Bode plot for lead compensator and lag compensator

关于超前补偿器和滞后补偿器的理解：

关于超前补偿器和滞后补偿器的理解_zmhzmhzm的博客-CSDN博客_超前补偿和滞后补偿

根轨迹的一般形式原理方框图如下 根轨迹描述的是该系统 闭环传递函数的根随着增益K从零到无穷变化时而作出的图像，以上图传递函数为例，设系统传递函数为 G ( s ) G(s) ，则该系统闭环传递函数为 G c l o s e ( s ) = K ∗ G ( s ) 1 + K ∗ G ( s ) G_{close}(s)=\frac{K*G(s)}{1+K*G(s)} ，那么它的根就是令 1 + K ∗ G ( s ) = 0 1+K*G(s)=0 ，即 1 + K ∗ 1 s 2 + 2 s = 0 1+K*\frac{1}{s^2+2s}=0 ，当K从0变化到无穷时，求出s的值，并在复平面做出轨迹，此轨迹即为根轨迹，上述例子在Matlab下用rlocus作图可直接得出根轨迹，如下：

https://blog.csdn.net/zmhzmhzm/article/details/107816221

Bode plot for lead compensator and lag compensator

Lead Compensator:

1. Characteristics:
• A lead compensator typically has a transfer function of the form , where and .
• It introduces a phase lead (positive phase shift) over a certain frequency range.

2. Bode Plot:
• Magnitude: Initially, the magnitude plot of a lead compensator is flat. As frequency increases, there is a range where the magnitude increases (due to the zero of the compensator), but it eventually levels off or decreases slightly at higher frequencies.
• Phase: The phase plot shows an increase in phase (phase lead) starting before the zero frequency () and peaking near the middle of the zero and pole frequencies ( and ), then decreasing back to a smaller phase lead at higher frequencies.

Lag Compensator:

1. Characteristics:
• A lag compensator has a transfer function of the form , where T > 0 and .
• It introduces a phase lag (negative phase shift) over a certain frequency range.

2. Bode Plot:
• Magnitude: The magnitude plot of a lag compensator decreases at the zero frequency () and then increases back to its original level at the pole frequency (). The overall effect is a decrease in magnitude over a certain frequency range.
• Phase: The phase plot shows a decrease in phase (phase lag) starting before the zero frequency () and reaching a minimum near the middle of the zero and pole frequencies, then increasing back to a smaller phase lag at higher frequencies.

Key Differences:

• Phase Effect: The most significant difference is in the phase behavior. Lead compensators introduce a phase lead, which can help improve system stability and transient response. Lag compensators introduce a phase lag, which is often used to improve steady-state accuracy or reduce sensitivity to disturbances.

• Magnitude Effect: Lead compensators can provide a temporary boost in magnitude (gain), which can be beneficial in crossover frequency and phase margin improvement. Lag compensators typically reduce the gain over a certain frequency range, which can be useful in attenuating noise or reducing overshoot.

Application:

• Lead Compensators: Often used to increase the phase margin and improve the transient response of a system, making it more stable and responsive.

• Lag Compensators: Typically employed to improve steady-state accuracy and reduce the sensitivity of the system to steady-state errors.