A kind of resonance DC/DC converter and control method thereof
[technical field]
The present invention relates to the DC power supply converter technique, specifically relate to a kind of resonance DC/DC converter and control method thereof.
[background technology]
Miniaturization and high frequencyization are the trend of current power supply development, but the rising of switching frequency has brought the excessive problem of switching tube loss, and this is that traditional B UCK converter is insurmountable, and controlled resonant converter then can address this problem preferably.
With the series resonant converter is example, series-resonant direct-current/direct-current (DC/DC) converter using resonant transformation technology, because resonant element is operated in the sinusoidal resonance state, the voltage natural zero-crossing on the switching tube can realize that no-voltage is open-minded, and the loss of power is very little.This topology adopts Frequency-variable Modulation (Pulse frequencymodulation is called for short PFM) mode usually, comes regulated output voltage by changing operating frequency.Fig. 1 is the citation form of full-bridge SRC series-resonant direct-current/direct-current converter, when this circuit is adopted PFM control, the complementary symmetry of two groups of switching tube S1, S4 and S2, S3 drives, the switch periods of each conducting 50% (this be an ideal value, should be as the setting in consideration dead band and is slightly less than 50%).The pass of electric power output voltage gain M and operating frequency f is:
Wherein, V
oWith V
InBe respectively output, input voltage, f is an operating frequency,
f
rBe resonance frequency, L
rBe resonant inductance value, C
rBe resonant capacitance value, P
oBe power output.
From formula (1), can find, when operating frequency f greater than resonance frequency f
rThe time, operating frequency is high more, and voltage gain M is low more; In like manner, when operating frequency f less than resonance frequency f
rThe time, operating frequency is low more, and voltage gain M is low more.Series resonance topology control frequency f and output voltage V
oRelation curve as shown in Figure 2.Can find that by Fig. 2 main difficult point problem of series resonant converter is that output voltage is difficult to stablize under underloading and the idle condition.When control frequency greater than resonance frequency f
rThe output voltage of series resonance topology reduces along with the rising of control frequency, when load is decreased to light condition, output voltage tends towards stability, for burning voltage, operating frequency need rise very highly like this, but the problem that the wide meeting of operating frequency range brings magnetic device to be difficult to optimize, and operating frequency is high more, and circuit loss is also big more; In addition, approaching unloaded when load, output voltage might rise on the contrary, causes carrying out negative feedback control.Therefore in the power supply industry, there is the people to add fixing load, utilizes this method regulated output voltage under underloading and idle condition, but can increase no-load loss like this, reduce power-efficient at output.
In a word, simple variable frequency control can cause operating frequency range wide even lost efficacy, and the problem of bringing magnetic element to be difficult to optimize, and the FEEDBACK CONTROL problem that is difficult to design are so simple frequency modulation control can't satisfy underloading or the requirement of output voltage stabilizing when unloaded.
Be that example has illustrated the defective that the frequency modulation control mode exists with the full-bridge series resonance circuit above, same, theoretically, all there is similar problem in the resonant circuit of all frequency modulation control.
[summary of the invention]
The object of the present invention is to provide a kind of control method and device thereof of resonance DC/DC converter, adopt frequency modulation control to have the problem that is difficult to voltage stabilizing in underloading or when unloaded to solve in the prior art loaded work piece.
To achieve these goals, the technical solution used in the present invention is: a kind of control method of resonance DC/DC converter, be to regulate output voltage by the turn-on frequency that changes its resonant circuit input switch pipe, also the feedback signal according to load circuit makes converter enter phase-shift control mode, adjust the phase shifting angle of described switching tube, make the voltage output range of resonant circuit expand.
Described resonant circuit input switch pipe can be controlled through drive circuit by driving pulse, and the acquisition of described driving pulse comprises the steps,
1) obtains the feedback signal of load circuit;
2) judge according to described feedback signal whether load is operated in underloading or Light Condition;
3) the common pulse signal of regulating makes resonant circuit be operated in phase shifting control and controls mixed mode with PFM as the driving pulse of drive circuit if loaded work piece at underloading or Light Condition, then changes frequency and phase shifting angle with feedback signal; Otherwise the pulse signal that the stable and frequency of phase shifting angle is changed with feedback signal makes resonant circuit be operated in the PFM control model as the driving pulse of drive circuit.
Described step 1) can comprise the steps:
1a) sampling feedback voltage from load circuit;
1b) described feedback voltage is carried out the negative feedback compensation operation and obtain feedback signal.
PFM control is preferably realized by following process with the selection of PFM+ phase shifting control in the above-mentioned control method: described step 2) feedback signal is obtained frequency change control signal and phase-shifted control signal respectively after calculation process; The calculation process process of described phase-shifted control signal comprises with first reference voltage comparing to judge whether load is operated in the step of underloading or Light Condition, and described first reference voltage is determined according to the electrical characteristic of described load; When loaded work piece during, promptly produce in the described step 3) by frequency change control signal and the common driving pulse of regulating of phase-shifted control signal at underloading or Light Condition; Otherwise, produce the stable and driving pulse that regulate separately by frequency change control signal of phase shifting angle.
Also can further increase the selection of independent employing phase shifting control in the above-mentioned control method: the calculation process process of frequency change control signal described step 2) comprises comparing with second reference voltage judges whether load is operated in the step of approximate Light Condition, described second reference voltage is determined according to the electrical characteristic of described load, and described second reference voltage satisfies, if load then must be operated in underloading or Light Condition according to the judgement of first reference voltage according to the judgment task of second reference voltage at approximate Light Condition; When loaded work piece during, promptly produce frequency stabilization and the driving pulse regulated separately by phase-shifted control signal in the described step 3) at approximate Light Condition.
Be purpose of the present invention, a kind of resonance DC/DC converter also is provided, comprise
Drive circuit and resonant circuit, described drive circuit is according to the input switch pipe of the driving pulse control resonant circuit of input, and resonant circuit offers load circuit with the power supply after the conversion under the control of described drive circuit, and
Negative feedback compensated regulator, the feedback voltage that will sample from load circuit carry out obtaining feedback signal behind the negative feedback compensation operation;
Discriminator is according to the load condition of the feedback signal judgement load circuit of importing, output driving pulse modulation signal;
Drive-pulse generator, according to the driving pulse modulation signal modulation and the output driving pulse of input: if loaded work piece at underloading or Light Condition, the frequency of the driving pulse of output and phase shifting angle change with feedback signal and common adjusting; Otherwise the stable and frequency of the phase shifting angle of the driving pulse of output changes with the variation of feedback signal.
A kind of possibility that above-mentioned resonance DC/DC converter is carried out PFM control and PFM+ phase shifting control is to adopt two relatively independent drive signal generation circuit modules, that is, described drive-pulse generator comprises frequency conversion control circuit and variable frequency control+phase-shift control circuit; If described discriminator is judged loaded work piece at underloading or Light Condition, then, control its output frequency and phase shifting angle and change and the common driving pulse of regulating with feedback signal to variable frequency control+phase-shift control circuit output driving pulse modulation signal; Otherwise,, control the driving pulse that the stable and frequency of its output phase shifting angle changes with the variation of feedback signal then to frequency conversion control circuit output driving pulse modulation signal.
A kind of preferred version that above-mentioned resonance DC/DC converter is carried out PFM control and PFM+ phase shifting control then can only adopt a drive signal generation circuit module: described discriminator comprises first computing circuit and 2, described computing circuit is set with first reference voltage, and described driving pulse modulation signal comprises frequency change control signal and phase-shifted control signal; Described feedback signal through first computing circuit carry out computing, with first reference voltage relatively after, be output as phase-shifted control signal; Described feedback signal carries out being output as frequency change control signal after the computing through second computing circuit; Described first reference voltage determines according to the electrical characteristic of load, makes that the phase-shifted control signal of output changes with the variation of feedback signal when loaded work piece during at underloading or Light Condition; Otherwise, then keep stable; Described drive-pulse generator comprises frequency conversion control circuit and phase-shift control circuit, described frequency conversion control circuit produces the variable-frequency pulse of frequency by frequency change control signal control, and described phase-shift control circuit carries out described variable-frequency pulse to export driving pulse after the phase shifting control according to phase-shifted control signal.
Described phase-shift control circuit can be with phase-shifted control signal and the relatively more synthetic back output of variable-frequency pulse driving pulse.
Also can be, described variable-frequency pulse is imported the synchronous end of described phase-shift control circuit, described phase-shift control circuit carries out exporting driving pulse after the phase shifting control according to the variable-frequency pulse of phase-shifted control signal to input.
For realizing the pure phase shifting control control under the approximate Light Condition, further preferably, described second computing circuit is set with second reference voltage; Described feedback signal through second computing circuit carry out computing, with second reference voltage relatively after, be output as frequency change control signal; Described second reference voltage determines according to the electrical characteristic of load, makes that the frequency change control signal of output keeps stablizing when loaded work piece during at approximate Light Condition; Otherwise then the variation with feedback signal changes; And described second reference voltage satisfies, if load then must be operated in underloading or Light Condition according to the judgement of first reference voltage according to the judgment task of second reference voltage at approximate Light Condition;
Described frequency conversion control circuit can adopt such structure, comprises the voltage-frequency oscillator and the triangular-wave generator that connect successively; Described frequency change control signal input voltage-frequency oscillator is controlled its frequency of oscillation, the controlled variable-frequency pulse of described triangular-wave generator output frequency.
Adopt technique scheme, beneficial technical effects of the present invention is:
1) in the control mode of resonance DC/DC converter, introduce phase-shift control mode, realize PFM and two kinds of control modes of PFM+ phase shifting control, when the power work frequency is low, adopt PFM control, introduce the PFM+ phase-shift control mode when power work frequency is too high.Also can be when frequency be higher fixed frequency, enter phase-shift control mode, thereby solved the difficult point problem of controlled resonant converter, promptly unloaded and problem that underloading is difficult to voltage stabilizing, greatly increase the output voltage regulating power of resonant circuit, effectively expanded voltage output range.
Traditional phase whole-bridging circuit is a kind of very outstanding DC-DC variator, utilizes the resonant inductance energy to realize the zero voltage switch of switching tube equally, has reduced the switching loss of switching tube.Advantages such as it has circuit and control is simple, switching tube is realized soft switch easily, circuit efficiency height, EMI are little, be described as one of best DC to DC converter, and phase shifting control is under the situation of underloading and zero load, input voltage and load are fixed, and phase shifting angle and output voltage have monotonic relationshi preferably.Though this kind circuit is in order to realize soft switch, need bigger output differential mode inductance to realize the soft switch of leading-bridge, and for the volume that guarantees resonant inductance can not be excessive, can only on a fixed load (as 50% load), could realize the soft switch of lagging leg, on off state before is hard switching, cause soft switch condition harshness, efficient and thermal design are in hard switching attitude difficulty, but because the present invention also keeps the part characteristic of resonance DC/DC converter, therefore its soft switch condition is still relatively easy, and can not bring difficulty owing to the introducing of phase-shift circuit.
2) PFM control logically is optional relation in parallel with PFM+ phase shifting control dual mode, but can adopt the mode of preferred frequency conversion control circuit and phase-shift control circuit series connection to realize, the preferred circuit of realizing has been avoided the use of many logic gating devices and multichannel negative feedback compensated regulator, circuit is simple, and two kinds of control modes are switched smoothly, and its reliability and dynamic characteristic are very good.
The present invention is suitable for the distortion topology of resonant circuits such as full-bridge series and parallel resonant circuit and LLC, and stronger engineering significance is arranged.
The present invention is described in further detail below in conjunction with the drawings and specific embodiments:
[description of drawings]
Fig. 1 is a kind of existing typical full-bridge series resonance circuit diagram
Fig. 2 is the output characteristic curve figure under the circuit PFM control mode among Fig. 1.
Fig. 3 is that the output characteristic curve after the circuit employing control method of the present invention compares schematic diagram among Fig. 1.
Fig. 4 is a kind of theory diagram of DC to DC converter of the present invention.
Fig. 5 A is the refinement block diagram of circuit in Fig. 4 frame of broken lines.
Fig. 5 B is the physical circuit instance graph of pi regulator among Fig. 5 A, first computing circuit and second computing circuit.
Fig. 5 C is that phase shifting angle produces circuit diagram.
Fig. 6 be among Fig. 4 DC to DC converter feedback signal and frequency and phase shifting angle concern schematic diagram.
Fig. 7 is drive pulse waveform figure.
Fig. 8 is the theory diagram of another kind of DC to DC converter of the present invention.
[embodiment]
A kind of control method of resonance DC/DC converter, be to regulate output voltage by the turn-on frequency that changes its resonant circuit input switch pipe, also adjust the phase shifting angle of described switching tube, make the voltage output range of resonant circuit expand according to the feedback signal (obtaining) of load circuit from the output voltage or the output current sampling of controlled resonant converter.
Be example still with the full-bridge series resonance circuit among Fig. 1, curve A 1~A3 among Fig. 3 adopts independent PFM control mode under the different loads situation, phase shifting angle is the output voltage V o of 0 degree and the characteristic curve schematic diagram of frequency f, as can be seen, along with alleviating of load, Vo tends towards stability, and is difficult to stablize by the mode that improves operating frequency.Curve B 1 among Fig. 3, B2, B3 be respectively the switching frequency of curve A 1~A3 when being elevated to fa in the PFM control mode characteristic curve schematic diagram of output voltage V o and phase shifting angle α and frequency f under the comprehensive system mode after the incision phase shifting control; Control to phase shifting angle α is spent till 180 degree since 0.By curve B 1, B2, B3 as can be seen, after having introduced the PFM+ phase shifting control, output voltage can begin greater than fa at control frequency, to control frequency fb rapidly to 0, compare output characteristic with simple PFM control mode and obtained very big improvement.
Control to resonant circuit input switch pipe is carried out through drive circuit by driving pulse, therefore the switching from PFM to the PFM+ phase shifting control realizes by the variation of driving pulse, because the switching of control mode is directly related with loading condition, so following method is adopted in the acquisition of driving pulse:
1) obtain the feedback signal of load circuit, described feedback signal is obtained through the negative feedback compensation operation by the feedback voltage of sampling from load circuit;
2) judge the operating state of load according to described feedback signal, this deterministic process by to the computing of feedback signal and with relatively the carrying out of reference voltage: feedback signal is obtained frequency change control signal and phase-shifted control signal respectively after calculation process.The calculation process process of phase-shifted control signal is feedback signal to be carried out compare with first reference voltage of setting after the computings such as ratio or plus-minus, (or be higher than reference voltage if be lower than reference voltage, logic behaviour according to circuit is determined, here suppose that feedback signal is directly proportional with load voltage, feedback signal is low more, and load is light more) then directly export feedback signal after the computing as phase-shifted control signal.Therefore otherwise export the first stable reference voltage as phase-shifted control signal, because the phase-shifted control signal of this moment do not change, corresponding to the situation of simple PFM control.First reference voltage need be determined according to the electrical characteristic of described load, makes whether the reaction load that can be in the main true of relatively judging of carrying out based on first reference voltage is operated in light condition.
The calculation process process of frequency change control signal can adopt dual mode, the one, directly feedback signal is carried out exporting as frequency change control signal after the computings such as ratio or plus-minus, like this, frequency change control signal changes with the variation of feedback signal all the time, so resonant circuit has remained the PFM control mode.Another is the compute mode of copying above-mentioned phase-shifted control signal, after computings such as ratio or plus-minus, also second reference voltage of result and setting is compared, (or be lower than reference voltage if be higher than second reference voltage, logic behaviour according to circuit is determined, here suppose that feedback signal is directly proportional with load voltage, feedback signal is low more, and load is light more) then directly export feedback signal after the computing as frequency change control signal.Therefore otherwise export the second stable reference voltage as frequency change control signal, because the frequency change control signal of this moment do not change, corresponding to the situation of simple phase shifting control.Second reference voltage need be determined according to the electrical characteristic of described load equally, makes whether the reaction load that can be in the main true of relatively judging of carrying out based on second reference voltage is operated in approximate Light Condition.And, in order to guarantee to control to PFM+ phase shifting control taking over seamlessly to phase shifting control again from PFM, if load then must be operated in light condition according to the judgement of first reference voltage according to the judgment task of second reference voltage at approximate Light Condition, this has also just guaranteed in any case, has at least one to be that variation with feedback signal changes in phase-shifted control signal and the frequency change control signal.
3) according to step 2) judged result, if loaded work piece is in non-light condition, phase-shifted control signal stabilizes to first reference voltage, and frequency change control signal changes with the variation of feedback signal, therefore what export under the control of these two signals is stable and the driving pulse that frequency changes with feedback signal of phase shifting angle, and this moment, resonant circuit was operated in the PFM control model; If loaded work piece is in light condition, phase-shifted control signal and frequency change control signal all change with the variation of feedback signal, what therefore export under the control of these two signals is that frequency and phase shifting angle all change and the common driving pulse of regulating with feedback signal, and this moment, resonant circuit was operated in PFM+ phase shifting control pattern; If loaded work piece is at approximate Light Condition, phase-shifted control signal changes with the variation of feedback signal, and frequency change control signal stabilizes to second reference voltage, therefore what export under the control of these two signals is frequency stabilization and driving pulse that phase shifting angle changes with feedback signal, and this moment, resonant circuit was operated in the phase shifting control pattern;
Introduce a kind of resonance DC/DC converter that adopts above-mentioned control method below in detail,, comprise in conjunction with Fig. 4 and Fig. 5 A, 5B, 5C
Drive circuit and resonant circuit, described drive circuit is according to the input switch pipe of the driving pulse control resonant circuit of input, and resonant circuit offers load circuit with the power supply after the conversion under the control of described drive circuit, and
The negative feedback compensated regulator is served as by pi regulator, and the given voltage of the feedback voltage that will sample from load circuit and setting carries out obtaining feedback signal behind the negative feedback compensation operation;
Discriminator comprises first computing circuit and 2, and described first computing circuit and 2 is set with first reference voltage and 2 respectively; Feedback signal through first computing circuit carry out computing, with first reference voltage relatively after, be output as phase-shifted control signal; Feedback signal is carried out computing, is output as frequency change control signal with second reference voltage after relatively through second computing circuit; First reference voltage and second reference voltage all determine that according to the electrical characteristic of load its setting principle is identical with the description in the aforementioned control method, does not repeat them here.Its establishing method can adopt measurement method, such as: make loaded work piece at underloading and approximate unloaded two states artificially respectively, measure the size of the input feedback signal of comparing with first reference voltage and second reference voltage respectively in first computing circuit and second computing circuit under this two states respectively, this value promptly can be used as the value of first reference voltage and second reference voltage.When loaded work piece during at approximate Light Condition, frequency change control signal (i.e. second reference voltage) that discriminator output is stable and the phase-shifted control signal that changes with feedback signal; When loaded work piece during in light condition, phase-shifted control signal and frequency change control signal that discriminator output all changes with feedback signal; In addition, the frequency change control signal that changes with feedback signal of discriminator output and stable phase-shifted control signal (i.e. first reference voltage).
Drive-pulse generator comprises frequency conversion control circuit and phase-shift control circuit.Frequency conversion control circuit comprises voltage-frequency oscillator and the triangular-wave generator that connects successively; Phase-shift control circuit is that phase shifting control produces circuit.Frequency change control signal input voltage-frequency oscillator is controlled its frequency of oscillation, the voltage-frequency oscillator becomes the square-wave signal that output frequency changes with the voltage signal of frequency change control signal, and the square-wave signal that triangular-wave generator receives this frequency change produces the triangular signal of frequency change and outputs to the synchronous end that phase shifting control produces circuit; Phase shifting control produces circuit according to the phase-shifted control signal V from the input of comp end
CompFrequency conversion triangular signal to input carries out exporting driving pulse after the phase shifting control.Work as V
CompStablize when being output as first reference voltage, the phase shifting angle of driving pulse no longer changes, and has only frequency change, promptly enters PFM control; Work as V
CompWhen all changing output with frequency change control signal, the phase shifting angle and the frequency of driving pulse change simultaneously, promptly enter the PFM+ phase shifting control; When frequency change control signal is stable when being output as second reference voltage, the frequency of driving pulse no longer changes, and has only phase shifting angle to change, and promptly enters phase shifting control.
Generally make reference signal 1 be half of triangular wave peak value, the driving pulse of exporting when the PFM control model has 50% duty ratio like this.Certainly, according to application need, also can cancel independent PWM control mode, make the output frequency of driving pulse relevant with feedback signal all the time, this just is equivalent to reference signal 2 is set to 0.
The module that can realize frequency conversion control circuit and pulse-width modulation circuit function is integrated in the chip piece usually, difference according to use chip internal function structure, physical circuit relation between them can have more situation, and for example pulse-width modulation circuit also can be with pulse-width signal V
CompWith the relatively more synthetic back output of frequency conversion triangular signal driving pulse.That but simplifies it seems that frequency conversion control circuit and pulse-width modulation circuit always may be summarized to be the circuit relationships of series connection, and promptly frequency conversion control circuit produces pulse signal and controls its frequency, and pulse-width modulation circuit is then further determined the duty ratio of pulse signal.
As shown in Figure 6, be the feedback signal V of the negative feedback compensated regulator output of above-mentioned resonance DC/DC converter
fWith the functional relation of resonant circuit operating frequency f and resonant circuit output V,
Wherein, transverse axis is represented the feedback signal V of negative feedback compensated regulator output
f, and the highest output of supposition negative feedback compensated regulator 12v.The operating state of above-mentioned resonance DC/DC converter is described below:
When feedback signal when a~b changes, control frequency is constant, is f=fmax, phase shifting angle changes to α from 180 degree
0Degree; Resonant circuit output V changes to V1 from 0; Mode of operation is a phase shifting control.
When feedback signal when b~c changes, control frequency and duty ratio change simultaneously: phase shifting angle is from α
0Degree changes to 0 degree, causes output voltage to continue to rise; Simultaneously, control frequency also is reduced to f0 from fmax, and this variation also causes rise of output voltage; When loop voltage rose to b, phase shifting angle reached maximum 0 degree, and output voltage rises to V2; Working method is the PFM+ phase shifting control.
When feedback signal when b~12V changes, phase shifting angle is constant, control frequency continues to drop to fmin from f0, output voltage continues to rise, when f=fmin, output voltage reaches maximum V3; Mode of operation is PFM control.
As can be seen from Figure 6, feedback signal raises, and output voltage also raises, so can do closed-loop control.The corresponding reference signal 1 of determining first computing circuit of c among Fig. 6, the corresponding reference signal 2 of determining second computing circuit of b; When control system does not comprise pure pwm pattern, b=0.
The waveform of driving pulse as shown in Figure 7 under the different control models, waveform A is the drive pulse waveform that stabilizes to 0 degree phase shifting angle under the PFM control, waveform B is the drive pulse waveform under the PFM+ phase shifting control, the phase shifting angle of driving pulse and frequency change with output voltage regulates, and is 180 degree up to phase shifting angle.
Above-mentioned resonance DC/DC converter has adopted the frequency conversion control circuit and the phase-shift control circuit structure of connecting on the circuit to realize in logic optionally PFM control and PFM+ phase shifting control scheme side by side, in fact also can adopt alternative dispensing means, as shown in Figure 8, promptly adopt two relatively independent drive signal generation circuit modules, set up frequency conversion control circuit and variable frequency control+phase-shift control circuit respectively; Make discriminator optionally control one of them output then and satisfy the driving pulse that the present invention controls requirement according to the load judgment situation, that is: if described discriminator judges that loaded work piece is at underloading or Light Condition, then, control its output frequency and phase shifting angle and change and the common driving pulse of regulating with feedback signal to variable frequency control+phase-shift control circuit output driving pulse modulation signal; Otherwise,, control the driving pulse that the stable and frequency of its output phase shifting angle changes with the variation of feedback signal then to frequency conversion control circuit output driving pulse modulation signal.This possibility can be realized the object of the invention equally, only compares with aforementioned preferred version to have more complicated form.
Used negative feedback compensation operation is not only can realize with pi regulator among the present invention, also available PID adjuster and more flexibly adjuster wait and realize.
Control mode of the present invention is applicable to the circuit that uses resonance principle work, comprises series resonance, parallel resonance, series parallel resonance etc., and circuit topology can be a full-bridge.When the operating frequency of resonant circuit is hanged down, make converter be operated in the variable frequency control mode; And when operating frequency is higher, make converter be operated in variable frequency control+phase-shift control mode, and so just avoided under unloaded and the underloading condition, the problem that switching frequency is too high, output voltage is stable when helping underloading.The present invention can be realized that when circuit switched, switching was level and smooth, had guaranteed the reliability of circuit working by simple circuit configuration between the state of different control modes.
Variable frequency control+phase shifting control of the present invention, variable frequency control and discriminator circuit etc. both can adopt hardware circuit to build also and can realize by the chip with the corresponding function module is programmed with software according to control method of the present invention, this part work be those of ordinary skill in the art according to above-mentioned technical scheme institute easily derivation come out.