Abstract: Mainly introduced the Bode theorem, take this as the rationale, introduced the invertor modelling, the electricity pressure ring reaction control design and so on.
Key word: Bode theorem; Bode chart; Loop gain
1 control theory foundation
1.1 loop gains
Regarding the general negative feedback control system, its closed-loop system block diagram as shown in Figure 1. Closed loop transfer function C (s) /R(s)=G(s)/[1 G(s)H(s)], its characteristic equation is F(s)=1 G(s)H(s)=0, the characteristic equation root namely for system’s closed-loop pole. From the equation may see the G(s)H(s) item, it contained has possessed about the closed-loop pole information, generally said that G(s)H(s) was the loop gain. In the practical application, may through design system’s compensating network to the loop gain Bode chart analysis, meets the closed-loop system stability requirements.
1.2 Bode theorems
The Bode theorem regarding the determination so-called minimum phase system’s stability as well as the seeking stability margin is very useful. Its content is as follows:
1) the linear minimum phase system’s magnitude-phase characteristics is 11 correspondences, to be specific, when assigns in the entire frequency sector the logarithm amplitude-frequency characteristic (precise characteristic) when slope, in the identical sector’s logarithm phase-frequency characteristic is only also determined; Similarly, when assigns in the entire frequency sector the phase-frequency characteristic, in the identical sector’s logarithm amplitude-frequency characteristic is only also determined;
2) in some frequency (e.g. cutting frequency ωc) on phase displacement, mainly decides in the identical frequency the logarithm amplitude-frequency characteristic slope; Is farther to this slope, the slope paraphrase displacement’s influence is smaller; In some frequency’s phase displacement and the identical frequency’s logarithm amplitude-frequency characteristic slope’s approximate corresponding relationships are, ±20ndB/dec the slope corresponds to approximately ±n90° phase displacement, n=0,1,2,….
For example, if in the cutting frequency ωc on logarithm amplitude-frequency characteristic gradually coil in’s slope is - 20dB/dec, then ωc on phase displacement approximately close - 90°; If ωc on frequency approach line’s slope is - 40dB/dec, then should light the phase displacement approaches - 180°. In the latter kind of situation, the closed-loop system or is unstable, or only has not the big stability margin.
In actual project, to enable the system to have the suitable phase margin, often like this designs the split-ring transfer function, even if frequency approach line by - 20dB/dec slope through cutting spot, and at least in cutting frequency about, from ωc/4 to 2ωc this section of frequency range in maintains the above approach line slope is invariable.
2 invertor electricity pressure ring transfer function (modelling)
Invertor’s cocurrent input voltage 24V, exchange output voltage 110V, frequency 400Hz, terminal switch frequency 40kHz, power 500W. Its control to output entire electricity pressure ring’s circuit structure as shown in Figure 2. Presently strives for its loop gain.
2.1 driving signal d (s) to outputs Vo? s) transfer function
1) driving signal d is the SPWM pulse modulating wave, adds in the IGBT tube’s electronics grid (G) on, but inputs bus bar voltage Vin to add in pipe’s collecting electrode (C) and the emitter electrode (E) the both sides, the structure according to shown in Figure 2, output voltage Vd with actuates between d to differ a scale-up factor, supposes is K1, then K1=. In the concrete invertor electric circuit, bus bar voltage Vin is ±200V, the driving signal is 12V, the substitution may result in K1=400/12=33.33. 2) the LC low-pass filtering network transfer function inferential reasoning may result in =Vo(s)/Vo’(s) =1/(s2LC 1), L=3mH, C=2μF.
In the synthesis, driving signal d (s) to outputs Vo (s) the transfer function is Vo(s)/d(s)=G1(s)=K1/(s2LC 1);
2.2 output Vo (s) to feedback signal B (s) transfer function H (s)
1) the output voltage sampling transformer’s transfer function is a scale-up factor, namely it changes compared to, supposes is K2, namely V’o(s)/Vo(s)=K2, in concrete electric circuit, K2=18/110=0.164.
2) resistance electric capacity bleeder network like chart 2 the dashed line frame shows, its transfer function is =B(s)/V’o(s)=1/(sR1C2 R1/R2 1), R1=820Ω, R2=5.1kΩ, C2=10nF.
In synthesis, Vo (s) to B (s) transfer function H(s)=B(s)/Vo(s)=K2/(sR1C2 R1/R2 1);
2.3 PDM keyers (PWM) transfer function Gd(s)
Generally the PWM modulator’s transfer function is Gd(s)==, Vm is the triangular wave maximum oscillation amplitude. In the concrete electric circuit, the feedback signal and the datum sine wave signal sends in the differentiator amplifier, the outlet error signal compares again with the standard triangular wave, produces the SPWM driving signal. Here uses the triangular wave the oscillation amplitude is Vm=3V.
In synthesis, before has not joined the compensating network, the entire loop gain is
G (s) =G1(s)H(s)Gd(s)
=K1/(s2LC 1)[K2/(sR1C2 R1/R2 1)(1/Vm)
=1.569/[(6×10-9s2 1) (7×10 -6s 1)
Draws up its frequency Bode chart, as shown in Figure 3.
3 compensating network design
By the fore-mentioned Bode theorem, after the compensating network joins the loop gain should satisfy, the frequency approach line passes through cutting by - the 20dB/dec slope (ωc spot), and in cuts about at least the frequency from to 2ωc the scope in maintains this slope is invariable.
From the request, first chooses the cutting frequency. In the practical application, chooses fc=fs/5 is suitable, fs is the invertor operating frequency or the switching valve turn-on frequency. In the concrete invertor, the turn-on frequency is 40kHz, then fc=40/5=8kHz.
Before has not added the compensating network loop gain Bode chart as shown in Figure 3, in the fc=8kHz place gain for - 20.17dB, from this, the compensating network should satisfy the following condition, namely in the fc=8kHz place gain is 20.17dB, the slope is 20dB/dec, moreover, this slope, in fc/4=2kHz and 2fc=16kHz (take 15kHz) in the scope maintains invariable. Compensating network’s Bode chart as shown in Figure 4 (a frequency).
May result in by Figure 4: f1=2kHz place, G (Omega) =20lg (2πf1) =8.129dB or 2.55 (multiple) =AV1, f2=15kHz place, G (Omega) =20lg (2πf2) =25.63dB or 19.12 (multiple) =AV2, two zero value correspondence frequency is fz1=fz2=2kHz, an extreme value in fp1=15kHz place, another extreme value in fp2=20kHz place. The consideration selects when as shown in Figure 5 the compensated amplifier, its resistance electric capacity parameter value may calculate as follows:
Takes R3=5.1kΩ, R0=39kΩ, then R2=R3AV2=97.5kΩ, C2=1/(2πfp2R2) =81.6pF, C1==816pF, R1=1/(2πfp1R3) =39kΩ, C3=π=2040pF.
In the actual electric circuit, takes R2=100kΩ, C2=100pF, C1=800pF, R1=39kΩ, C3=2200pF.
4 experimental results
Joins after the above compensating network, the invertor may have the full load and the steady work, its IGBT tube both sides voltage vCE and output voltage vo profile as shown in Figure 6, the electric circuit working condition is: Power P=500W (full load), bus bar voltage Vin=±180V.
5 conclusions
The experimental result indicated that control theory’s frequency response law will apply in the invertor voltage simple ring reaction control design has its direct-viewing simple merit, simultaneously realizes easy. After the invertor electric circuit joins the compensating network, its stability has the improvement. The deficiency lies, the output wave shape when the misalignment load and the variation of load is big the distortion is obvious, needs to seek the better adjustment method to improve.
