## 2 The Principle of Boost-TL Converter  
Figure 2 shows the main circuit of the boost-TL converter. \(S_{1}\) and \(S_{2}\) are the main power switches, \(L\) is the boost inductor, \(D_{1}\) and \(D_{2}\) are for the freewheeling diode, \(C_{1}\) and \(C_{2}\) are the voltage divider capacitors, and \(R\) is the load; \(V_{c1}\) and \(V_{c2}\) are the capacitor voltage, \(i_{L}\) is the current flowing through the boost inductor, and \(Vo\) is the output voltage.  
It is assumed that the switch, the diode, the inductor, and the capacitors of the circuit are ideal components. The drive signals \(S_{1}\) and \(S_{2}\) have a phase difference of \(180^{\circ}\) to reduce the inductor current ripple and the output voltage ripple. \(C_{1}=C_{2}\), and it is large enough to divide the supply voltage evenly, and the ratio of ripple to the output voltage is small enough to be negligible.  
When the duty cycle of the switch is \(d<0.5\) or \(d>0.5\), the operation mode of the converter is different. The following are analyzed separately. In a switching cycle, the four operating modes of the circuit are described as follows:  
1. Mode 1: switch \(S_{1}\), \(S_{2}\) are on, diode \(D_{1}\), \(D_{2}\) are off, inductance current increases linearly, the inductor stores energy, and the capacitors \(C_{1}\), \(C_{2}\) supply energy to a load, as shown in Figure 3a.
2. Mode 2: switch \(S_{1}\) is on, \(S_{2}\) is off, diode \(D_{1}\) is off, \(D_{2}\) is on, the input voltage \(V_{in}\) provides energy to the load R through a loop composed of inductor L, \(D_{2}\), and \(S_{1}\), as shown in Figure 3b.
3. Mode 3: switch \(S_{1}\) is off, \(S_{2}\) is on, diode \(D_{1}\) is on, \(D_{2}\) is off, the input voltage \(V_{in}\) provides energy to the load R through a loop composed of inductor L, \(D_{1}\), and \(S_{2}\), as shown in Figure 3c.  
Figure 1: BTL and T-type three-level inverter.  
Figure 2: Main circuit of the boost-TL converter.  
(4) Mode 4: switch \(S_{1}\) is off, \(S_{2}\) is off, diode \(D_{1}\) is on, \(D_{2}\) is on, the input voltage \(V_{in}\) provides energy to the load R through a loop composed of inductor L, \(D_{1}\), and \(D_{2}\), as shown in Figure 3d.  
It can be seen that when the duty cycle \(d<0.5\), the operating modes include modes 2, 3, and 4, and there is no mode 1; when the duty cycle \(d>0.5\), modes of operation include modes 1, 2, and 3, and no mode 4. When the switch duty cycle \(d<0.5\), according to the voltage voltage-second balance at both ends of the inductor during a switching cycle, the following results can be obtained:  
\[(V_{in}-V_{c2})d_{1}+(V_{in}-V_{c1})d_{2}+(V_{in}-V_{c1}-V_{c2})(1-d_{1}-d_{2} )=0 \tag{1}\]  
From Equation (1):  
\[V_{in}=(1-d_{1})V_{c1}+(1-d_{2})V_{c2} \tag{2}\]  
When the switch duty cycle \(d>0.5\), for the same reason, according to the voltage voltage-second balance at both ends of the inductor during a switching cycle, the following results can be obtained:  
\[V_{in}(d_{1}+d_{2}-1)+(V_{in}-V_{c2})(1-d_{2})+(V_{in}-V_{c1})(1-d_{1})=0 \tag{3}\]  
According to the above formula, which can be introduced into the relationship between the input and output and is still satisfied by \(V_{in}=(1-d_{1})V_{c1}+(1-d_{2})V_{c2}\), which means when the duty cycle \(d>0.5\) and \(d<0.5\), the converters have the same input-output relationship. If the output capacitance is equally divided, the output voltage, which is \(V_{c1}=V_{c2}=\frac{V_{0}}{2}\), and \(d_{1}=d_{2}\), can be further obtained:  
\[\frac{V_{0}}{V_{in}}=\frac{1}{1-d} \tag{4}\]  
It can be seen from the above formula that the boost-TL converter has the same input and output voltage relationship as the traditional boost converter.  
Figure 3: Four operating modes of boost-TL converters.