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Lead Lag

The document discusses lead-lag compensation in control systems, detailing the mathematical relationships between various components such as resistors and capacitors. It explains how the lead-lag network combines phase lead and phase lag for improved system stability by adjusting gain and phase crossover frequencies. Key parameters like time constants and their relationships are outlined to illustrate the design constraints for effective compensation.

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kingston8629
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31 views3 pages

Lead Lag

The document discusses lead-lag compensation in control systems, detailing the mathematical relationships between various components such as resistors and capacitors. It explains how the lead-lag network combines phase lead and phase lag for improved system stability by adjusting gain and phase crossover frequencies. Key parameters like time constants and their relationships are outlined to illustrate the design constraints for effective compensation.

Uploaded by

kingston8629
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Lead-lag Compensation

𝑅1

𝐶2
𝑉𝑖 𝐶1 𝑉𝑂

𝑅2

1
𝑍𝑖 = ( + 𝑆𝐶1 )−1
𝑅1
1 + 𝑆𝐶1 𝑅1 −1 𝑅1
𝑍𝑖 = ( ) =
𝑅1 1 + 𝑆𝐶1 𝑅1
1 1 + 𝑆𝐶2 𝑅2
𝑍0 = (𝑅2 + =( )
𝑆𝐶2 𝑆𝐶2
𝑅1 1 + 𝑆𝐶2 𝑅2
𝑍𝑇 = +
1 + 𝑆𝐶1 𝑅1 𝑆𝐶2
𝑉0 𝑍0 1 + 𝑆𝐶2 𝑅2 ⁄𝑆𝐶2 (1 + 𝑆𝐶2 𝑅2 )(1 + 𝑆𝐶1 𝑅1 )
= = =
𝑉𝑖 𝑍𝑇 (𝑅1⁄ 1 + 𝑆𝐶2 𝑅2
⁄𝑆𝐶 ) 𝑆𝐶2 𝑅1 + (1 + 𝑆𝐶1 𝑅1 )(1 + 𝑆𝐶2 𝑅2 )
1 + 𝑆𝐶1 𝑅1 ) + ( 2

𝑉0 (1 + 𝑆𝐶2 𝑅2 )(1 + 𝑆𝐶1 𝑅1 )


=
𝑉𝑖 𝑆𝐶2 𝑅1 + 1 + 𝑆𝐶2 𝑅2 + 𝑆𝐶1 𝑅1 + 𝑆 2 𝐶1 𝐶2 𝑅1 𝑅2
𝑉0 (1 + 𝑆𝐶2 𝑅2 )(1 + 𝑆𝐶1 𝑅1 )
=
𝑉𝑖 1 + 𝑆(𝐶1 𝑅1 + 𝐶2 𝑅2 + 𝐶2 𝑅1 ) + 𝑆 2 (𝐶1 𝐶2 𝑅1 𝑅2 )
The general form is:
𝑉0 (1 + 𝑠𝜏1 )(1 + 𝑠𝜏2 )
=
𝑉𝑖 (1 + 𝑠𝛼𝜏1 )(1 + 𝑠𝛽𝜏2 )
Where we get:
𝜏1 = 𝐶1 𝑅1
𝜏2 = 𝐶2 𝑅2
𝛼𝛽𝜏1 𝜏2 = 𝑅1 𝑅2 𝐶1 𝐶2

Page 1 of 3
𝛽𝜏2 + 𝛼𝜏1 = 𝑅1 𝐶1 + 𝑅2 𝐶2 + 𝑅2 𝐶1
From the above we get 𝛼𝛽 = 1.
Hence there is no independence choice of 𝛼 and 𝛽. For our design and 𝛽 must be
reciprocal of each other.
We have: 𝛼𝜏1 𝛽𝜏2 = 𝜏1 𝜏2
𝛼𝜏1 𝛽𝜏2 = 1
𝛼𝜏1 > 𝜏1 > 𝜏2 > 𝛽𝜏2

𝑀 𝑑𝐵 1 1
1 1
𝜏2 𝛽𝜏2
𝛼𝜏1 𝜏1

∅0

𝐿𝑒𝑎𝑑

𝐿𝑎𝑔 1 1
𝜔= =
𝜏1 √𝛼 𝜏2 √𝛽

The lead-lag network is a combination of the phase lag network and phase lead
network. The phase symmetry has odd symmetry about the center frequency.
1 1
𝜔= =
𝜏1 √𝛼 𝜏2 √𝛽

Page 2 of 3
The network when employed for compensation is used primarily for its attenuation
in the region of the center frequency in combination with positive phase shift above
1 1 1 1
the center frequency. The frequencies , , 𝑎𝑛𝑑 are chosen such that the
𝜏1 𝛼𝜏1 𝜏2 𝛽𝜏2
system will decrease the system gain cross-over frequency and simultaneously
increase the phase cross-over frequency. This results in improvement of the system
stability.
We have:
𝑅2
𝛽=
𝑅1 + 𝑅2
𝑅1 + 𝑅2
𝛼=
𝑅2

Page 3 of 3

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