The K Multiplier - An Economical Supercell Circuit

The basic circuit is at once simple and super-effective. Film and low-ESR decoupling can be used with it and it will not oscillate. Is there another circuit that is as effective with a comparable parts count?

Last updated April 22, 2014
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In professional circles, designing analog circuits for good supply noise tolerance is the accepted norm.  But home experimenters have found that low-noise power supplies present the ultimate freedom to test out their inexhaustible creativity. The extra degree of freedom gained from clean power allows for less distractions from the ultimate goal of a good-sounding circuit. So audio hobbyists have been hard at work for decades refining the art of quiet, inert power supplies. Along these lines several super-high performance solutions are available from the DIY community. Many of them are very complex, needing performance opamps and many discrete devices. But as is often the case, increasing complexity tends to give diminishing returns. For this reason the simplest solutions are popular to hobbyists. My circuit is a bit simpler than most, but I see few competitors where parts count, cost and ease of assembly are concerned.

Kmultiplier

I'll use the positive circuit to explain. Q1 and Q2 can be recognized by most poweramp designers as a Complementary Follower Pair, CFP for short. This arrangement provides dramatically reduced output impedance and equivalent current gain to a Darlington arrangement - just what we want for a power source. Both transistors need at least 1.2V Vce to work best, which is what the triple diode string D1 is for. Q2 needs more than Q1, so I've chosen 3 diodes for 1.2V. This may seem wrong to you, but consider that the only current through these diodes is the base current of Q1. Diode Vfb is roughly 400mV in the uA range, following the 60mV/decade rule. Q1 is biased at about 15mA, which generally gives the lowest output impedance. D2 and R8 determine the turn-on time, and limit how much current the circuit will dump into proceeding supply capacitors. This was important for one builder who employed a morbidly gratuitous reservoir array. The first time he flipped the power switch the transistors would blow from the inrush current! It took a few tries for me to realize the problem... If your circuit does not have inrush current over 1A, and this can be verified in simulation, you do not need R8 and can bypass it. D3 discharges C1 when power is lost so it doesn't discharge through Q1 - do not omit! The BC337/327 pair are crucial to this design - they set the upper limit on input rejection. I got input rejection using these of about 66db for the positive circuit and 54db for the negative circuit (measured in real life). BC550C/560C are second-best in this regard. ALL high-voltage transistors I have seen have bad quasi-saturation behavior and should be avoided for low-Vce applications (2N5551, 2SC1845, etc).


Circuit Specs:


Advantages of the Kmultiplier


If we say that the output impedance of the Kmultiplier is equivalent to that of electrolytics, we establish a baseline to consider its advantages. One incontrovertible advantage over electrolytics is that the Kmultiplier has low impedance into the subsonic range. A comparable lytic in this regard would be huge, and would necessitate soft-start circuitry to prevent the massive inrush from destroying the preceeding wiring. Clearly the Kmultiplier provides an expediency that no capacitor can replace.

The average electrolytic 470uF and up has lower than 50mR ESR. I have measured this figure repeatedly on lytics pulled out of broken modern entertainment devices such as computer monitors, TVs, and power amps. The best have less than 20mR. So the output impedance of this very simple circuit fares very well in practical terms. The fact that this circuit has output impedance comparable to modern lytics AND will tolerate film and low-ESR bypass shows that I have struck a good balance - output impedance is not any lower than is actually needed, but is low enough to augment and seamlessly integrate with a good bypassing and decoupling scheme.

The voltage overhead of the Kmultiplier is low, making it a persuasive drop-in addition to circuits which don't need DC regulated rails but would benefit from heavy power filtering. In this case the Kmultiplier replaces an oversized transformer and banks of capacitors, making it a truly economical and expedient solution.

Because the Kmultiplier has no more feedback than needed and is self-limiting in bandwidth, the compensation capacitors are merely those parasitically included with the transistor junctions. This meager amount of capacitive stress means that the circuit is unlikely to glitch even with very fast load transients. Many regulators will emit a highly distorted output signal when stressed with fast load current swings, which can get into circuitry and make things worse than if the supply was not regulated. The Kmultiplier however remains faithful into the hundreds of KHz. Of course, why even risk high-speed signals at the regulator when you can apply liberal decoupling across the load without fear of oscillation?


Design Considerations


Troubleshooting


Making Changes


If you want to change something, beware that it may change troubleshooting procedures and design considerations. That said, if you are confident you do not need guidance for your application, here are some ideas and helpful information. Also, discuss your modifications with us in the Kmultiplier thread.

Other adaptions of the Kmultiplier


This version is good for the frontend of very high-power amps with 100V rails, or any application where you need more than 45V output. It can be used at any rail voltage, regardless of the transistors' individual voltage limits, as long as the capacitors are up to it. The only difference is the startup is immediate rather than delayed, and it needs a lot of diodes.

High Voltage Kmultiplier


This version is for tube preamps. R5 limits output current but is necessary to ensure safe inrush.

High Voltage Kmultiplier


Is there a Kmultiplier I can use for a power amp?


Adapting the Kmultiplier for higher currents is not so simple. An increased number of active devices is required, and new feedback loops must be added. This creates a pandora's box of new difficulties which are not in the original design simply because of its simplicity. A more powerful version needs compensations and very careful dimensioning to achieve the benefits of low voltage drop, low output impedance and compatibility with well-decoupled designs. The resulting circuit is not so elegant in its presentation or its performance, but I think it can be done, and I may have done it. I will link to it tentatively.


Theory of Operation


The design of this circuit begun when I was thinking about the capacitance multiplier circuit block, shown below. The C-multiplier is such a useful, underused circuit. It can replace banks of caps, given enough voltage headroom. Furthermore its input isolation can be a big bonus for RF filtering provided you select transistors carefully. However it seemed to me no one had thought about this circuit for decades. Relegated to ancient history, no one seemed to grasp it's potential when revamped with modern transistors and design expertise. Here is the whole story.

Capacitance Multiplier

It is called the capacitance multiplier because, (according to theory) it appears to the load as if C1 has been multiplied by the Hfe of Q1. This particular transistor has an Hfe of 150 or so, which means our load sees a supply capacitance of about 1500uF.

Those who have gotten this far probably understand that the reality is less pretty. After all, the BE junction of Q1 acts like a silicon diode in series with the load. This means the output impedance of this filter circuit is very high, and very nonlinear. The impedance of general purpose silicon follows the rule R=.033/Id, where R is the small-signal resistance of the junction and Id is it's forward bias current in the given application. So if our load draws about 40mA, then our filter has an output impedance of tentatively .825 ohms. This is abysmal.

As seen by the load, R1 is also divided by the Hfe of Q1, and this is in series with the "diodic" output impedance - 825mR+(100R/150Hfe)=1.492R. Even worse. But this is followed by a sigh of relief when we realize that at AC the transistor's base sees a short through C1, returning the output impedance to 825mR.

Even so the nearly doubled output resistance at DC looms over us.
Lets set up a comparison to give a sense of reality and scale to these figures. This circuit tries to emulate a 1500uF capacitor. A standard electrolytic capacitor in this range has around 50mR or .05 ohms. Furthermore, the transistor only conducts one way, unlike the capacitor which can absorb negative current surges. So as you can see, the circuit does not really compare to a real capacitor. They are so different they cannot be treated like equivalents. The drawbacks are listed as follows:
That aside, the circuit still has one considerable benefit over a bare 1500uF electrolytic. This is ripple rejection and input isolation. As Q1 is in the emitter follower configuration, the emitter follows the base voltage, and the base is fed by an RC lowpass filter of 100R*100uF.  We don't get the benefit of this RC arrangement with just a 1500uF capacitor - where the only R is that of the rectifier diodes and the transformer winding. The input isolation is limited only by the RC corner frequency and transistor leakage. Depending on the transistor, you may expect AC input rejection from 40-70db. This is where transistor selection makes all the difference. I won't go into detail on all the ways to improve this dinosaur, but here are a number of ways for you to consider if you don't really need the performance that can be gained from an extra transistor:
At this point there are many, many things we could try to tailor the performance in many directions. Of these, the options adding another active device tend to become a bit less flexible. If we replace Q1 with a Darlington pair for instance (diagram below), you will need to draw enough current at all times to keep the driver transistor on. An extra diode drop is added, but possibly offset by a lower R1 voltage drop. Even so, the performance gains can be dramatic. Due to the greatly decreased base current, we can increase R1 by several times. This allows output filtering to the subsonics, or alternatively less resistive output impedance at DC. However AC output impedance can be made worse. Say for instance our driver transistor is biased at 2mA and our output transistor has an Hfe of 100. The load draws 102mA. Following the diodic output impedance rule, Q1 defines most of the output impedance at 330mR. It's base current is 100mA/100, 1mA. The diodic resistance of Q2's emitter is divided by the output's Hfe like in the original C-multiplier, and comes to 165mR. So the total output impedance, neglecting R1, is 495mR. Often times designers neglect to give the driver any bias current at all except the base current of the  output. Because the output's base current rises proportionally with load current, and the diodic emitter resistance decreases proportionally with emitter current, the net result is that proportions cancel, and the Darlington output resistance gains the nonlinearity of 2 diodes in series - 660mR. Neat, right? Of course, you had better decouple the supplies well, because any fast load signals will saturate and pump the driver transistor and result in nasty glitching.

Darlington C-multiplier

Ultimately, the Darlington C-multiplier still leaves us wanting more. Many designers don't feel like pushing the limit -  the C-multiplier was never the Rolls Royce of supply solutions anyway. Why not just use an LM7812? But because of reasons mentioned at the beginning of this page, even that is not a satisfying option. Is there a middle ground between simplicity, expediency and performance?

Most designers already know about the benefits of a CFP over a Darlington, even though few seem to have thought to apply it to a C-multiplier. The CFP has higher transconductance in common emitter form, which translates to lower output impedance in common collector form. However the general consensus on the CFP from amplifier designers is that it is unstable and risky. Many early amplifier designs featuring CFP output stages were found to up and blow up one day for no apparent reason. Eventually it came out that the cause was that it has a tendency to oscillate, causing the transistors to dissipate a lot more power than they should have. For well-trained amplifier designers, the problem is just a matter of engineering, but the circuit in question must be measured in the prototype; RF parasitics depend so heavily on wiring that it is easier just to probe with a signal generator. Truth be told many designers don't have the background necessary to understand the problem.

I can go into detail on what the problem is in another article, but here I will discuss the principle of operation.

Here again is the Kmultiplier diagram for you to refer to throughout my explanation:

Kmultiplier

Another name for the CFP could be the "G-multiplier pair". In essence, the output conductance is the conductance of Q1 multiplied by the  current gain of the Q2/R1 arrangement. Let's analyze the situation and get an idea of how this works.

Q1 is biased into it's most linear range by R1. So Ic(Q1) is roughly .68/47=~15mA. This is a bit low to accommodate for a nominal max of 5mA Ib(Q2). In the nominal range of loading Ic(Q1) ranges from 15mA to 20mA. So, once more following the diode rules, the output resistance of Q1 varies between 2.2R and 1.65R. Now lets include Q2. Lets say our load current is 116mA. This sets the Ic of Q1 and Q2 to 100mA and 16mA respectively, accounting for Q2's Hfe of about 100. At 100mA, Q2's Re is about 330mR. For every 1mA of loading, Vbe(Q2) increases 330uV. This increase in voltage across R1 results in a 7uA increase in it's current. This is added to the increase in Ib(Q2) of 10uA and we get an increase in Ic(Q1) of 17uA. 17uA across Q2's Re of ~2.2R gives us a final 37uV output drop per 1mA. 37uV/1mA gives us 37mR as the entire arrangement's output impedance. This is close enough for horseshoes to the measured values.

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