Technical Bulletin No.1
Application Notes on Rectifier Transformers

A large percentage of our transformers are used in rectifiers of the type illustrated in Figure 1, so we decided to dedicate this first edition of TRANSFORMER FACTS to rectifier transformers, and provide some practical tips for power supply designers.

The opening statement is true if we compare average currents (I_m) on both the AC and the DC side of the rectifier. But AC current is always measured as average current (I_RMS), while DC current is always measured as average current (I_m). The opening statement is false if we compare I_RMS on the AC side with I_m on the DC side of the rectifier.

The RMS current (I_RMS) is always larger than the average current (I_m) because of the peaked shape of the AC current. When I_RMS is divided by I_m we obtain a measure of the peakedness of the current, which is called Form Factor. (F = I_RMS/I_m). The sharper the peaks are, the larger is the value of F.

The heating effect of an electrical current in wiring, resistors, and transformer windings is proportional to the square of the RMS current. The heating effect of the AC current in a rectifier circuit is accordingly proportional to I_S^œ = (F‡I_m)^œ = (F‡I_L )^œ, or the square of the DC current multiplied by F squared. The temperature rise in a given rectifier transformer is thus heavily dependent of the value of the Form Factor (F), and the required size of a rectifier transformer cannot be determined until the actual value of the Form Factor is known.

In a rectifier of the type shown in Figure 1, F has a value somewhere between 1.11 and 5.0 depending on the relative values of the impedances before and after the diode bridge. When these impedances are known, it is possible to calculate F (and U_C) using graphical methods published by Schrade in 1943. But at that point the power supply designer usually has in his hand a prototype transformer, so U_C and I_S can be determined quickly by bench tests. (Be careful to measure I_S with a meter measuring true RMS current. Most AC current meters measure I_m but are graduated in I_RMS, assuming F=1.11 which is only true for sinewave.

An accurate and simple method for determining the Form Factor (F) from oscilloscope readings with the aid of graphs developed by Toroid Corporation of Maryland is described below.

Let us assume we observe the current and voltage waveforms in different parts of the circuit of Figure 1 on a CRT oscilloscope, so we can compare waveforms before and after the diode bridge. Diagrams I-III show oscillograms for different values of the capacitor (C), assuming a transformer with negligible series inductance, such as a toroidal transformer.

The desired effect of the capacitor is to smooth the DC voltage, but at the same time it causes the AC current to flow in short bursts, which means higher F and larger RMS current in the transformer. The "conduction angle" (a) of the rectifier can be measured directly off the oscillogram - just remember that a full halfcycle is 180þ.

It is clear that the Form Factor (F) must depend on the conduction angle (a). We have calculated the exact relationship between F and a for toroidal transformers, and the result is shown here in this graph. By measuring the conduction angle (a) off the oscilloscope a very accurate value for the Form Factor (F) can be read off the graph. Variations in DC load will change the conduction angle, and the corresponding changes in the Form Factor can easily be determined.

The graph sheet includes more information, which can serve as an aid to evaluating the trade-offs in power supply design. In the comments to the diagrams we have defined h=U_DC/Ç_o, which relates the DC voltage to the peak no-load voltage of the transformer secondary. The flattening of the tops of the AC voltage waveform is caused by the voltage drop in the total impedance ahead of the diode bridge, so it is reasonable to assume that h must vary with the conduction angle (a). We have calculated this relationship also, again assuming toroidal transformers, and the result is shown in the graph sheet enclosed as a broken curve.

An important use of the graph for h is to determine the DC load regulation of a rectifier. DC load regulation DU_DC/U_DC = (1-h) ‡ 100%. Remember that the diode voltage drops are included in the valued for U_DC. Each diode voltage drop can be assumed to be constant = 1V at all loads. The net load regulation is accordingly slightly worse than 1-h, especially for low DC voltages.

It is important to note that better efficiency of voltage conversion (as measured by h) can only be obtained at the cost of higher Form Factor, and conversely lower Form Factor can only be obtained at the cost of poorer DC load regulation.

The size of the transformer supplying the rectifier is proportional to the product of no-load voltage (U_o) and current capacity (I_S), which we call P_o. The dotted line in the graph sheet represents the lowest value of P_o required for any value of a (of any corresponding value of F or h) for a given DC power. (P_o/P_DC = F/‹2).

The transformer has a minimum size (P_o) of about 1.52 ‡ P_DC (total DC power including diode losses) for a = 75þ, where h= .8 and F = 1.7. Unfortunately it is not possible to stay near the minimum at all times, partly because better DC regulation than 20% is often required, and partly because the load regulation of transformers vary widely with transformer size. DC load regulation and transformer load regulation are not proportional, but they generally increase and decrease together, so very small transformers tend to work at larger than optimal values of a, and very large transformers work at smaller than optimal values of a.

The design of a rectifier transformer to meet specific requirements for U, U, DC regulation, temperature rise, etc., requires exact data for both Form Factor (F) and rectification efficiency (h). But F and hare in turn determined by the data of the not-yet-designed transformer, so the power supply designer is caught in a "Catch-22" situation. One way out is to grab some old design, modify it some, and pray that the prototype will work.

Another way out is to let Toroid Corporation of Maryland do the transformer design. Our application engineers have solid backgrounds in transformer and power supply design, and they have at their disposal interactive computer programs for transformer optimization, so they can design not only a transformer that will work, but also the most economical transformer that will do the job.

	Comments:
	IL= ¡ ‡ œ/p (average)
	IS = ¡ ‡ 1/‹2 (RMS)
	F = IS/IL= p/(2 ‡ ‹2) = 1.1107
Diagram I. C = 0. No regulator

	Comments:
	UDC = UC + UDIODES
	UDC/Ço = h= h(a)
	a = conduction angle
	F = IS/IL= f (a)
Diagram II. C ’ Æ

	Comments:
	Ripple Voltage (Ur) Symmetrical on UC
	UDC to center of slope - otherwise as in II.
Diagram III. C = normal (Ur /UC<10%). Regulator working

 O.H. Schrade: "Analysis of Rectifier Operation" Proceedings of the I.R.E., July 1943, pp.341-361. (This paper is difficult to find even in libraries. The important parts of the graphs are reprinted in MOTOROLA Silicon Rectifier Manual.)