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• Thomas P John

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Basic Principle of Boost The boost is a popular non-isolated power stage topology, sometimes called a step-up power stage. Power supply designers choose the boost power stage because the required output is always higher than the input voltage. The input current for a boost power stage is continuous, or non-pulsating, because the output diode conducts only during a portion of the switching cycle. The output capacitor supplies the entire load current for the rest of the switching cycle. Figure 1 shows a simplified schematic of the boost power stage. Inductor L and capacitor C make up the effective output filter. The capacitor equivalent series resistance (ESR), RC, and the inductor dc resistance, RL, are included in the analysis. Resistor R represents the load seen by the power supply output.

Figure 1. Boost Power Stage Schematic A power stage can operate in continuous or discontinuous inductor current mode. In continuous inductor current mode, current flows continuously in the inductor during the entire switching cycle in steady-state operation. In discontinuous inductor current mode, inductor current is zero for a portion of the switching cycle. It starts at zero, reaches peak value, and return to zero during each switching cycle. It is desirable for a power stage to stay in only one mode over its expected operating conditions because the power stage frequency response changes significantly between the two modes of operation.

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Boost Steady-State Continuous Conduction Mode (CCM) In continuous conduction mode, the boost power stage assumes two states per switching cycle. In the on state, Q1 is on and D1 is off. In the off state, Q1 is off and D1 is on. A simple linear circuit can represent each of the two states where the switches in the circuit are replaced by their equivalent circuit during each state. Figure 2 shows the linear circuit diagram for each of the two states.

Figure 2. Boost Power Stage States The duration of the on state is D×Ts=TON, where D is the duty cycle set by the control circuit, expressed as a ratio of the switch on time to the time of one complete switching cycle, Ts . The duration of the off state is TOFF.

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Figure 3. CCM Boost Power Stage Waveforms Refer to Figures 1 and 2. The inductor-current increase can be calculated by using a version of the familiar relationship: diL V ⇒ ∆I L = L ∆T dt L The inductor current increase during the on state is given by: VL = L ×

∆I L ( + ) =

Vi − (VQ + I L × RL )

× TON L The quantity ∆IL(+) is the inductor ripple current. During this period, all of the output load current is supplied by output capacitor C.

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The inductor current decrease during the off state is given by: ∆I L ( −) =

(VO + Vd + I L × RL ) − Vi × TOFF L

The quantity ∆IL(-) is also the inductor ripple current. In steady-state conditions, the current increase, ∆IL(+), during the on time and the current decrease, ∆IL(-), during the off time are equal. Therefore, these two equations can be equated and solved for VO to obtain the continuous conduction mode(CCM) boost voltage conversion relationship: VO = (Vi − I L × RL ) × (1 +

TON T ) − Vd − VQ × ( ON ) TOFF TOFF

And, D=

TON T = ON TON + TOFF TS

(1 − D ) =

TOFF TS

The steady-state equation for VO is: VO =

Vi − I L × RL D − Vd − VQ × 1− D 1− D

*Notice that in simplifying the above, TON+TOFF is assumed to be equal to Ts. This is true only for CCM mode.

The above voltage conversation relationship for VO illustrates that VO can be adjusted by adjusting the duty cycle, D, and is always greater than the input because D is a number between 0 and 1. A common simplification is to assume VQ, Vd, and RL are small enough to ignore. The above equation simplifies considerably to :

VO =

Vi 1− D

I O = (1 − D ) × I i

A simplified, qualitative way to visualize the circuit operation is to consider the inductor as an energy storage element. When Q1 is on, energy is added to the inductor. When Q1 is off, the inductor and the input voltage source deliver energy to the output capacitor and load. The output voltage is controlled by setting the on time of Q1. For example, by increasing the on time of Q1, the amount of energy delivered to the inductor is increased. More energy is then delivered to the output during the off time of Q1 resulting in an increase in the output voltage.

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To relate the inductor current to the output current, refer to Figure 2 and 3. Note that the inductor delivers current to the output only during the off state of the power stage. This current averaged over a complete switching cycle is equal to the output current because the average current in the output capacitor must be equal to zero. The relationship between the average inductor current and the output current for the CCM mode is given by: I L ( Avg ) ×

TOFF  I  = I L ( Avg ) × (1 − D ) = I O ⇒ I L ( Avg ) =  O  TS 1 − D 

Boost Steady-State Discontinuous Conduction Mode (DCM) Figure 4 shows the inductor current condition where the power stage is at the boundary between continuous and discontinuous mode. This is where the inductor current just falls to zero and the next switching cycle begins immediately after the current reaches zero. From the charge and discharge of output capacitor, the output current is given by: I O × (TON + TOFF ) =

2 × IO I PK × TOFF ⇒ I PK = 2 1− D

Figure 4. Boundary Between Continuous and Discontinuous Mode Further reduction in output load current puts the power stage into discontinuous current conduction mode(DCM). The discontinuous mode power stage input-tooutput relationship is quite different from the continuous mode.

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Figure 5. Discontinuous Current Mode The duration of the on state is TON=D×TS, where D is the duty cycle set by the control circuit. The duration of the off state is TOFF=D2×TS. The idle time is the remainder of the switching cycle and is given as TS-TON-TOFF= D3×TS. These times are shown with the waveforms in Figure 6. The inductor current increase during the on state is given by: Vi V × TON = i × D × TS = I PK L L The ripple current magnitude, ∆IL(+), is also the peak inductor current, Ipk, because in discontinuous mode. The current starts at zero each cycle. The inductor current decrease during the off state is given by: ∆I L ( + ) =

VO − Vi V − Vi × TOFF = O × D 2 × TS L L As in the continuous conduction mode case, the current increase, ∆IL(+), during the on time and the current decrease during the off time, ∆IL(-), are equal. So, ∆I L ( −) =

VO = Vi ×

TON + TOFF D + D2 = Vi × TOFF D2

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Figure 6. Discontinuous Mode Boost Power Stage Waveforms Now calculate the output current. It is the average over the complete switching cycle of the inductor current during the D2 interval. IO =

VO 1 1 = × ( × I PK × D 2 × TS ) R Ts 2

VO 1  1 Vi  = ×  × ( × D × TS ) × D 2 × TS  R TS  2 L  V × D × D 2 × TS = i 2×L

IO =

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Now solve two equations, IO and VO, the discontinuous conduction mode boost voltage conversion relationship is given by: 1+ 1+ VO = Vi ×

4 × D2 K

D = K × M × ( M − 1)

2 Where K is defined as: K=

2×L R × TS

M =

VO Vi

Critical Inductance The previous analyses for the boost power stage have been for continuous and discontinuous conduction modes of steady-state operation. The conduction mode of a power stage is a function of input voltage, output voltage, output current, and the value of the inductor. A boost power stage can be designed to operate in continuous mode for load currents above a certain level usually 5 to 10% of full load. Usually, the input voltage range, output voltage, and load current are defined by the power stage specification. This leaves the inductor value as the design parameter to maintain continuous conduction mode. The minimum value of inductor to maintain continuous conduction mode can be determined by the following procedure. First, define IOB as the minimum output current to maintain continuous conduction mode, normally referred to as the critical current. This value is shown in Figure 4. In boundary between CCM and DCM, D 2 = (1 − D ) Vi = VO × (1 − D )

I OB =

Vi × D × D 2 × TS VO × D × (1 − D ) 2 × TS = 2×L 2×L

Lmin ≥

VO × D × (1 − D ) 2 × TS 2 × I OB

The worst case condition for the boost power stage is at an input voltage equal to onehalf of the output voltage because this gives the maximum ∆IL Lmin ≥

VO × TS 16 × I OB

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Output Capacitor In switching power supply power stages, the function of output capacitor is to store energy. The output capacitance for a boost power stage is generally selected to limit output voltage ripple to the level required by the specification. The series impedance of the capacitor and the power stage output current determine the output voltage ripple. The three elements of the capacitor that contribute to its impedance (and output voltage ripple) are equivalent series resistance (ESR), equivalent series inductance (ESL), and capacitance (C). The voltage variation due to the inductor current flow in the output capacitor is approximately:

For CCM Mode: ∆VO =

2 I PK ×L 2 × C × (VO + Vd − VIN )

C≥

For DCM Mode  2×L I O (max) × 1 − R × Ts  C≥ f s × ∆V o

   

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I O max × Dmax f S × ∆VO

The above equation is based on the assumption that all inductor ripple current flows through the capacitor and the ESR is zero. Now, assuming that the capacitor is very large, the ESR needed to Limit the ripple to ∆VOmax is: For CCM Mode: ∆VO max

ESR ≤ (

I O (max) 1 − D Max

+

∆I O ) 2

=

∆VO max I PK

For DCM Mode: ESR ≤

∆VO max ∆VO max = ∆I O I PK

*The output filter capacitor should be rated at least 10~20 times the calculated capacitance and 30 to 50 percent lower than the calculated ESR.

The RMS value of the ripple current flowing in the output capacitance(CCM) is given by: I CRMS = I O ×

D 1− D

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Synchronous Rectifier Synchronous rectification allows for high efficiency by reducing the losses associated with the Schottky rectifiers.

The Schottky rectifier D1 conducts during the time that MOSFET Q2 is on, which improves efficiency by pre-venting the synchronous-rectifier MOSFET Q2 loss body diode from conducting.

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Boost DC/DC Converter Small Signal Model (Transfer Function):

∧

∧

VO ( s ) = Vref ( s ) T ( s) =

∧ ∧ GVg ( s ) Z ( s) 1 T ( ) + V g ( s )( ) − iload ( s )( OUT ) H 1+ T 1+ T 1+ T

H ( s )GC ( s )GVd ( s ) = loop VM

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gain

For CCM Mode:

GVd ( s ) =

GVg ( s ) =

Vg

(1 − D )2

1 (1 − D )2

  1 +    1+ 

s  s  1 −  wZ 1  wZ 2   2   s   s +  w0Q  w0  

    1   2  1 + s +  s    w0Q  w0  

2   s   s +   1 + w1Q1  w1    Z OUT ( s ) = Req  2   1 + s +  s    w0Q  w0  

Q=

w0

w0 =

RL 1 + L C ( R + RC )

wZ 1 =

1 RC C

wZ 2

RL  D  Req = + R 2 (1 − D )  1 − D  C

w1Q1 =

1 LC

RL + (1 − D ) 2 R (1 − D ) ≅ R LC

2 ( 1 − D ) R − RL =

w1 =

L

1− D LC

≅

(1 − D ) 2 R L

Req RC

1 L + RC C (1 − D ) 2 Req

* Two Pole fLC , One Zero fESR for GVd(s) and One Right-Half-Plane zero

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z From a practical view, at RHP zero frequency, the loop gain starts increasing at a 20dB/decade rate but the loop phase decreases by –45 degrees (in a normal, LHP zero, the loop phase will increase by +45 degrees). This imposes the restriction that the gain be rolled off to 0dB before encountering the RHP zero. z The output inductor, capacitor and the capacitor’s ESR must be selected so that the double pole occurs first and then the output capacitor zero and then the RHP zero. This assures that the loop gain crosses 0dB at a slope that is first order (20dB/decade) and that the instability inherently associated with the RHP zero is circumvented by crossing 0dB before the RHP zero frequency occurs.

Compensate rule: 1. Decrease the double pole influence. Æ f Z ( compensator ) ≅

1 1 ~ ) fS 10 6

2. Crossover frequency fC Æ f C < f ESR

fC ≤ (

3. Decrease the RHP zero influence Æ f C