ch4-DCDC变换器反馈控制设计课件

1、Introduction Modeling of CCM DC/DC ConverterModeling of DCM DC/DC ConverterCurrent Programmed ControlFeedback Control DesignModeling and Control of Single Phase InverterContents 第1页，共76页。 Chapter 4Feedback control design 第2页，共76页。第3页，共76页。Typical variation in vg(t): 100Hz ripple, produced by rectifi

2、er circuit.Load current variations: a significant step-change in load current, such as from 50% to 100% of rated value.Circuit elements are constructed to some specified tolerance. Effects to poor output第4页，共76页。Negative feedback: switching regulator systemObjective: maintain constant output voltage

3、 v(t) = V, in spite of disturbances in input and loadA typical output voltage regulation specification: 2% for example 5V 0.1V.Introduce feedback control第5页，共76页。Negative feedbackconverter第6页，共76页。Small signal model of converterOutput voltage can be expressed asLoad disturbance model第7页，共76页。DC/DC c

4、onverter system dynamic modelUse small-signal converter modelPerturb and linearize remainder of feedback loop第8页，共76页。DC/DC converter system small-signal block diagram叠加原理求解第9页，共76页。Solution of block diagramLoop gain T(s) = products of the gains around the negative feedback loop.Changed into form第10

5、页，共76页。Terminology: open-loop vs. closed-loopOriginal transfer functions, before introduction of feedback (open-loop transfer functions):Upon introduction of feedback, these transfer functions become(closed-loop transfer functions):The loop gain:第11页，共76页。Feedback improve the line regulationopen-loo

6、p line-to-output transfer function:With addition of negative feedbackFeedback reduces the line-to-output transfer function by a factor ofIf T(s) is large in magnitude, then the line-to-output transfer functionbecomes small.第12页，共76页。Closed-loop output impedanceOriginal (open-loop) output impedance:W

7、ith addition of negative feedback, the output impedance becomes:Feedback reduces the output impedance by a factor of- 第13页，共76页。Closed loop gainClosed-loop transfer function If the loop gain is large in magnitude, i.e., | T | 1, then T/(1+T) T/T = 1. The transfer function then becomeswhich is indepe

8、ndent of the gains in the forward path of the loop.This result applies equally well to dc values:Output is not sensitive to parameter variation in the forward path第14页，共76页。Open loop gain example 第15页，共76页。Approximating 1/(1+T) and T/(1+T)第16页，共76页。Example: construction of T/(1+T)第17页，共76页。At freque

9、ncies sufficiently less that the crossover frequency, the loopgain T(s) has large magnitude. The transfer function from thereference to the output becomesThis is the desired behavior: the output follows the reference according to the ideal gain 1/H(s). The feedback loop works well at frequencies whe

10、re the loop gain T(s) has large magnitude.At frequencies above the crossover frequency, | T | 1. The quantity T/(1+T) then has magnitude approximately equal to 1, and we obtainAt frequencies where | T | 1, the loop has essentially no effect on the transfer function from the reference to the output.D

11、iscussion第18页，共76页。Example: construction of 1/(1+T)第19页，共76页。How does the loop reject disturbances?Below the crossover frequency: f 1 Then 1/(1+T) 1/T, anddisturbances are reduced in magnitude by 1/| T |Above the crossover frequency: f fc and | T | 1 Then 1/(1+T) 1, and The feedback loop has essenti

12、ally no effect on disturbances第20页，共76页。inputoutputTransfer functionTransfer function第21页，共76页。Factorize the denominator and nominatorPoles:Zeros:Fold freq.Fold freq.第22页，共76页。Frequency characteristicsBode plots:Amplitude plotAngle plotUnit: dBAmplitude plot is a Folding line graphMultiplying factor

13、s become addition operation in amplitude Bode plot.第23页，共76页。Meet a pole, Fold down with a rate of 20dB/dec Meet a zero, Fold up with a rate of +20dB/dec Amplitude Bode plot20dB/dec 40dB/dec 20dB/dec 第24页，共76页。If meet a pole , increase phase delay 90 degree occurring between fp/10 and 10fpIf meet a

14、zero ,lead phase angle with 90 degree occurring between fz/10 and 10fz Angle Bode plot-90-1800第25页，共76页。+-Closed loop transfer functionCharacteristic equationIf all roots are in the left half plane, stableIf a root in the right half plane, unstable.Root:Stability of closed loop第26页，共76页。Contains all

15、 the information about the roots of Therefore we can know the stability of the closed loop system by studying Bode plot is used to analysis the stability of the system相位裕量增益裕量增益交越频率相位交越频率Bode graph第27页，共76页。+-Oscillation conditionandOscillation conditionTo Break the oscillation condition, it is requ

16、iredwhenwhen第28页，共76页。gain margin: GM （dB） Relative stabilityphase margin: PM 相位裕量增益裕量增益交越频率相位交越频率回路增益函数的相位为180时的频率称为相位交越频率 增益裕量是指当回路增益函数的相位为180时，回路增益函数所能容许增加的回路增益 +-第29页，共76页。相位裕量增益裕量增益交越频率相位交越频率相位裕量:当闭环系统达到不稳定之前，其回路内所能容许增加的相位。当回路增益函数的幅值为零分贝（单位增益）时，回路增益函数的相移与180之差。 Phase margin+-第30页，共76页。相位裕量乃是在平面

17、上为了使轨迹的增益交越点通过（1，j0）点，则图必须以原点为中心顺时针所须旋转角度.增益裕量就是在平面上相位交越点对（1, j0）点接近程度的一种量度 Naquist Graph相位裕量增益裕量增益交越频率相位交越频率-1第31页，共76页。 The relationship between phase marginand closed-loop damping factorHow much phase margin is required?A small positive phase margin leads to a stable closed-loop system, which has

18、 complex poles near the crossover frequency with high Q. The transient response exhibits overshoot and ringing.Increasing the phase margin reduces the Q. To obtain real poles with no overshoot and ringing requires a large phase margin.第32页，共76页。Consider the case where T(s) can be well-approximated i

19、n the vicinity of the Crossover frequencyA simple second-order system第33页，共76页。Closed-loop responseExpressed byWherewhereis substituted第34页，共76页。Low-Q case第35页，共76页。High-Q case第36页，共76页。Solve for exact crossover frequency, evaluate phase margin, express as function of QQ vs. phase margin 第37页，共76页。Q

20、 vs. phase margin 第38页，共76页。Transient response vs. damping factorUnit-step response of second-order system T(s)/(1+T(s)where第39页，共76页。Transient response vs. damping factor第40页，共76页。Regulator designTypical specifications:Effect of load current variations on output voltage regulation, which is a limit

21、 on the maximum allowable output impedanceEffect of input voltage variations on the output voltage regulation, which limits the maximum allowable line-to-output transfer functionTransient response time, which requires a sufficiently high crossover frequencyOvershoot and ringing. An adequate phase ma

22、rgin must be obtainedThe regulator design problem: add compensator network Gc(s) to modify T(s) such that all specifications are met.第41页，共76页。+-反馈信号参考信号误差信号DC/DC power converter systemPWM调制器的传递函数为 传递函数为至输出的传递函数第42页，共76页。Buck变换器占空比至输出的传递函数Buck变换器系统原始回路增益函数 原始回路增益函数是一个二阶系统，有两个极点。 幅频图在低频段为水平线，幅值为高频段以4

23、0dB/dec斜率下降，转折点由LC滤波器的谐振频率决定。第43页，共76页。设Buck变换器系统的参数为：输入电压输出电压，输出负载输出滤波电感值，电容值开关频率kHz，即开关周期PWM调制器中锯齿波幅值反馈分压网络传递函数可求出工作占空比： Buck变换器占空比至输出的传递函数 第44页，共76页。原始回路增益函数 幅频图的转折频率为： 幅频图低频段为幅值约为20dB水平线,高频段为斜率穿越0dB线的折线。Hz 第45页，共76页。幅频图相频图 增益交越频率， 相位裕量 第46页，共76页。原始回路增益函数频率特性的相位裕量太小。虽然系统是稳定的，但存在较大的输出超越量和较长的调节时间。

24、通常选择相位裕量在45左右，增益裕量在10dB左右。因此需要加入补偿网路 一般原始回路增益函数不能满足系统静态和动态特性的要求，为了使系统满足静态和动态的指标，需要加入补偿网路。 虽然，补偿网路只是系统中极小的一部分，但是，对系统静态和动态特性而言却是非常重要部份，它会影响到系统的输出精度、电压调整率、频带宽度以及暂态响应。第47页，共76页。最小相位系统的幅频特性和相频特性之间存在一一对应关系，如幅频图中斜率为-20dB/dec折线对应相频图中相移为斜率为-40dB/dec折线对应相频图中相移为斜率为+20dB/dec折线对应相频图中相移为知道了幅频特性也就知道了相频特性，反之也然。最小相位

25、系统理论水平线对应相频图中相移为第48页，共76页。为使DC/DC变换器系统满足稳定性要求，可以通过外加补偿网络使DC/DC变换器系统的回路增益函数的幅频图在增益交越频率处（增益为零dB）的斜率为-20dB/dec。相位裕量PM大于零 因为根据最小相位系统的性质，幅频图的斜率为-20dB/dec折线对应相移为-90度第49页，共76页。还须验证在相位交越频率处（相位在180时）若相位裕量与增益裕量的值，只是稍稍大于零时，虽然对系统而言也是稳定的，不过却会具有较大的超越量和调节时间。通常选择相位裕量在45左右，增益裕量在10dB左右。 增益裕量必须大于零。回路增益函数第50页，共76页。补偿网路

26、设计 （1）超前补偿网路（2）滞后补偿网路；（3）超前滞后补偿网路。 第51页，共76页。Lag (PI) compensationImproves low-frequency loop gain and regulationSuitable to the 1st order system第52页，共76页。Lag Compensation Examplecompensator:original loop gainis selected less than fo to maintain adequate phase margin is selectedPIFirst select crosso

27、ver frequency of closed loop第53页，共76页。Lead (PD) compensatorImproves phase marginSuitable to the 2nd order system第54页，共76页。Lead compensator: maximum phase leadThe frequency with maximum phase leador55 deg第55页，共76页。Lead compensator designTo obtain a compensator phase lead of at frequency fc, the pole

28、and zero frequencies should be chosen as follows:If it is desired that the magnitude of the compensator gain at fc be unity, then Gc0 should be chosen as第56页，共76页。Lead compensation example第57页，共76页。超前滞后补偿网络 （a）电路 （b）零极点分布超前滞后补偿网络的传递函数 “超前滞后”补偿网路输出正弦信号的相位在不同频率范围有落后又有超前于正弦输入信号的特性，它结合超前补偿与滞后补偿的特性，发挥滞后补

29、偿特性提高静态或稳态性能，利用超前补偿特性提高相对稳定性和动态性能。第58页，共76页。，项产生超前补偿效果项产生滞后补偿效果 (c)第59页，共76页。有源超前滞后补偿网路1 RC网路所组成的超前滞后补偿网络的增益只能衰减不能增加，增益设计不便。因此，一般采用运算放大器构成有源的超前滞后补偿网路。第60页，共76页。-20dB/dec20dB/dec补偿网路增益 利用补偿网络幅频特性图低频的积分特性，可以使经补偿后的系统成为无差系统，使静差为零，同时减少了低频误差。利用补偿网络幅频图在 至 之间斜率为20 dec/dB上升特性，补偿原始回路函数 以斜率 穿越0dB线的特性，使补偿后的回路函数

30、 以 穿越0dB线，这样才能使DC/DC变换器系统具有较好的相对稳定性。因此，一般将补偿后系统的增益交越频率 设定在补偿网路的 与 之间。第61页，共76页。有源超前滞后补偿网络2 零点为：极点为：与有源超前滞后补偿网路1的差异是在高频部份增加了一个极点fp3，而使其向下反折为20dB/dec。 第62页，共76页。频率fz1与fz2之间的增益可近似为 在频率 与 之间的增益则可近似为一般也将补偿后系统的增益交越频率 设定在补偿网路的 与 之间 20dB/dec-20dB/dec-20dB/dec第63页，共76页。下面以有源超前滞后补偿网络2为例介绍补偿网络的设计方法。 补偿网络的设计方法选择补偿后的回路函数 的增益交越频率fg 补偿后的回路函数增益交越频率fg愈大，变换器系统动态响应愈快。将补偿网络 两个零点频率设计为原始回路函数