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AND8112/D A Quasi-Resonant SPICE Model Eases Feedback Loop Designs
Prepared by: Christophe Basso Prepared by: Joel Turchi ON Semiconductor http://onsemi.com
Within the wide family of Switch Mode Power Supplies (SMPS), the Flyback converters represent the structure of choice for use in small and medium power applications. For compact designs and radio-frequency sensitive applications, e.g. TV sets or set-top boxes, Quasi- Resonant power supplies start to take a significant market share over the traditional fixed frequency topology. However, if the feedback loop control is well understood with this latter, for instance via a comprehensive literature and SPICE models, the situation differs for self-oscillating variable switching frequency structures where no model still exists. This article will show how a simple large-signal averaged SPICE model can be derived and used to ease the design work during stability analysis. Quasi-Resonant Operation It is difficult to abruptly dig into the analytical analysis without giving a basic idea of the operation of a converter working in Quasi-Resonance (QR). Figure 1 depicts a typical FLYBACK converter drain-source waveform as you probably have already observed. When the switch is closed, the drain-source voltage VDS is near 0 V and the input voltage Vg appears across the primary inductance LP: the current inside LP ramps up with a slope of
SON + Vg LP
(eq. 1)
Leakage Inductance Plateau: (Vout + Vf)/N Core is Reset
Vin
ON
OFF Valleys
Figure 1. A Typical FLYBACK Drain-Source Waveform
Ipeak Soff = (V + Vf) / (LP x N) Son = Vg/LP
ON
OFF
IP = 0, Reset
When the controller instructs the switch opening, the drain-source quickly rises and the energy transfer between primary and secondary takes place: the secondary diode conducts and the output voltage flies back on the primary side, over LP. This "Flyback" plateau is equal to Vg + (V + Vf) / N, where N is the secondary to primary turn ratio, V the output voltage and Vf the diode forward voltage drop. During this time, the primary current decreases with a slope now imposed by the reflected voltage
SOFF + (V ) Vf) N LP
(eq. 2)
0
Figure 2. The Primary Current Ramps Up and Down to Zero in DCM
Figure 2 zooms on the simulated primary current (actually circulating in the magnetizing inductor), showing how it moves over one switching cycle.
When the primary current reaches zero, the transformer core is fully demagnetized: we are in Discontinuous Conduction Mode (DCM). The primary inductance LP together with all the surrounding capacitive elements Ctot create a LC filter. When the secondary diode stops conducting at IP = 0, the drain branch is left floating since the MOSFET is already open. As a result, a natural oscillation occurs, exhibiting the following frequency value:
Fring + 1 2 @ p @ LP @ Ctot
(eq. 3)
(c) Semiconductor Components Industries, LLC, 2003
1
October, 2003 - Rev. 1
Publication Order Number: AND8112/D
AND8112/D
As in any sinusoidal signal, there are peaks and valleys. When you re-start the switch in one valley, where the voltage is minimum, the MOSFET is no longer the seat of heavy turn-on losses engendered by capacitive effects: this is the so-called Quasi-Resonance operation where the switching frequency depends on the peak current, the various slopes ON and OFF and the number of valleys you choose after the core reset. In our study, we will first concentrate on a simplified SPICE version where the power switch is actuated right after the reset detection point (parasitic ringings are neglected) and later on, a more sophisticated declination will incorporate parasitic delays. Modeling the Switch Network Figure 3 depicts a Flyback topology where the switching elements generating the above waveforms have been highlighted: the power switch (usually a MOSFET transistor) and a diode, performing a rectification job. During the converter operation, the Pulse Width Modulator controller (PWM) instructs the transistor to turn ON, in order to store energy in the primary side. The primary current builds up until the setpoint imposed by the feedback loop is reached. At this time, the controller toggles the transistor to the OFF state and energy transfers to the secondary side. If the ON and OFF states can be described by a set of linear equations, there exists a discontinuity linking these two events. Despite the presence of linear elements in the converter (capacitors, inductors and resistors), the presence of the commuting switch clearly introduces the non-linearity that prevents us from directly writing the small-signal equations... When learning electronic circuits at school, there were some exercises in which we were asked to reveal the transfer function of bipolar amplifiers. At that time, we learned to replace the transistor symbol by its equivalent small-signal model: the schematic turned into the simple association of current and voltage sources that greatly simplified the analysis. In the average circuit modeling technique, we also follow the same philosophy: the exercise lies in isolating and replacing the switch network with a set of current and voltage sources whose electrical architecture do not vary with time. Therefore, plugging the equivalent model back into the converter of interest allows us to resolve its transfer characteristics.
IL(t)
1
IL(t)
Vg LP
IP
V N @ LP
*
I1(t)
Vg LP
IP
(V(LP) Vg
* V N 2 @ LP
I2(t)
IP/N
V1(t)
V ) Vg N
*V N d @ TS TS
V2(t)
N @ Vg ) V
d @ TS
TS
Figure 4. 0and Individual Signals Separately Plotted.
Deriving Equations The object of deriving a model consists in writing the equations that describe the switch network averaged input and output quantities that a) depend from each other b) obey to the control input. Let us draw the various waveforms before starting any line of algebra (Figure 4). From this picture, we can develop equations that will finally describe the averaged evolution of the values of interest, the input / output voltage and current of our switch network:
I t I1(t) u+ P 2 t I2(t) u+ IP 2 N N Vg ] V)N d d Vg d d
(eq. 4) (eq. 5)
1:N LP
t V1(t) u+
+ Vg I1(t) d
2
(eq. 6) (eq. 7)
I2(t) a
3
t V2(t) u+ [V ) N
V1(t) s Control k
5
V2(t)
C
R
V(t)
where: V is the output voltage, Vg the input voltage, IP the primary peak current, N the NS / NP turn ratio and d the duty-cycle (d = 1 -d). Please note that in this first approach, we do not consider any delay occurring at the switch opening or induced by equation 3. These events will considered later on, in a more complex model.
Figure 3. A Flyback Power Supply where Switches have been Isolated0
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Averaging Input / Output Voltages From the inductor volt-second balance approximation, we know that the average voltage across an inductor operated in a steady-state converter is null. By looking at the V(LP) sketch, we obtain the following equation:
t V(LP) u+ d(t) t Vg(t) u * d (t) t V(t) u + d(t) N t Vg(t) u * (1 * d(t)) N t V(t) u +0
(eq. 8)
which lets us extract the classical output / input voltage ratio
t V(t) u +N t Vg(t) u d(t) (1 * d(t))
(eq. 9)
and as a result, the duty-cycle expression:
d(t) + tV t V(t) u (t) u ) N t Vg(t) u
(eq. 10)
Now, by plugging equation 10 in equation 6, we obtain the average voltage across the primary switch terminal: =
t V(t) u ) N N t Vg(t) u (1 * d(t)) + t V(t) u ) N N t Vg(t) u 1* t V(t) u +t Vg(t) u t V(t) u ) N t Vg(t) u
(eq. 11)
which agrees with the inductor volt-second balance approximation (from Figure 1 since, by definition, = 0, then Vg appears across the switch terminals). To reveal , let us plug equation 10 into 7: =
[t V(t) u ) N t Vg(t) u] d(t) + [t V(t) u ) N t Vg(t) u] t V(t) u +t V(t) u t V(t) u ) N t Vg(t) u
(eq. 12)
which again could be deduced from Figure 3 since the average voltage across the secondary inductance is zero... Averaging Input / Output Currents The peak inductor current depends on the time during which Vg is applied over LP. If we recall that this time (actually ton ) is d x TS, then:
Vg IP + LP d TS
(eq. 13)
TS t I2(t) u+ 1 TS I2(t) @ dt + 1 2 d.TS IP N d (t)
(eq. 16)
plugging equation 13 in 16 leads to:
t I2(t) u+ t Vg(t) u 2 d(t) N (1 * d(t)) LP (1 * d(t)) LP TS TS +
(eq. 17)
From Figure 4, the average current can be obtained by evaluating the triangular area (charge in Coulomb) and dividing by the switching period. This is expressed by equation 4. Now plugging equation 13 in 4, we obtain:
t I1(t) u+ 1 2 t Vg(t) u LP t Vg(t) u 2 d(t) d(t) 2 LP TS TS d(t) +
(eq. 14)
t V1(t) u
d(t) 2N
however, from equation 11, we know that = thus equation 14 turns into:
t I1(t) u+ t V1(t) u 2 d(t) 2 LP TS
(eq. 15)
A 100% Efficiency Power Transfer... Assuming that 100% of the primary stored energy is released to the secondary side, then we can use equations 11 and 12 to write:
= X = X
(eq. 18)
Applying the same technique to the secondary current I2(t), leads to:
From equation 15, we can see that a current is generated by a voltage multiplied by a term. This term is obviously homogenous to the inverse of an impedance. By re-arranging equation 15, we obtain:
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Re(d) + t V1(t) u 2 LP + t I1(t) u d(t) 2 TS
(eq. 19)
where the input impedance depends on the duty-cycle d(t). However, in quasi-resonant converters, the power transfer adjusts by varying the peak current IP which finally imposes the operating frequency. Since by definition ton = d x TS, we can re-arrange equation 9 to reveal
TS + ton (N Vg) ) V V
(eq. 20)
Our switch network can thus modeled according to the so-called loss-free network where all the power developed across an input resistance transfers to the output without any loss (Figure 6) [1]. A More Complex Model Including Parasitic Effects The above simplified model assumes that there are no transient times between the conduction and demagnetization phases. A more precise modelling approach requires that the two following delays Dt1 and Dt2 are taken into account, as highlighted by Figure 7 and 8:
By finally plugging equation 10 and 20 into 19, we obtain a ton -dependent input effective resistance definition:
Re(ton) + 2 LP (V ) N ton V Vg)
(eq. 21)
Core Is Reset!
that Figure 5 portrays:

Re(ton) Dt1 Dt2
Figure 5.
The average input waveforms of the switch can be modeled via the above equivalent network.
Figure 7.
The presence of a capacitive node slows-down the VDS rising and makes the drain sinusoidally ring at the core reset...
Re(ton)

1. At the end of the ON-time, the power switch opens but the energy transfer to the secondary side does not start immediately. The primary inductor current (IP) that cannot flow through the power switch, charges the surrounding capacitive elements (Ctot) until Ctot voltage exceeds
Vg ) V N
(eq. 23)
Figure 6.
The two-port loss-free network where all the input power transforms into output power.
As reference [1] details, the apparent power consumed by Re, Pin, is entirely transmitted to the output since we assume a 100% efficiency. Therefore, equation 19 can be re-written by:
t P(t) u+t V1(t) u t V2(t) u t I1(t) u+ t V1(t) u 2 t I2(t) u+ Re(ton)
(eq. 22)
At that moment, the secondary diode starts to conduct and current feeds the output capacitor. One can assume that the Ctot charging time (Dt1) is short enough to consider that the primary inductor current stays equal to IP during this interval. Then, Dt1 is the time necessary to charge the capacitor Ctot with a current IP from zero to
Vg ) V N
, i.e.: Dt1 + Ctot
V Vg ) N IP
(eq. 24)
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D1 V + 1 2 Vg 3 4 NCP1207 8 7 6 5 Ctot <=> Ctot Cout Vg
Figure 8.
When the power switch turns off, the primary inductor behaves like a current source that charges the Ctot capacitor. This sequence ends when voltage developed across Ctot exceeds [Vg+(V+Vf)/N], that is when the secondary diode D1 starts to conduct.
2. At the end of the core reset, both switches (power switch and secondary diode) are off. The primary inductor LP together with Ctot form a LC network. Ctot voltage (and thus the drain source voltage of the power switch) oscillates around the input voltage Vg between a peak value (the initial level:
Vg ) V ) and a valley value Vg * V , the damping N N
replacing d(t) by its novel value as expressed by equation (27):
t V(LP) u+ d Vg * 1*d*
Dt1)Dt2 TS
V
(eq. 28)
N
+0
Re-arranging equation (28), one can unveil the duty cycle expression:
d+ 1* (N
Dt1)Dt2 TS
V
(eq. 29)
effects being neglected. To benefit from the quasi- resonant mode, it is recommended to turn the power switch on in the valley, where the drain-source voltage is minimized. This naturally reduces the dV/dt and switching losses to a minimum (in practice, an appropriate delay inserted after the core reset detection provides an effective method to synchronize the power switch turn on with the valley event). A simple look at Figure 7 shows that the valley occurs at half the oscillation period. Therefore, the delay Dt2 between the core reset completion and the optimal turn on time is given by the following equation:
Dt2 + p LP Ctot
(eq. 25)
Vg) ) V
The switching period is the sum of the on-time, the core reset time (tdemag), Dt1 and Dt2:
TS + ton ) tdemag ) Dt1 ) Dt2
(eq. 30)
The time tdemag can be easily deducted from the Figure 4 sketch. Since the core reset is the time necessary to discharge the primary inductor from IP to zero with a (V+Vf)/N slope, it comes:
N Vg L IP tdemag + P + ton V VN
(eq. 31)
Substitution of equation (31) into (eq. 30), leads to the following expression where TS is a function of ton :
TS + Dt1 ) Dt2 ) [(N Vg) ) V] N ton
(eq. 32)
Once these delays are defined, it is about time to revise the previous equations in order to include Dt1 and Dt2 effects. The main parameters of interest are the average input and output currents, the equivalent resistance and the switching period. If we combine equations 24 and 13 that express IP as a function of the input voltage, the inductor value and the ON-time leads to:
Dt1 + LP Ctot Vg ) V Vg
N
Equation (15) that defines the average input current as a function of the input voltage, the duty cycle, the inductor value and the switching period, still holds. Substitution of equation (29) into equation (15) leads to:
t I1(t) u+ Vg 2 1* LP
Dt1)Dt2 TS
ton
(eq. 26)
[(N
V ton Vg) ) V]
(eq. 33)
Replacing TS by its equation 32 expression, it comes:
(eq. 34)
If (d x TS) depicts the core reset time, Dt1 and Dt2 times require to change (d=1-d) into:
D ) Dt2 d + 1 * d * t1 TS
(eq. 27)
Vg t I1(t) u+
1* 2 LP
Dt1)Dt2
Dt1)Dt2) (N Vg))V V ton
V
ton
[(N
The inductor volt-second balance approximation of equation 8 still holds. However, it must be revised by
Vg) ) V]
Re-arranging this equation, one can obtain:
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(eq. 35)
t I1(t) u+ Vg 2 LP
ton 2 Dt1 ) Dt2 )
(N Vg))V V ton
Substitution of equation (32) giving TS into equation (37), leads to: =
Vg 2 ton LP ton Dt1 ) Dt2 )
(N Vg))V V ton
Vg V
(eq. 38)
Similarly to the simplified model analysis, one can note that the average input current is proportional to the input voltage. The effective resistance Re is thus: Re(ton) =
2 LP ton 2 Dt1 ) Dt2 ) [(N Vg ) ) V] V ton
(eq. 36)
This expression can be simplified as follows: =
Vg Re(ton) Vg +t I1(t) u V ) Vf Vg V ) Vf
(eq. 39)
It is pleasant to confirm that if Dt1=Dt2=0, the Re(ton ) expression reduces to equation 21... Then, the equivalent circuit depicted in Figure 6 and based on the loss-free resistor Re(ton ) can be applied. To complete the model, let's calculate by combining equations (17) where d is taken equal to (tdemag/TS), (13) and (31):
t I2(t) u+ 1 N Vg 2 ton LP ton TS N V Vg
(eq. 37)
The model assumes a 100% efficiency power transfer. To better stick to reality, the above expression should be multiplied by the estimated efficiency to obtain the final equation:
t I2(t) u+t I1(t) u Vg V ) Vf eff
(eq. 40)
where eff is the efficiency.
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The following table summarizes the main equations upon which our model is based:
Delay between the power switch opening and the start of the energy transfer to secondary side:
Dt1 + LP
Ctot
V Vg ) N Vg @ ton
Delay between the core reset completion and the next turn on of the power switch (Note 1): Equivalent input resistance:
Dt2 + p 2 LP Re(ton) + ton 2 fSW +
LP
Ctot [(N Vg) ) V] V ton
Dt1 ) Dt2 ) 1 Dt1 ) Dt2 )
(N
Switching Frequency:
Vg))V V
ton
Average input current: Average output current:
t I1(t) u+ Vg Re(ton) t I2(t) u+ Vg V t I1(t) u eff
NOTE: : even if the proposed value appears to us as the optimal one, SMPS designers might make a different choice for Dt2. That is why, if the model
LP Ctot as default proposes Dt2 + p value, you can modify this simulation parameter to stick to your application in case valley switching is not considered.
The input voltage source being supposed to emulate a resistance, its expression shall be in the form of: Et = I1 x Re where Re is simply equation 22, thus:
Et + I1 2 LP (V ) N ton V Vg)
(eq. 41)
Implementing the SPICE Model with the Loss-Free Network As exemplified by Figure 6, the model shall emulate an input resistor being ton dependent and then transmit a power following equation 23. Different ways exist to implement this topology in Spice. INTUSOFT's IsSpice authorizes behavioral resistors, e.g. following any particular ohmic evolution with time, voltage, current etc. For instance, the following code would be accepted by the simulator: R1 1 2 R = 2.0 * v(1)^0.5 + 3.0*v(2)*time + v(2)*sqrt(temp) Unfortunately, despite its obvious interest, this code is not very portable and would constrain the model usage to IsSpice only. Figure 9 offers a more practical association using behavioral voltage and current sources [2]:
I1 I2
where ton is an input port of the model, imposed by the control loop. In the final model, this value will be derived from LP and the peak current given by the error voltage divided by Rsense, where V, Vg and N are to be passed or sensed by the model. The output current source together with V2 shall deliver the output power as imposed by equation 22. Thus, i2 generation shall follow:
(eq. 42)
Rin(eq) I2 + v2
i1 2
+
2
LP
(V ) N ton V
Vg)
i1 2 v2
Also, one can introduce the efficiency by simply multiplying the I2 current source by {eff}, where eff is a parameter entered by the user in the model. Hence, I2 can be written as:
I2 + 2 LP (V ) N ton V Vg) i1 2 v2 eff
(eq. 43)
V1
Et
Gd
V2
Figure 9. Implementing the DCM Model via Two Controlled Elements
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Operating Parameter Calculation
FB R9 1 Meg errint B8 Voltage V(FB)/3 > 1 ? 1 : V(FB)/3 < 10 m ? 10m : V(FB)/3 System Parameter calculation toff BIp Voltage V(ton)*V(13)/{Lp} 1
1
ton Bton Voltage V(errint)*{Lp}/({Rs}*V(13))
X2 XFMR RATIO = N
13
13 FSW Bfreq Voltage (1/(V(ton)+V(toff)))/1 k 3
3
IP
BToff Voltage {Lp}*V(Ip)*{N}/V(3)
Lm {Lp}
12
Rs {Rlf} 1 4
VI1
7
Bclamp Current I(VI1)>0 ? I(VI1) : 0
20
V6 BGd Current (((2*{Lp}*(V(3)+{N}*V(13))/(V(ton)*V(3)+1u))*I(V6)^2)/(V(3,4)+1u))*{eff}
4
BEt Voltage (2*{Lp}*(V(3)+{N}*V(13))/(V(ton)*V(3)+1u))*I(V6) Gnd 4
Figure 10. The final simplified model implementation where added sources reveal operating parameters such as Ton, FSW and the peak current IP
Figure 10 portrays the final simplified model subcircuit where all relevant sources appear, among them, the switching frequency, peak current and Ton calculations. For the extended model, only BGd and BEt sources need to be changed. As you can see, there are plenty denominator expressions where a variable such as Ton appears. If during the bias point calculation SPICE Ton starts or goes close to zero, the simulator can fail to converge (or find a wrong bias point which is worse). To avoid this potential problem, a trick consists in inserting a fixed value, small enough like 1 m or less, to clamp the maximum value the source can take if Ton becomes null. To the opposite, the frequency expression modeled by a voltage source can deliver kV to express kilo Hz. The simulator dynamic being bounded, mixing values of a few mV with sources delivering kV can puzzle the bias point calculation. Again, a division by 1000 will limit the range. The FB pin undergoes a division by 3 to be further clamp by a 1 V limiter, a classical circuitry found in most PWM controllers (IP max = 1 V / Rsense). DC-bias calculation always represents a difficult task for SPICE simulators running averaged models. In order to enhance the extended model robustness (the one including parasitic effects), we have constrained the BGd source to be positive only by using a simple in-line equation that differs depending on the simulator syntax: IsSpice BGd 4 3 I= ((2*{Lp}/V(ton)) * ( ({N}*V(13)+V(3))/(V(3)+1u) + {DEL}/V(ton) + +({Lp}*{Ctot}/V(ton))*(1+(V(3)/{N})/V(13)) ) * I(V6)^2)/(V(3,4)+1u)*{EFF} < 10m ? 10m : +((2*{Lp}/V(ton)) * ( ({N}*V(13)+V(3))/(V(3)+1u) + {DEL}/V(ton) + +({Lp}*{Ctot}/V(ton))*(1+(V(3)/{N})/V(13)) ) * I(V6)^2)/(V(3,4)+1u)*{EFF} PSpice Gd 4 3 TABLE { ((2*{Lp}/(V(ton)+10n)) * ( ({N}*V(13)+V(3))/(V(3)+1u) + {DEL}/(V(ton)+10n) + +({Lp}*{Ctot}/(V(ton)+10 n))*(1+(V(3)/{N})/V(13)) ) + * I(V6)^2)/(V(3,4) + 1u)*{EFF} } ( (10m,10m) (1000,1000) ) Finally, the model comes with two different names: .SUBCKT QuasiFly 13 FB GND 3 IP Ton FSW params: LP = 3.22 m RS = 0.5 N = 0.06 eff = 0.86 the simplified model version QuasiFlyDel 13 FB GND 3 IP Ton FSW params: LP = 3.22 m RS = 0.8 N = 0.06 eff = 0.86 Ctot = 100 p the complete model including parasitic effects Passed parameters are: Ctot, the lump parasitic component present on the drain. LP, the primary inductance Rlf, the ohmic losses of the primary winding N, the NP : NS ratio with NP=1 Eff, the circuit estimated efficiency Please note that for the sake of simplicity, both models do not account for the secondary rectifier forward drop Vf whose effect is nevertheless negligible in our approach.
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Putting the Model to Work Different ways exist to test the validity of a model. The first one is by using SPICE only, where one can compare the transient response of the averaged model versus that of the equivalent cycle-by-cycle. The other one implies the comparison of the averaged model results versus a real board measurement. In this paper, we will depict both approaches, using our simplified cycle-by-cycle transient model. The averaged template is depicted by Figure 11 where Figure 10 sources have been pushed into a single graphic
0.868 iP Vin 8.68 m ton 80.7 fSW Vout R6 10 m 24 16.8 16.8 R7 150 m 16.8
25
symbol. The symbol must be fed by Lp, Efficiency, Rsense, transformer turn ratio and the primary inductance ohmic loss. The FB pin goes to a component arrangement particular to the NCP1207 series from ON Semiconductor where the optocoupler collector is internally pulled-up to a reference voltage.
IP ton fSW (kHz) Flyback FreeRunning Averaged model OUT IN FB GND
L3 2.2 m 16.8
3
Vout
120
9
R8 60 m
17
Vg 120 AC = X2 QuasiFly LP = 1.2 m RS = 0.5 N = 0.06 eff = 0.91 Rlf = 0.5
FB
16.8
C3 1 mF IC = 16
C4 220 m IC =
Rload 8.5
Vout V9 4.8
6
FB R10 20 k
Fb 1.30 16.7
R1 1k
1
4.80 C5 1n 15.8
X1 SFH615AGR
2
D4 BV = 15.6
Figure 11. Averaged Model Template
The averaged model template featuring DC bias points which confirms the correct bias point calculation In Figure 11, once the simulation has been done, DC points are reflected to the schematic and confirm the validity of the original calculation. The feedback loop is made of a simple Zener diode to avoid any long feedback time constants as with a standard TL431. The cycle-by-cycle circuitry uses our simplified QR transient model which emulates a free-running controller such as ON Semiconductor NCP1207 or NCP1205 [3] (Figure 12). The output stage and feedback configuration conforms to Figure 11 in order to compare similar topologies. The first test consists in testing the input audio susceptibility by stepping the input voltage from 200 V to 350 VDC. Figure 13 reveals the good agreement between the averaged response and the cycle-by-cycle one. The next experience will step load the converter output from light to heavy load in a few ms. Figure 14 testifies for the right behavior on both configuration, average or cycle-by-cycle. On the static point of view, the following data compare numbers given by the averaged model and the cycle-by-cycle one: IP AVG = 868 mA / IP TRAN = 858 mA Ton AVG = 8.68 ms / Ton TRAN = 8.78 ms FSW AVG = 80.7 kHz / FSW AVG = 77.8 kHz
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X4 XFMR-AUX D1 RATIO_POW = -0.06 RATIO_AUX = -0.06 MBR20100CT
21 11
Iout L3 2.2 m
13 15
Rprim 0.5
9
R5 10 m Vout + Vout
7
Icoil Lprim 1.2 m
5 31
12
VCoil V3 120 AC =
Resr1 60 m
14
D
R6 100
IDiode dem C6 100 p
Cout1 1 mF IC = 16.5
R7 150 m
16
R1 1k dem fb
FreeRun
4
Lleak 1p 8 7 6 5
3 24 6
C1 220 m IC = 16.5 Vout
V1 X3 PSW1
1 2 3 4
Vdrain
+ ID X2 PSW1
1
Feedback fb IReso Rled 1k
17
X1 FreeRunDT toff(min) = 1 m
X5 SFH615AGR Creso 10 p C5 1n
18
Simplified simulation of a NCP1207-based board
Rsense 0.5
D4 BV = 15.6
Figure 12. This Simplified Transient Model Will Help to Check the Averaged Results
16.88
16.86
16.84
16.82 Cycle-by-Cycle Vout 16.80 2.57 m 3.11 m 3.65 m 4.19 m 4.73 m Averaged Vout
Figure 13. Audio Susceptibility with a Line Step (200 to 350 V)0
16.95 Averaged Vout
Real World Confrontation Even if the above paragraph gives us the assumption that our model sticks to reality quite well, nothing replaces a real board measurement with a network analyzer. However, on the NCP1207, the collector of the optocoupler is directly internally pulled-up to a reference via a resistor, it thus becomes difficult to open the loop via the series transformer method. We thus went back to a simple open-loop configuration where a DC source fixes the expected operating point. It does not cause any problem in our case since the overall gain Vcontrol to Vout is low. The AC injection is then made via a 1000 mF capacitor. Figure 15 depicts the adopted configuration on the bench, but also replicated on the averaged model.
1 2 3
4
8 7 6 5
5
4 NCP1207
16.85 D = 20 mV 16.75
C1 1000 m Vstim AC = 1
R3 1k
3 1
Vbias 1.57
Rsense
16.65 Cycle-by-Cycle Vout
Figure 15.
16.55 2.48 m 3.04 m 3.60 m 4.16 m 4.72 m
Figure 14. 0 and Load Step Response Comparison Between Models (50 W to 0.5 W)
The AC measurement is obtained once the proper operating point is reached by adjusting Vbias. The gain being low, there is no problem of output runaway as long as Vbias is slowly increased.
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The bandwidth measurement has been carried on a board further to a 15 mn warm-up. This board does not use any clamping network but a 800 V MOSFET instead and a large capacitor connected between drain and ground Figure 16 and 17 compare the obtained results with the averaged model including valley and turn-off delays:
Mag (dB) 20 10 High Line 0 Low Line -10 -20 -30 -40 -50 High Line -60 1 10 100 -180 Low Line 1k 10 k F0 dB = 49 Hz 45 0 -45 -90 -135 Phase () 180 135 90
Figure 16. Bode Plot Captured with a Network Analyzer
20.0
180 Open-loop Gain: High Line Low Line
0
90.0 F0 dB = 46 Hz
-20.0
0
-40.0 -90.0 Open-loop Phase: High Line Low Line -60.0 -180
1
10
100
1k
10 k
100 k
Figure 17. Bode Plot Obtained with the Averaged Model
One can detect a slight gain difference (around 3.5 dB) in DC but the overall simulated shape stays in good agreement with the real measurement. The phase dips are imputed to the presence of the LC network whose cut-off frequency obviously affects the results. The small-signal analysis details are available in [5].
Finally, a step-load response was performed on a real board fed back by a TL-431 network and compared to our SPICE model, also implementing the same control loop structure. Results prove that the proposed model accuracy is acceptable to predict board stability and final transient response:
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16.93 30 ms/div
16.87
16.81
16.75 2 ms/div 16.69 4.50 m 8.50 m 12.5 m 16.5 m 20.5 m
Conclusion A SPICE model dedicated to the AC analysis of free-running topologies was missing. The simple model presented in this article shows that loop stabilization of QR converters becomes easy thanks to the simulation. Furthermore, the good agreement between simulated results and hardware measurements will surely diminish the prototype development time. As usual, the application templates of the paper examples are available to download from the author's website [5] in both Intusoft's IsSpice and OrCAD's PSpice. References: 1. B. Erickson, D. Maksimovic, "Fundamentals of Power Electronics", Kluwers Academic Publishers, ISBN 0-7923-7270-0 2. B. Erickson, D. Maksimovic, Advances in Averaged Switch Modeling and Simulation", CoPEC. http://schof.Colorado.EDU/~pwrelect/publications .html 3. C. Basso, "Determining the Free-Running Frequency for QR Systems", ON Semiconductor, AND8089/D 4. J. Chen, B. Erickson, D. Maksimovic, "Averaged Switch Modeling of Boundary Conduction Mode Dc-to-Dc converters", the 27th Annual Conference of the IEEE Industrial Electronics Society. 5. http://perso.wanadoo.fr/cbasso/Spice.htm
Figure 18. The Simulated Step-Load Response On a TL431-based Feedback Loop0
Figure 19. 0versus Real Board Oscilloscope Shot
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