frequency response analysis closed loop frequency … · mcg 3302 () 1 cs gs rs gs = + frequency...

. MCG 3302

FREQUENCY RESPONSE ANALYSISClosed Loop Frequency Response

The Bode plot is generally constructed for an open loop transfer function of a system. Inorder to draw the Bode plot for a closed loop system, the transfer function has to bedeveloped, and then decomposed into its poles and zeros. This process is tedious andcannot be carried out without the ais of a very powerful calculator or a computer.

Nichols, developed a simple process by which the unity feedback closed loop frequencyresponse of a system can be easily deduced from the open loop transfer function. Hisapproach will be outlined on the next few slides.

While the Nichols approach is derived strictly for a unity feedback system, any non-unityfeedback system can be transformed to two cascaded open loop systems as follows:

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Consider the following non unity feedback system:


+ - G(s)

H(s)

C(s)R(s)

+ - G(s)C(s)R(s)

1/H(s)

It can be transformed into the following system composed of a simple block cascaded to aunity feedback system:

The frequency response can be then obtained using the additive feature of the Bode plots.

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( )( )

( )( )1

C s G sR s G s

=+


+ - G(s)C(s)R(s)

With reference to the generic unity feedback systemblock diagram and its polar plot; the system transferfunction is given by:

The dashed line in the polar plot is the trace of the tip ofthe vector OA which represents the system. The length ofthe vector measure the magnitude of the system at a givenfrequency T, and the angle N represents the phase shift.

O

A

P (-1,0)

Re

Im

N

"=N-2

2

Polar Plot

M

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( )

( )( )( )

represents

represents -1

But 1

1

OA G j

OP

PA PO OAG j

G jOAG jPA

ω

ω

ωω

= +

= +

∴ =+

α φ θ= −


O

A

P (-1,0)

Re

Im

N

"=N-2

2

Polar Plot

M

With reference to the polar plot;

The phase shift angle of the closed loop system is the included angle between the vectorsOA and PA, i.e.;

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( )G j X jYω = +

( )( )

( )

( )

2 22

2 2

1

1

1

G jM

G j

X jYX jY

X YMX Y

ωω

=+

+=

+ +

+∴ =

+ +

FREQUENCY RESPONSE ANALYSISClosed Loop Constant Magnitude Locii

Since the open loop transfer function is a complex quantity that can be expressed as:

It follows that the magnitude of the closed loop system can be expressed as follows:

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( ) ( )2 2 2 2 2 21 2 1 0X M M X M M Y− − − + − =

12X = −

2 22 2

2 2

2 01 1

M MX X YM M

+ + + =− −

Expanding and collecting terms, the previous equation gives:


For M=1 the above equation reduces to;

For M…1, the equation can be written in the form:

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( )2

22 1

M

M −

( )

22 22

22 21 1

M MX YM M

⎛ ⎞+ + =⎜ ⎟−⎝ ⎠ −

2

2 2center: , 0 and radius: 1 1

M MM M

⎛ ⎞−⎜ ⎟− −⎝ ⎠

The preceding equation canot be factored as it stands. However by adding;

to both sides and factoring gives:


This is an equation of a circle with radius and center as follows:

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The plot of the previous equation is shown below for different magnitudes M:


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1 1

1

tan tan1

j X jYeX jY

Y YX X

α

α − −

+=

+ +

∴ = −+

1 1tan tan tan1

Y YNX X

− −⎡ ⎤= −⎢ ⎥+⎣ ⎦

The phase of the closed loop system can be expressed as follows:

FREQUENCY RESPONSE ANALYSISClosed Loop Constant Phase Locii

Now let " = N, then;

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( )

( ) ( )( )( )

tan tantan1 tan tan

11 1

A BA BA B

Y YX XNY Y

X X

−− =

+

− +∴ =+ +

2 2

2 2

or1 0

YNX X Y

X X Y YN

=+ +

+ + − =

But

Simplifying gives:


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( )21 14 2N+

2 2 21 1 1 12 2 4 2

X YN N

⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + − = +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( )21 1 1 1center: , and radius: 2 2 4 2N N

⎛ ⎞ +⎜ ⎟⎝ ⎠


The preceding equation canot be factored as it stands. However by adding;

to both sides and factoring gives:

This is an equation of a circle with radius and center as follows:

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The plot of the previous equation is shown below for different phase angles ":

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( ) ( )( )503 6

G ss s s

=+ +

FREQUENCY RESPONSE ANALYSISClosed Loop Frequency Response from

Open Loop Response

Using the constant magnitude and phase circles and the polar plot of the open loop toobtain the closed loop frequency response.

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FREQUENCY RESPONSE ANALYSISThe Nichols Chart

Nichols superimposed the log of the constant magnitude and phase circles on the log-magnitude phase plot to create the Nichols chart.

The magnitude and phase of the open loopvalues are plotted on the rectilinear log-magnitude phase grid, and the closed loopfrequency response is read from thecurvilinear grid.

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( ) ( )( )1

1 0.2 1G j

j j jω

ω ω ω=

+ +


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( ) ( )2

0.641

G jj j

ωω ω ω

=+ −


frequency response analysis closed loop frequency … · mcg 3302 () 1 cs gs rs gs = + frequency...

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