Approximations of Bode diagrams by asymptotes (review)

Theoretical aspects

Each transfer function can be written as a ratio of polynomials given in the product form where real

singularities and pairs of complex conjugates singularities are involved.

Using approximations for magnitude and phase for terms of first order and second order results in good

approximation for the overall magnitude and phase for the considered system.

To obtain a good approximation, student must be able to understand and then apply the next

“remarkable” results:

- pole in the origin, 1

𝑠, with the magnitude: a slope of -20dB with the cut off frequency in 1 rad/sec

and a phase shift of −900 for the entire positive range of frequencies;

Figure 1 Bode Approximation for the “integrator”, 𝑯(𝒔) =𝒌

𝒔

- real and negative pole, 1

𝑇𝑠+1, with magnitude: a slope of -20dB after the corner frequency,

1

𝑇, and

phase having 3 points of interest: (0.11

𝑇, −50) , (

1

𝑇, −450) , (10

1

𝑇, −850) , whit an entire phase

shift over the positive range of frequencies: (00, −900);

Figure 2 Bode approximation for the “first order element”, 𝑯(𝒔) =𝒌

𝑻𝒔+𝟏

- pair of complex pole , 1

𝑠2

𝜔𝑛2 +

2𝜉𝑠

𝜔𝑛+1

, with magnitude: a slope of -40dB after the corner frequency,

ωn, and phase having 3 points of interest: (0.1𝜔𝑛, −100), (𝜔𝑛, −900), (10𝜔𝑛, −1700) , whit an

entire phase shift over the positive range of frequencies: (00, −1800);

Figure 3 Bode approximation for the “second order element”, 𝑯(𝒔) = 𝟏

𝒔𝟐

𝝎𝒏𝟐+

𝟐𝝃𝒔

𝝎𝒏+𝟏

Theoretical requests 1. Sketch on your laboratory workbooks the Bode approximations presented in Figure 1, 2 and 3.

(10 minutes)

2. Compute the cut off frequency for the real integrator in Figure 1. (2 minutes)

3. For the Bode diagram in the next figure (Figure 4) read and mention on your workbook (10

minutes) the value of:

a. the cut-off frequency (𝜔𝑐 =?𝑟𝑎𝑑

𝑠𝑒𝑐. )

b. the BandWidth frequency (𝜔𝐵𝑊 =?𝑟𝑎𝑑

𝑠𝑒𝑐. )

c. the low-frequency asymptote slope (? 𝑑𝛽)

d. the high-frequency asymptote slope (? 𝑑𝛽)

e. the frequency when the phaseshift is -135° (𝜔−135° =?𝑟𝑎𝑑

𝑠𝑒𝑐).

Figure 4 Bode Diagram (Matlab plot)

4. Recognize the transfer function from its Bode diagram in the figure below (Figure 5)

a. read from the phase characteristic the frequency 𝜔−90° =?𝑟𝑎𝑑

𝑠𝑒𝑐.

b. write down on your workbooks the resulted transfer function.

c. mention the matlab script used for plotting figure 5.

Figure 5 Bode diagram for a second order element with positive gain

5. Recognize the transfer function from its Bode diagram in the figure below (Figure 6)

a. read from the magnitude characteristic the cutoff frequency 𝜔𝑐 =?𝑟𝑎𝑑

𝑠𝑒𝑐.

b. write down on your workbooks the resulted transfer function.

c. mention the matlab script used for plotting figure 6.

Figure 6 Bode diagram for an integrator with positive gain.

6. Recognize the transfer function from its Bode diagram in the figure below (Figure 7)

a. read from the phase characteristic the frequency 𝜔−45° =?𝑟𝑎𝑑

𝑠𝑒𝑐.

b. write down on your workbooks the resulted transfer function.

c. mention the matlab script used for plotting figure 7 (use logspace function in Matlab).

Figure 7 First order element, Bode diagram

Problems Approximate Bode diagrams by asymptotes, for the next transfer functions:

1. 𝐻(𝑠) =1800

𝑠(𝑠+60);

2. 𝐻(𝑠) = 2𝑠+2

(3𝑠+1)(2𝑠+1);

3. 𝐻(𝑠) = 410𝑠+1

(3𝑠+1)(2𝑠+1);

4. 𝐻(𝑠) =12𝑠

36𝑠2+12𝑠+1

5. 𝐻(𝑠) =𝑠2+144

(𝑠+4)(𝑠+20)

6. 𝐻(𝑠) =2

𝑠(0.5𝑠+1)(𝑠+2)

Solve problems (in class)