Lead correction of series correction

Circuit configuration of "passive lead correction" and "active lead correction"

The essence of advance correction

Moreover, the advance correction can increase the cut-off frequency Wc, so that the response speed and overshoot of the system are improved (increased Wc will cause the system response speed ts to be shortened, and the overshoot σ% decreases, which is beneficial to the stability of the system).

Note: Steady-state performance and stability are not the same. When we look at steady-state performance, we generally look at the steady-state error ess of the system. When looking at system stability, we generally look at the overshoot σ% of the system and the resonance peak value of the system. Mr.

Advance correction steps

example:

① Calculate the open-loop gain K according to the steady-state error given in the title, and then draw the Bode plot and obtain the frequency characteristic, that is, perform the drawing operation according to the following expression:

② Find the open-loop cut-off frequency ωc , and the argument angle and phase angle margin γ when ω=ωc

③ Find the required maximum lead ∆∅=∅m according to the phase angle margin required in the title , and the maximum lead has come out, we can then solve the parameters in the following equation:

If we want to make the γ phase angle margin meet the meaning of the question after compensation, we must take into account the “phase angle loss caused by the increase of the cut-off frequency” as part of the compensation.

By the formula:

Got,

Among them, the whole vicinity of Wc rises by 10lg(a), namely

④ According to the obtained α , we can find the new cut-off frequency:

From the above formula, we can find the " cutoff frequency Wc' obtained after correction according to the γ phase angle margin" as:

However, we find that the cut-off frequency Wc' obtained after correction according to the γ phase angle margin does not meet the meaning of the question:

⑤ Compare the solved ω c'with the cut-off frequency ω c'' required in our problem, and take the final cut-off frequency as

If , that is, the cutoff frequency we finally find does not meet the requirements, then we choose ω c'' in the title as our new corrected cutoff frequency;

According to the comparison between the cutoff frequency Wc' obtained after correction according to the γ phase angle margin and the cutoff frequency Wc'' required in the title, we can know:

⑥ Find the new α and the time constant Ta according to the new cut-off frequency ;

⑦ Calculate the phase angle margin at the cut-off frequency ω c'' according to the final results , and see if it meets the meaning of the question. If there are other required parameters, such as "amplitude margin", generally follow the above 7 The final calculated result of each step generally meets the meaning of the question, if it does not meet the meaning of the question, you can appropriately increase the cut-off frequency for calculation.

The results of the above network correction are as follows:

Application fields of advance correction network

① First of all, we must clarify the essence of the advanced correction network: make full use of the phase angle advance characteristics of the mid-frequency band to compensate for the phase angle margin at W=Wc (cutoff frequency) of the uncorrected network, and at the same time, due to its own in the mid-band The amplitude-frequency characteristic is positive, and the advance correction can increase the cut-off frequency in a small range, so that the cut-off frequency of the network after correction is greater than the cut-off frequency of the network before the correction, for example:

In the above question, the cut-off frequency of the network before uncorrected is Wc=3.16rad/s, and the cut-off frequency of the network after correction is Wc''=6rad/s, but Wc is appropriately increased, and it is not increased by 10~20rad/s all at once. s, this is caused by the nature of the lead correction network, which is equivalent to the main function of the lead correction network is to "provide the system with lead through its own leading phase angle", and the auxiliary function is to "use yourself to increase the Wc cutoff frequency" Features to improve the speed and stability of the system".

② There is an upper limit for the amount of advance provided to the system, that is, when the phase angle of the system is near W=Wc, there will be a cliff-like decline, then the advance correction network is obviously not enough, although sometimes we may have a small probability through the series connection of multiple correction networks To make the network barely meet the requirements of the question, but such a system is extremely unstable, and a slight change in frequency immediately does not meet the meaning of the question.

③ When the question asks to increase the phase angle margin, while reducing the cut-off frequency, "shrinking" is obviously contrary to the "increasing Wc cut-off frequency" characteristic of the leading correction network. At this time, the leading correction network is not applicable.

Matlab view stability margin

Uncorrected network parameters

The open loop transfer function of the system before correction (the default feedback transfer function is 1)

Amplitude margin Gm, phase angle margin Pm, amplitude crossover frequency Wcg, phase angle crossover frequency Wcp

The open-loop transfer function of the system after leading correction (the default feedback transfer function is 1)