Frequency response methods

Prof. Muhammad Abid

Basic Tools for analysis and design

Bode diagrams

Polar plots

Nyquist plots
Polar Plots
Plot of frequency response on a single graph
𝐺 ( 𝑗 𝜔 )=|𝐺 ( 𝑗 𝜔 )|∠( 𝐺( 𝑗 𝜔))

Phase frequency response

Magnitude frequency response

Locus of vectors as is varied from zero to infinity

Conveys the same information as the Bode
diagram
Polar plots are less detailed
Polar Plots ℑ

Polar plot for

𝜔=0
ℜ
𝐺 ( 𝑗 𝜔 )= 𝑗 𝜔

¿ 𝜔 ∠ 90𝑜
Polar plot for
ℑ

1 −𝑗
𝐺 ( 𝑗 𝜔 )= ¿
𝜔→0
𝑗𝜔 𝜔 ℜ

1 𝑜
¿ ∠ − 90
𝜔
Polar Plots
ℑ
Polar plot for
𝜔=0
𝜔→∞ ℜ
1 1
𝐺 ( 𝑗 𝜔 )=
−1
¿ ∠ − tan 𝜔𝑇
1+ 𝑗 𝜔 𝑇 √ 1+ 𝜔 𝑇
2 2

𝜔=0 𝐺 ( 𝑗 𝜔 )=1 ∠ 0 𝑜

𝜔→∞ 𝐺 ( 𝑗 𝜔 )=0 ∠ −90 𝑜

Polar Plots
ℑ
Polar plot for
𝜔=0
𝜔→∞ ℜ
1 1
𝐺 ( 𝑗 𝜔 )=
−1
¿ ∠ − tan 𝜔𝑇 1
𝜔=
1+ 𝑗 𝜔 𝑇 √ 1+ 𝜔 𝑇
2 2
𝑇

𝜔=0 𝐺 ( 𝑗 𝜔 )=1 ∠ 0 𝑜 1 1− 𝑗 𝜔𝑇
𝐺 ( 𝑗 𝜔 )= ×
1+ 𝑗 𝜔 𝑇 1 − 𝑗 𝜔 𝑇
𝜔→∞ 𝐺 ( 𝑗 𝜔 )=0 ∠ −90 𝑜 1− 𝑗 𝜔𝑇
¿ 2 2
1+ 𝜔 𝑇
1 1 𝑜
𝜔= 𝐺 ( 𝑗 𝜔 )= ∠ − 45 −𝜔𝑇
𝑇 √2 ℑ ( 𝐺 ( 𝑗 𝜔 )) = =0
2 2
Crossing on real axis 1+ 𝜔 𝑇

⇒ 𝜔=0 , 𝜔=∞
Polar Plots
Polar plot for
ℑ

√
𝐺 ( 𝑗 𝜔 )=1+ 𝑗 𝜔 𝑇 ¿ 1+ 𝜔 2 𝑇 2 ∠ tan −1 𝜔 𝑇
𝜔=0
0 1 ℜ

𝜔=0 𝐺 ( 𝑗 𝜔 )=1 ∠ 0 𝑜

𝜔→∞ 𝐺 ( 𝑗 𝜔 )=∞ ∠ 90 𝑜

1
𝜔=
𝑇 𝐺 ( 𝑗 𝜔 )= √ 2∠ 45𝑜
Polar Plots
Polar plot for
ℑ

𝜔2𝑛 1 𝜔=∞ 1 ℜ
𝐺 ( 𝑗 𝜔 )= ¿
( )( )
( 𝑗 𝜔 )2 +2 𝜁 𝜔𝑛 ( 𝑗 𝜔 ) +𝜔 2𝑛 𝑗𝜔 𝑗𝜔
2
𝜔=0
1+2 𝜁 +
𝜔𝑛 𝜔𝑛
𝜔=𝜔𝑛
𝑜
𝜔=0 𝐺 ( 𝑗 𝜔 )=1 ∠ 0

1
𝜔→∞ 𝐺 ( 𝑗 𝜔 )→ ¿ 0 ∠ −180 𝑜
( )
2
𝑗𝜔
𝜔𝑛

1 1 𝑜
𝜔=𝜔𝑛 𝐺 ( 𝑗 𝜔 )= ¿ ∠ − 90
1+ 𝑗 2 𝜁 −1 2𝜁
Polar Plots
Polar plot for
ℑ

1 𝜔=∞ ℜ
𝐺 ( 𝑗 𝜔 )=
𝑗 𝜔 ( 𝑗 3 𝜔+1 ) ×
𝜔=0 1
𝐺 ( 𝑗 𝜔 )= ¿ ∞ ∠ − 90𝑜 ≈
𝑗0
1 𝑜
𝜔→∞ 𝐺 ( 𝑗 𝜔 )→ =0 ∠ −180
( 𝑗 ∞) ( 𝑗∞ ) 𝜔=0

1 ( − 𝑗 3 𝜔+1 ) ( − 𝑗 3 𝜔 +1 ) ( −3 ) 1
𝐺 ( 𝑗 𝜔 )= × ¿ ¿ − 𝑗
𝑗 𝜔 ( 9 𝜔2 +1 ) ( 9 𝜔 2+1 ) 2
𝑗 𝜔 ( 𝑗 3 𝜔+1 ) ( − 𝑗 3 𝜔+1 ) 𝜔( 9 𝜔 +1)
 Polar Plots As

Polar plot for −3 − 𝑗 ( 1− 2 𝜔2 )

𝐺 ( 𝑗 𝜔 )= +
( 𝜔 +1 ) ( 4 𝜔 +1 ) 𝜔 ( 𝜔 2 +1 )( 4 𝜔2 +1 )
2 2

1 − ( 1 −2 𝜔 2 ) 1
𝐺 ( 𝑗 𝜔 )= =0 𝜔= , 𝜔=∞
𝑗 𝜔 ( 𝑗 𝜔+1 ) ( 𝑗 2 𝜔 +1 ) 𝜔 ( 𝜔 +1 )( 4 𝜔 +1 )
2 2
√2
ℑ

1 2 −0.67
𝜔=0 𝐺 ( 𝑗 𝜔 )= ¿ ∞ ∠ − 90𝑜 𝐺 ( 𝑗 𝜔 ) |𝜔=1 / √ 2=− 𝜔=∞ ℜ
𝑗0 3
1 𝑜
𝜔→∞ 𝐺 ( 𝑗 𝜔 )→ =0 ∠ − 270
( 𝑗 ∞ ) ( 𝑗 ∞ )( 𝑗 ∞ )
≈
( − 𝑗 𝜔+1 ) (− 𝑗 2 𝜔+1 ) ¿ ( 1 − 2 𝜔 − 𝑗 3 𝜔 )
2
1
𝐺 ( 𝑗 𝜔 )= ×
𝑗 𝜔 ( 𝑗 𝜔+1 ) ( 𝑗 2 𝜔 +1 ) ( − 𝑗 𝜔+1 ) (− 𝑗 2 𝜔+1 ) 𝑗 𝜔 ( 𝜔 2 +1 )( 4 𝜔2 +1 )
𝜔=0
 Polar Plots 𝐺 ( 𝑗 𝜔 )=
( 8 − 𝜔 2 )+ 𝑗 ( 6 𝜔 )
( 𝜔2 + 4 ) ( 𝜔2 +16 )
Polar plot for ( 8− 𝜔 2 )
( 𝜔 + 4 )( 𝜔 +16 )
2 2
=0 𝜔= √ 8 , 𝜔=∞
1 ( 6 𝜔)
𝐺 ( 𝑗 𝜔 )=
( 𝑗 𝜔+2 ) ( 𝑗 𝜔+ 4 ) ( 𝜔 + 4 )( 𝜔 +16 )
2 2
=0 𝜔=0 , 𝜔=∞

𝜔=0 𝐺 ( 𝑗 𝜔 )=
1
8 ¿ 0.125 ∠ 0 𝑜
𝐺 ( 𝑗𝜔 ) |𝜔= √ 8=− 𝑗 0.056
ℑ
1
𝜔→∞ 𝐺 ( 𝑗 𝜔 )→ ¿ 0 ∠ −180 𝑜
( 𝑗 𝜔 )( 𝑗 𝜔 )2 𝜔=∞ 𝜔=0 ℜ

0.125 ∠0 𝑜
1 ( 𝑗 𝜔 +2 ) ( 𝑗 𝜔+ 4 )
𝐺 ( 𝑗 𝜔 )= × − 𝑗 0.056
( 𝑗 𝜔+2 ) ( 𝑗 𝜔+ 4 ) ( 𝑗 𝜔 +2 ) ( 𝑗 𝜔+ 4 ) 𝜔= 𝑗 √ 8
 Polar Plots
Polar plot for

1
𝐺 ( 𝑗 𝜔 )=
( 𝑗 𝜔+2 ) ( ( 𝑗 𝜔 )2 + 𝑗 2 𝜔+2 )

ℑ
1
𝜔=0 𝐺 ( 𝑗 𝜔 )= ¿ 0.25 ∠ 0 𝑜
4
𝜔=∞ 𝜔=0 ℜ
1
𝜔→∞ 𝐺 ( 𝑗 𝜔 )→ 2 ¿ 0 ∠−270
𝑜
( 𝑗 𝜔 )( 𝑗 𝜔 )
 1
𝐺 ( 𝑗 𝜔 )=
( 𝑗 𝜔+2 ) ( ( 𝑗 𝜔 )2 + 𝑗 2 𝜔+2 )

1 (− 𝑗 𝜔 +2 ) ( 2− 𝜔 2 − 𝑗 2 𝜔 ) ( 4 − 4 𝜔 2) + 𝑗 𝜔 ( 6 − 𝜔 2)
𝐺 ( 𝑗 𝜔 )= × ¿
( 𝑗 𝜔+2 ) ( 2− 𝜔 + 𝑗 2 𝜔 ) (− 𝑗 𝜔 +2 ) ( 2− 𝜔 − 𝑗 2 𝜔 )
2 2
( 𝜔2 +4 ) ( ( 2 − 𝜔 )2 + 4 𝜔 2 )

𝜔 ( 6 − 𝜔2 )
( 𝜔 + 4 ) ( (1 − 𝜔 ) + 4 𝜔 )
2 2 2
=0 ⇒ 𝜔=0 , √ 6 , ∞ 𝐺 ( 𝑗𝜔 ) |𝜔= √ 6=−0.05

( 4 − 4 𝜔2 )
( 𝜔 + 4 ) ( ( 2− 𝜔 ) + 4 𝜔 )
2 2 2
=0 ⇒ 𝜔=1 , ∞ 𝐺 ( 𝑗𝜔 ) |𝜔=1 =− 𝑗0.2
 Nyquist Diagram
0.25

0.2

Polar Plots 0.15

0.1

Imaginary Axis
0.05

Polar plot for

0

-0.05

-0.1

-0.15

1 -0.2
𝐺 ( 𝑗 𝜔 )=
( 𝑗 𝜔+2 ) ( ( 𝑗 𝜔 )2 + 𝑗 2 𝜔+2 ) -0.25
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4
Real Axis

ℑ
1
𝜔=0 𝐺 ( 𝑗 𝜔 )= ¿ 0.25 ∠ 0 𝑜
4 𝜔= √6
−0.05 𝜔=∞ 𝜔=0 ℜ
1 𝑜
𝜔→∞ 𝐺 ( 𝑗 𝜔 )→ =0 ∠ −270
( 𝑗 𝜔 )( 𝑗 𝜔 )2

𝐺 ( 𝑗𝜔 ) |𝜔=1 =− 𝑗0.2 𝐺 ( 𝑗𝜔 ) |𝜔= √ 6=−0.05 − 𝑗0 .2

𝜔=1