel along the ordinate (vertical axis) and phase angle along the abscissa (horizontal axis).

For Constant M Circle

For a system with unity feedback having transfer function

T(s)= G(s)/1+G(s)

Replace s=jω

T(jω) = G(jω)/1+G(jω)

But we can write G(jω) = X(jω) +jY(jω)

The closed loop transfer function T(s) is

T(jω) = X+jY/1+X+jY= Mejα

M= |X+jY|/|1+X+jY|

M2=X2+Y2/(1+X2) + Y2

Solving above equation we get

[X-(M2/1-M2)]2 + Y2 = ([M/1-M2)]2

This is equation of circle with centre (M2/1-M2),0 and radius M/1-M2

When M<1, M decreases the radius of the circle decreases and centres located on positive real axis shifts towards origin. When M=0 the radius becomes 0 and the centre (0,0). This is shown in figure above.

Fig 20 Constant M circle