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Switched-Mode Power Supply Theory

Field: Power Electronics

Sequence of Expressions

Definition

Duty Cycle (D)

Let TsT_s be the total switching period and TonT_{on} be the duration during which the switch is ON. The duty cycle DD is defined as the dimensionless ratio:\nD=TonTs such that 0 and D are constrained by 0 and 1D = \frac{T_{on}}{T_s} \text{ such that } 0 \text{ and } D \text{ are constrained by } 0 \text{ and } 1
Define the switching period TsT_s as the inverse of the switching frequency fsf_s. The relationship is given by:\nfs=1Ts=1Ton+Tofff_s = \frac{1}{T_s} = \frac{1}{T_{on} + T_{off}}
For an inductor LL and capacitor CC operating under a voltage ripple ΔV\Delta V and current ripple ΔI\Delta I, the fundamental relationships are derived from the governing differential equations:\nΔV≈LCΔIVout (Approximation for ΔV based on ΔI and L/C ratio)\Delta V \approx \frac{L}{C} \frac{\Delta I}{V_{out}} \text{ (Approximation for } \Delta V \text{ based on } \Delta I \text{ and } L/C \text{ ratio)}\nΔI=Vout⋅DL⋅(1−D) (Approximation for ΔI based on Vout and D ratio)\Delta I = \frac{V_{out} \cdot D}{L} \cdot (1 - D) \text{ (Approximation for } \Delta I \text{ based on } V_{out} \text{ and } D \text{ ratio)}
For an ideal DC-DC buck converter operating in steady state, the average output voltage VoutV_{out} is determined by the duty cycle DD and the input voltage VinV_{in}:\nVout=D Vin (Ideal case)V_{out} = D \text{ } V_{in} \text{ (Ideal case)}\nIncluding efficiency η\eta, the regulated output voltage is:\nVout=D Vin ηV_{out} = D \text{ } V_{in} \text{ } \eta
For an ideal DC-DC boost converter operating in steady state, the output voltage VoutV_{out} is determined by the duty cycle DD and the input voltage VinV_{in}:\nVout=1−DD Vin (Ideal case)V_{out} = \frac{1-D}{D} \text{ } V_{in} \text{ (Ideal case)}\nThis relationship holds provided D∈(0,1)D \in (0, 1).
Maximum Power Point Tracking (MPPT) requires maximizing the instantaneous power PP with respect to the operating variable, typically the load resistance RLR_L or current II. The condition for maximum power is found by setting the derivative of power with respect to the variable to zero:\ndPdRL=0 or dPdI=0\frac{d P}{d R_L} = 0 \text{ or } \frac{d P}{d I} = 0 \nFor a solar cell model P(V)=V×I(V)P(V) = V \times I(V), the optimal operating point (VMPP,IMPP)(V_{MPP}, I_{MPP}) satisfies the condition dPdV=0\frac{d P}{d V} = 0.
Consider the closed-loop system described by the transfer function G(s)H(s)G(s)H(s). Stability is guaranteed if and only if all poles of the closed-loop characteristic equation 1+G(s)H(s)=01 + G(s)H(s) = 0 lie in the left half of the complex ss-plane (Re(s)<0\text{Re}(s) < 0). This is formally verified using the Nyquist stability criterion, which requires the encirclement of the critical point (−1,0)(-1, 0) by the locus plot G(jω)H(jω)G(j\omega)H(j\omega) to be zero, ensuring sufficient phase and gain margins.
Let VDS(t)V_{DS}(t) be the drain-source voltage across the switch. The dead time τd\tau_d is the mandatory non-conducting interval between the turn-off of one device and the turn-on of the complementary device. To prevent shoot-through, the voltage VDS(t)V_{DS}(t) must be maintained at zero during the interval [toff,ton][t_{off}, t_{on}], requiring the inclusion of τd\tau_d in the switching period TsT_s such that Ton+Toff+2τd=TsT_{on} + T_{off} + 2\tau_d = T_s.
For a transformer with primary and secondary windings, the relationship between the voltages VpriV_{pri} and VsecV_{sec} is governed by the turns ratio Nsec/NpriN_{sec}/N_{pri} and the mutual flux linkage Φ\Phi. The voltage transformation is given by:\nVsec=NsecNpriVpriV_{sec} = \frac{N_{sec}}{N_{pri}} V_{pri} \nFurthermore, the average power transfer PoutP_{out} is related to the primary power PpriP_{pri} by Pout=Nsec2Npri2Ppri (assuming ideal coupling)P_{out} = \frac{N_{sec}^2}{N_{pri}^2} P_{pri} \text{ (assuming ideal coupling)}.