Type-A chopper RLE load numerical

A type-A (first-quadrant, step-down) chopper applies $V_s$ to the RLE load during $T_{on}$; during $T_{off}$ the load current freewheels through the diode FD and $v_0 = 0$.

Given: $V_s = 220$ V, $T = 2000\ \mu$s, $T_{on} = 600\ \mu$s, $R = 1\ \Omega$, $L = 5$ mH, $E = 24$ V.

$$\begin{aligned} \alpha &= \frac{T_{on}}{T} = \frac{600}{2000} = 0.3 \\ T_a &= \frac{L}{R} = \frac{5\times10^{-3}}{1} = 5\ \text{ms} \\ \frac{T}{T_a} &= 0.4, \qquad \frac{\alpha T}{T_a} = 0.12 \end{aligned}$$

i) Continuity of load current

The current is continuous if $m = \dfrac{E}{V_s}$ is less than the critical value:

$$\begin{aligned} m &= \frac{24}{220} = 0.1091 \\ \frac{e^{\alpha T/T_a}-1}{e^{T/T_a}-1} &= \frac{e^{0.12}-1}{e^{0.4}-1} = \frac{1.12750-1}{1.49182-1} = 0.2592 \end{aligned}$$

Since $m = 0.1091 < 0.2592$, the load current is continuous.

ii) Average output current

$$\begin{aligned} V_0 &= \alpha V_s = 0.3 \times 220 = 66\ \text{V} \\ I_0 &= \frac{V_0 - E}{R} = \frac{66 - 24}{1} = 42\ \text{A} \end{aligned}$$

iii) Maximum and minimum steady-state current

$$\begin{aligned} I_{max} &= \frac{V_s}{R}\cdot\frac{1-e^{-\alpha T/T_a}}{1-e^{-T/T_a}} - \frac{E}{R} = 220\times\frac{1-0.88692}{1-0.67032} - 24 \\ &= 75.46 - 24 = 51.46\ \text{A} \\[4pt] I_{min} &= \frac{V_s}{R}\cdot\frac{e^{\alpha T/T_a}-1}{e^{T/T_a}-1} - \frac{E}{R} = 220\times 0.25923 - 24 = 33.03\ \text{A} \end{aligned}$$

$I_{min} > 0$ confirms continuous conduction. Ripple $= 51.46 - 33.03 = 18.43$ A.

Answer: continuous; $I_0 = 42$ A; $I_{max} = 51.46$ A; $I_{min} = 33.03$ A.

iv) Waveforms

v0  220|____       ____       ____
       |    |     |    |     |    |
     0 |____|_____|____|_____|____|___ t
        Ton  Toff
i0     |   /\        /\        /\     51.46 A
       |  /  \      /  \      /  \
       | /    \____/    \____/    \   33.03 A
       |______________________________ t
iT     |  /|        /|        /|      (= i0 in Ton)
       | / |       / |       / |
       |/  |______/  |______/  |____ t
vT  220|    ______     ______
       |   |      |   |      |       (Vs in Toff)
     0 |___|      |___|      |______ t
        0  0.6    2   2.6    4  (ms)

• $v_0$: 220 V during $T_{on}$ (0.6 ms), 0 during $T_{off}$ (1.4 ms).
• $i_0$: rises exponentially from 33.03 A to 51.46 A in $T_{on}$, decays back in $T_{off}$.
• $i_T$: equals $i_0$ during $T_{on}$, zero during $T_{off}$ (diode current $i_{FD} = i_0$ then).
• $v_T$: about 0 during $T_{on}$; equal to $V_s = 220$ V during $T_{off}$ (FD conducts, so $v_0 = 0$).