The boost converter is capable of providing an output voltage that is greater than the input voltage. In Fig. 12 [12], shows the circuit of a boost converter. During the continuous conduction mode (CCM) the inductor current conducts continuously and the minimum current is always larger than zero. In Fig. 13, shows the waveforms of a boost converter in CCM operation. Therefore, there are only two subintervals for switching converter in CCM operation. The two equivalent circuits of the first and second subintervals are as shown in Fig.
14.
Fig. 12. The boost converter with pulse width modulator
Fig. 13. Waveforms of a Boost Converter in CCM operation
Fig. 14(a) shows the first subinterval operation in CCM. When converter operating in first subinterval the low side NMOS turned on and the inductor current increased. During this subinterval the inductor voltage and capacitor current can be derived as Eq. (5) and (6).
( )
LFig. 14(b) illustrates the second subinterval operation in CCM. When the converter operates in the second subinterval the high side PMOS turned on and inductor current delivering to output. During this subinterval the inductor voltage and capacitor current can be derived as Eq. (7) and (8).
L( )
+V t − i tC( )
(b)
Fig. 14. (a) Equivalent circuit of the first subinterval in CCM. (b) Equivalent circuit of the second subinterval in CCM
Equation (9) is based on the inductor voltage second balance. The output voltage increases when D rises. In the ideal case, the conversion ratio tends to infinity when D is toward to 1. The steady-state current in the switching converter is based on the capacitor charge balance, as Eq. (10) shows. magnitude is greater than the load current. Combining Eq. (5) and (6) shows that the inductor current ripple and output voltage ripple can be calculated as Eq. (11) and (12), respectively:
2
2.3.2 Discontinuous Condition Mode (DCM)
When the output average current is smaller than the half of the inductor peak-to-peak ripple current, the voltage regulator is operated in DCM as shown in Fig. 15 [12]. Because the inductor current conducts discontinuously and the minimum current equals zero during this mode, this situation usually occurs under light load condition. This is why the boost converter has three subintervals. The first and the second subinterval structures are the same as depicted in Fig. 14 (a) and the Fig. 14 (b) respectively. The third subinterval for the boost converter in DCM is shown in Fig. 16.
Fig. 15. Waveforms of a Boost Converter in DCM operation
L
( )
+ V t − i t
C( )
Fig. 16. Equivalent circuit of the third subinterval in DCM
The inductor voltage and capacitor current during the first subinterval are given by:
( )
LThe inductor voltage and capacitor current during the second subinterval are given by
( )
LThe inductor voltage and capacitor current during the third subinterval are given by
L
0
In the steady-state, Eq. (13) to (18) can be written by the volt second theorem:( )
1 2The output current can be derived as follows:
Let Eq. (19) is equal to Eq. (20), it is possible to derive the expression of output voltage as follows:
Analyzing Eq. (21) and (9) reveals major differences between CCM and DCM operation.
In DCM operation, the voltage conversion ratio depends on the input voltage, duty cycle, power stage inductance, switching frequency, and output load resistance. In CCM operation, however, the voltage conversion ratio depends only on the input voltage and duty cycle.
2.3.3 Operation Theorem of Current Mode Control
The block diagram of the current mode boost converter is shown in Fig. 17 [12]. In this case, the switching converter has two control modes: one is the voltage mode controller, and the other is the current mode controller.
Voltage mode control uses a single voltage feedback loop to regulate the output voltage.
The duty cycle of pulse width modulation is produced by comparator output signal of error amplifier compares with a ramp signal of fixed frequency.
The current mode control method uses two control loops, an inner current control loop and an outer loop for voltage control. The block diagram of the current mode boost converter is shown in Fig. 17 [12]. The small duty ratio of the clock signal generates the PWM signal at the start of each switching period. In this state, the power MOSFET MN is turned on and the diode D is tuned off. The inductor current increases follow a raised slope which depends on the input voltage and the value of inductor. An artificial ramp prevents unstable oscillation
when the duty ratio is larger than 0.5. The output signal from the error amplifier is compared with the sum of ramp and sensed inductor current signal. When the sum of the ramp and sensed inductor current signal exceed than the control signal, the output of comparator produce high to reset the SR latch and turn off the power MOSFET MN and connect the diode D as shown in Fig. 18 [12].
Fig. 17. Block Diagram of current mode boost switching converter
Fig. 18. Inductor current waveform with compensation ramp.
2.3.4 Oscillation when Duty > 50% and Slope Compensation
The current mode controller encounters major instability problems when the duty ratio D is larger than 50%. Fig. 19 depicts the inductor current waveform; a small perturbation in the inductor current down slope is greater than the upslope. These perturbations could be due to noise or other changes in the operating environment.
In the current mode control, the inductor current changes with the rising and falling slopes for boost converter are as:
1
V
in,
2V
inV
outm m
L L
= − = −
( 2 2 )Assume that the inductor current is perturbed by an amount ∆ at the beginning of the I1 switching period; the perturbation ∆ for the following period is greater if the duty cycle is I2 greater than 50%. If the duty cycle is smaller than 50%, the successive periods attenuate the perturbation until it disappears. Mathematically, this can be stated as
2
2 1
1
I I m m
⎛ ⎞
∆ = −∆ ⎜ ⎟
⎝ ⎠ ; F o r s t a b l e c o n d i t i o n 2
1
m 1
m < (23) Equation (23) shows the stable condition. To maintain a stable operation, the duty cycle of the converter must remain below 0.5.
Fig. 19. (a) Waveform of I with perturbation L ∆ for D > 0.5 I1 (b) Waveform of I with perturbation L ∆ for D < 0.5 I1
The artificial ramp generator which prevents unstable oscillation is applied to the switching current sensing loop, as illustrated in Fig. 20 [12][14]. The relation of inductor current and perturbation iL
( )
0 is derived as Eq. (24) and (25).t
Fig. 20. Inductor Current Waveform with Slope Compensation