Question 10.SP.6: Determine a set of equations that describe the instantaneous......
Determine a set of equations that describe the instantaneous inductor current i_L(t) over a period T_s for the buck converter.
Refer to i_L and v_L of Fig. 10-3. For 0 \leq t \leq D T_{s}, application of (10.2) gives
V_{L} = \langle v_{L}\rangle = {\frac{1}{T_{s}}}\int_{0}^{T_{s}}\,v_{L}\,d t = {\frac{1}{T_{s}}}\int_{0}^{T_{s}}\left(L\,{\frac{d i_{L}}{d t}}\right)d t = {\frac{L}{T_{s}}}\int_{i_{L}(0)}^{i_{L}(T_s)}\,d i_{L} = 0 (10.2)
i_{L}(t) = {I}_{\mathrm{min}} + \frac{{V}_{1} – {V}_{2}}{ L}\;t (1)
For DT_s \leq t \leq T_s, create a t′-coordinate frame with origin at t = DT_s so that t^′ = t – DT_s. In the t′ frame, application of (2) from Example 10.1 yields
i_{L}(t^{\prime}) = I_{\mathrm{max}} – {\frac{V_{2}}{L}}\;t^{\prime} = I_{\mathrm{max}} – {\frac{V_{2}}{L}}\;(t – D T_{s}) (2)
To complete the work, expressions for I_{\mathrm{max}} and I_{\mathrm{min}} must be found. Evaluate (1) for t = DT_s to find
i_{L}(D T_{s}) = I_{\mathrm{max}} = I_{\mathrm{min}} + {\frac{V_{1} – V_{2}}{L}}\,D T_{s} (3)
Since \langle i_L \rangle = I_L = I_2, and since the i_L waveform of Fig. 10-3 is made up of straight line segments,
I_{2} = {\frac{I_{\mathrm{max}} + I_{\mathrm{min}}}{2}} = {\frac{V_{2}}{R_{L}}} (4)
Simultaneous solution of (3) and (4) result in
I_{\mathrm{max}} = {\frac{V_{2}}{R_{L}}} + {\frac{(V_{1} – V_{2})D}{2f_{s}L}} (5)
I_{\mathrm{min}} = {\frac{V_{2}}{R_{L}}} – {\frac{(V_{1} – V_{2})D}{2f_{s}L}} (6)
where f_s = 1/T_s.
