The semi-discrete design of a power-factor controller (PFC) uses a multiplier as a variable-gain amplifier (VGA). It amplifies the rectified sinewave voltage and applies it as the control input to the current loop. In this article, low-cost, low parts-count multiplier design is presented.

The sine-magnitude voltage from the rectifier is the input signal to the current loop. The PFC amplifies the voltage waveform as a scaled current waveform, thereby producing a resistive PFC input. The scaling is controlled by the PFC voltage loop, to control the dc output voltage. The sine-magnitude is unipolar and positive; therefore, the multiplier need operate only in the two positive quadrants. Such a multiplier can be implemented with a low parts count and low cost.

This approach to two-quadrant multiplier design applies the translinear principle that Barrie Gilbert discovered in the 1960s. The (inverting) translinear multiplier circuit is shown below.

The translinear cell is formed by Q1, Q2, Q4, and Q5. The ratio of currents i₄/i₅ is the same as the ratio  i₂/i₁. This can be derived beginning with the basic v-i relationship for a bipolar junction (diode):

where I_S is the junction saturation current, typically around the miniscule value of 10^–15 A. The base-emitter junctions of Q₄ and Q₅ are in series, connected at node 3 to current source I₁. Starting at the emitter of Q₄ and summing voltages to the emitter of Q₅, the Dv across Q₄ and Q₅ is:

For matched transistors, I_S4 = I_S5 and

A similar calculation for matched transistor pair Q₁, Q₂ results in:

The two transistor pairs are in parallel with the same Dv values. Equating the two Dv expressions,

Next, consider the current source I₁ supplying Q₄ and Q₅. Summing currents at node 3,

i₄ + i₅ = I₁

Similarly,

i₁ + i₂ = I₂

Now, it is a most interesting fact of algebra that if

then

Applying this algebraic transform (which is the key to translinear multipliers) to the above circuit,

where the differential output current,

i_O = i₁ – i₂

and the differential diode input current,

i_I = i₄ – i₅

The Q₁, Q₂ diff-amp output half of the translinear cell is combined at the collector of Q₂. Q₁ collector current, i₁, is flipped around by a ´1 current mirror (IcIs1) and added algebraically to i₂, producing a single-ended i_O at the output node (8), which develops an output voltage across R₄. At the input, i_I is controlled by input voltage Vs in the diagram.

Let's now examine these math results for their electronic meaning. The output current, i_O, is scaled by current source I₂. The transfer function equation can be written as:

The multiplier function is I₂×i_I. By making I₂ controllable by another input, that input multiplies with v_s. In addition, note that I₁ divides the result, so that if I₁ were controlled, this input would be a divisor.

Now that the basic theory of the Gilbert gain-cell multiplier has been presented, we will proceed to look at its behavior, both simulated and measured.

Ó Dennis L. Feucht, 2000