Transformers can be mysterious devices to electrical engineers, especially in a digital age. Basic concepts such as turns ratio, magnetic field density (B-field) saturation, and hysteresis are not too difficult, but are not enough when analyzing their behavior in power converters.
Three variations on one of the three basic second-order power-switch configurations (buck, boost, and buck-boost) are the Cuk (pronounced "chook", as in book), SEPIC and zeta topologies. (SEPIC stands officially for "single-ended primary inductance converter" but I prefer to call it a "SEcondary Polarity-Inverted Cuk" instead.) The difference between Cuk and SEPIC topologies is where ground is connected. Because the SEPIC does not invert, it tends to be the most popularly used. Neither converter will be explained here; there is explanation elsewhere in the power converter literature. For instance, see Lloyd Dixon's excellent write-up in the Unitrode (Texas Instrument) seminar book ("High Power Factor Preregulator Using the SEPIC Converter" in SEM-1100).
What this article focuses on is how the transformer of the SEPIC works when its two inductors are coupled. The basic SEPIC converter topology is shown below.
At turn-on, Cc will charge to Vg. When power-switch S turns on, the left end of Cc becomes zero volts, and its right-plate voltage, –Vc = –Vg is applied to the undotted end of the secondary winding – with the same polarity that the Vg source (at the converter input) is applied to the primary. Each winding attempts to induce the same voltage into the other. During the off time of S, D1 conducts, applying the output voltage (plus diode drop) to the primary winding through the secondary. This voltage, induced across the primary winding, happens to be the same voltage applied to it by the algebraic addition of the input source and Vc in series with Vout + VD1.
For an ideal transformer, this would cause an over-constrained electrical indeterminacy, like connecting current sources in series or voltage sources in parallel. What's going on here? What will the winding currents actually be?
To understand this transformer behavior, let's start with something more familiar: a resistive divider circuit with two equal voltage sources driving a common resistor, R3, as shown below.
Note that R1 and R2 are much smaller than R3 and connect to it at the center node. Consequently, a slight variation in either Vs1 or Vs2 will cause a large change in the fraction that each source supplies the common branch, R3.If Vs2 were to increase to 10.1 V, it would raise the voltage at the center node from a balanced value of 9.95 V to 10 V, causing the current through R1 to be zero. All R3 current would then be supplied by Vs2.
Similarly, the leakage inductance values of the SEPIC or Cuk transformer windings are much smaller than the mutual (magnetizing) inductance, M. The transformer model is shown below.
The secondary voltage source, Vs, induces n× Vs into the primary, as shown across the magnetizing inductance. For the interesting case, Vs = Vp = Vg, if the turns ratio, n, is increased slightly from unity, by 1/k (where k < 1 is the coupling coefficient between windings), then the voltage induced by Vs will increase the voltage at the center node to n× Vg, thereby "bootstrapping" Llp, just as R1 was bootstrapped. Because the voltage at each end of Llp is the same, its inductance is effectively infinite. Consequently, all variations in magnetizing current, (through M) due to a varying Vg is supplied from the secondary winding source. By symmetry, setting n = k causes the secondary-winding current to become constant while the primary source supplies the magnetizing-current variations.
This effect can be desirable because, for n = 1/k, it results in constant (dc) primary current. Noisy switching current does not appear at the converter input but is diverted instead to the secondary winding. However, typical values of k are slightly less than one, and turns ratios of nearly 1:1 may not be easy to wind. One simplification is to use a 1:1 transformer, such as a low-cost, commodity, common-mode power-line input-filter choke, and add a small additional inductance in series with the primary winding. This effectively increases Llp so that the same secondary-winding dominance of magnetizing current is obtained with n = 1.
