The inductance characteristic of the inductor in Example 10.3 satisfies the criteria for the inductance-current pairs in Example 10.7. The buck converter in Figure 10.23 has the specifications given in Table 10.6. Simulate the circuit and plot the inductor current for each of the following values

Question

The inductance characteristic of the inductor in Example 10.3 satisfies the criteria for the inductance-current pairs in Example 10.7. The buck converter in Figure 10.23 has the specifications given in Table 10.6.

Simulate the circuit and plot the inductor current for each of the following values of inductance:

1. constant inductance of 21.9 μH at solar insolation of 800 W/m²

2. constant inductance of 21.9 μH at solar insolation of 200 W/m²

3. variable inductance of Figure 10.6 at 200 W/m²

Accepted Answer

The circuit of Figure 10.23 has been simulated for the purposes of evaluating the response for the variable inductor. The method of simulation is based on the electromagnetic transient program (EMPT). The simulated circuit is shown in Figure 10.25.

The series coil resistance R, which has a damping effect on the inductor current, is included and, for convenience, the solar panels and MOSFET voltage are represented by a rectangular PWM waveform with a variable duty cycle. The appropriate value of the duty cycle was calculated in Example 10.7.

The general solution for the three nodes in Figure 10.25, recognizing that there are no input currents at node 2 and 3, is:

[latex]\left[\begin{array}{l}i_1\left(t_n ight) \ i_2\left(t_n ight) \ i_3\left(t_n ight)\end{array} ight]=\left[\begin{array}{lll}Y_{11} & Y_{12}&Y_{13} \Y_{21} & Y_{22} & Y_{23} \ Y_{31} & Y_{32} & Y_{33}\end{array} ight] \cdot\left[\begin{array}{l}v_1\left(t_n ight) \ v_2\left(t_n ight) \v_3\left(t_n ight)\end{array} ight]+\left[\begin{array}{l}I_1\left(t_{n-1} ight) \ I_2\left(t_{n-1} ight) \ I_3\left(t_{n-1} ight)\end{array} ight][/latex]

The terms in the admittance matrix are:

[latex]Y_{11}=\frac{1}{R} \quad Y_{12}=-\frac{1}{R} \quad Y_{13}=0[/latex]

[latex]Y_{21}=-\frac{1}{R} Y_{22}=\frac{1}{R}+\frac{\Delta t}{2 L_{ eff }} Y_{23}=-\frac{\Delta t}{2 L_{ eff }}[/latex]

[latex]Y_{31}=0 \quad Y_{32}=-\frac{\Delta t}{2 L_{ eff }} \quad Y_{33}=\frac{\Delta t}{2 L_{ eff }}+\frac{2 C}{\Delta t}[/latex]

[latex]t_n=n \Delta t[/latex]

[latex]L_{eff}[/latex] must be calculated at [latex]i_{1}(t_{n - 1})[/latex] according to Equation 10.15.

[latex]L_{ eff }=L(i)+i \frac{d L(i)}{d i}=L(i)+i \frac{d L}{d \mu_r} \frac{d \mu_r}{d i}[/latex] (10.15)

The current sources [latex]I_{1}(t_{n} - 1)[/latex] are known from past history:

[latex]I_1\left(t_{n-1} ight)=0[/latex]

[latex]I_2\left(t_{n-1} ight)=I_2\left(t_{n-2} ight)+\frac{\Delta t}{L_{ eff}}\left[v_2\left(t_{n-1} ight)-v_3\left(t_{n-1} ight) ight] \quad n=2,3 \ldots[/latex]

[latex]I_3\left(t_{n-1} ight)=I_{30}\left(t_{n-1} ight)-I_2\left(t_{n-1} ight)+\frac{v_3\left(t_{n-1} ight)}{R_L} \quad n=2,3 \ldots[/latex]

[latex]I_{30}\left(t_{n-1} ight)=-I_{30}\left(t_{n-2} ight)-\frac{4 C}{\Delta t} v_3\left(t_{n-1} ight) \quad n=2,3 \ldots[/latex]

We define the initial conditions of voltages and currents equal to 0.

If [latex]i_{1}[/latex] results in a negative value, it is reset to zero and [latex]I_{2} ext{ and } I_{3}[/latex] are adjusted. This is because, in reality, the diode D is in series with the PV panel, as shown in Figure 10.23, and will block the current flowing into the PV panel.

1. Figure 10.26 shows the inductor current for 21.9 μH, at 800 W/m² (this corresponds to point a in Figure 10.24 and the current is continuous).

2. Figure 10.27 shows the inductor current for 21.9 μH, at 200 W/m² (point b in Figure 10.24) and, as expected, the converter is operating in discontinuous conduction mode.

3. Figure 10.28 shows that the variable inductor whose characteristic is shown in Figure 10.6 restores continuous conduction. The current waveform has sharp peaks and wide valleys that arise because of the lower value of inductance at higher current.

The inductor current range is 2.27 A to 4.61 A, corresponding to a solar insolation level from 200 W/m² to 800 W/m². In this range, the minimum inductor falls from 111 μH to 21.9 μH. A conventional inductor would have 111 μH at 4.61 A, corresponding to a stored energy of 1.2 mJ. With a variable inductor, the stored energy at 111 μH and 2.27 A is 0.29 mJ and, at 21.9 μH with 4.61 A, the stored energy is 0.25 mJ. The size of an inductor is directly proportional to its stored energy, so that the variable inductor would occupy 25% of the volume of a conventional fixed value inductor.

Table 10.6 Specifications
Input voltage	41.6 V
Output capacitance	80 μF
Inductor resistance	1 Ω
Load resistance	8 Ω
Frequency	20 KHz

Question 10.8: The inductance characteristic of the inductor in Example 10....

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