Power Supply Design
The power supply design appears to be causing some confusion in the DIY community as well. I figured I'd shed some light on the topic by providing the math needed for the power calculations.
As mentioned on the Thermal Design page, the power drawn from the power supply, including the quiescent current, for a given output swing can be calculated as,
where P_{S} is the total supply power, I_{bias} is the quiescent or bias current, V_{OUTpeak} is the peak output voltage, R_{L} is the load impedance, and V_{CC} is the supply voltage. It is assumed that the amplifier operates from a symmetric power supply, i.e. V_{CC} = -V_{EE}. The maximum output swing is determined by the supply voltage and the output dropout voltage of the LM3886, which is found in the data sheet, as excerpted below.
The LM3886 clips asymmetrically with the negative swing being clipped before the positive. Thus, the peak undistorted swing possible for the LM3886 is 2.5 V less than the supply voltage. Similarly, the quiescent current is found in the data sheet. I will use the typical number rather than the worst case for this example.
Example: Supply voltage: ±25 V; Load resistance: 4 Ω. The power drawn from the power supply at the full undistorted output power can be calculated as:
The supply power for a range of common supply voltages and load impedances is tabulated below.
VCC | V_{OUTpeak} | R_{L} | P_{S} |
---|---|---|---|
±25 V | 22.5 V | 4 Ω | 92.0 W |
±25 V | 22.5 V | 8 Ω | 47.3 W |
±28 V | 25.5 V | 4 Ω | 116 W |
±28 V | 25.5 V | 8 Ω | 59.6 W |
±35 V | 32.5 V | 8 Ω | 94.0 W |
P_{S} is the total power drawn by the amplifier from the power supply. From this power, the VA rating of the power transformer needs to be determined. Had the amplifier presented a purely resistive load, this would have been a simple task, however, the load presented by a full-wave rectifier is by no means a "nice" load. Rather, the current through the rectifier is a pulse train. It is possible to find an analytical solution for this, but the math gets rather involved. For those interested in the math, I suggest consulting Blencowe, who suggests using a conversion factor of 1.5 to convert from resistive power to the VA rating of the transformer. I.e. for the ±25 V, 4 Ω example above, a transformer with a VA rating of 1.5 × 92.0 W = 138 VA should be specified.
To verify this rule of thumb, I simulated the class AB load current using LTspice. The sim sheet is shown below. The power transformer model is an Antek AS-2222 toroidal transformer.
The resulting VA rating of the power transformer can be found as the sum of the reactive power (V-A products) of the two secondary windings. This is plotted below.
The extreme power at the beginning of the simulation is caused by the current needed to charge the reservoir capacitors to the full supply voltage. After about 100 ms, the supply voltage is within 90 % of its final value and it has fully settled by 250 ms. This extreme power is no cause for alarm. It is, however, a good argument for using an in-rush limiter or soft-start circuit, in particular in supplies using high efficiency transformer types, such as toroids.
The VA product is a somewhat distorted, half-wave rectified, sine wave with a peak value of 400 VA. The average value works out to 141 VA - pretty darn close to the 138 VA obtained by Blencowe's rule of thumb. Note that the reactive power does depend on the size of the reservoir capacitor. The bigger the capacitor, the larger the conversion factor between resistive power (W) and reactive power (VA). Blencowe's rule of thumb appears to hold up well for reasonable values of reservoir capacitors of, say, 4700 µF - 22000 µF.
Conclusion: An LM3886 running on a ±25 V supply, delivering the largest possible undistorted sine wave into a 4 Ω load will need a 140 VA transformer. For a stereo amplifier a 280 VA transformer should be used. I would round up to the nearest available standard size, which, typically, is 300 VA.
The Crest Factor, Revisited
Anyone who's ever taken apart a commercially available amplifier will comment that none of the 65 W rated amplifiers they have looked at have contained transformers capable of supplying 300 VA. What's the deal...? It's the crest factor again (see the thermal design section for a more thorough treatment of the subject).
Assuming the amplifier is to be used for music reproduction rather than the reproduction of sine waves, the power transformer can be undersized quite a bit. Extremely compressed music, such as some heavy metal, has a crest factor of 5~6 dB. Classical music, lands at the other end of the spectrum with a crest factor of about 20 dB. This means the peak power of classical music is 100× higher than the RMS power. In an analysis of 4500 tracks performed by Sound on Sound Magazine, an average crest factor of 14 dB was found. The supply power and resulting power transformer VA ratings (per LM3886 channel) are tabulated below for a range of common crest factors.
Crest Factor (dB) | P_{L} RMS | P_{L} peak | VCC | R_{L} | P_{S} | Power Transformer VA Rating |
---|---|---|---|---|---|---|
3 (sine wave) | 63.3 W | 127 W | ±25 V | 4 Ω | 92.1 W | 138 VA |
6 | 31.8 W | 127 W | ±25 V | 4 Ω | 66.0 W | 98.9 VA |
10 | 12.7 W | 127 W | ±25 V | 4 Ω | 42.5 W | 63.8 VA |
14 | 5.04 W | 127 W | ±25 V | 4 Ω | 27.8 W | 41.6 VA |
20 | 1.27 W | 127 W | ±25 V | 4 Ω | 15.2 W | 22.7 VA |
As seen in the table, a 100 VA transformer can actually be specified for a stereo LM3886 amplifier, assuming the crest factor remains around 12~14 dB. A transformer providing 20~22 V RMS, such as the Antek AS-1220 or AN-1222 would be suitable choices.
Section 7: Rectification & Snubbers