On the morning of August 2, 1927, a young electrical engineer named Harold Black was riding the Lakawanna Ferry across the Hudson River on his way to work in Manhattan, where he was employed by Bell Laboratories. Black was pondering an important problem that he had wrestled with for several years without making any progress. Suddenly, as he later recalled, the solution came to him in a “flash of insight”, and he wrote some notes about his idea on the only paper he had at hand – his morning copy of the New York Times.

Black’s brilliant idea was for something called “negative feedback”, and it was, along with the transistor and the Integrated Circuit, one of the most significant technical inventions of the twentieth century. Almost every electronic device made since Black’s invention, from radios to computers, uses negative feedback.

Negative feedback is fairly easy to understand, with only some basic algebra required. Like many great inventions, the wonder is that it was not though of sooner.

Before describing how negative feedback works, we need to first understand the technical problem that it solves. In Black’s time, vacuum tubes were used to amplify the strength of signals, and almost everything electronic used them. Vacuum tubes had a problem however; the amplification is nonlinear. The graph below shows an example of vacuum tube output as a function of the input.

Note that the graph is not a straight line, so that the output is not exactly proportional to the input. The result is distortion, which is almost always undesirable. For example, if we amplify music or a voice, the output will be louder, but muddled.

In the case of the vacuum tube, the output is proportional to the 3/2 power of the input, and there is no way to change the tube’s design to make it otherwise – the physics of the device determine what the amplification equation will be. When transistors were invented, they too had non-linear amplification curves, though the form of the equation is different.

So it appears that we are stuck – if we amplify a signal, the output will be distorted. However, Black’s simple idea provides a way out, as we shall now see.

First, consider the general amplifier below.

Note that the amplification factor A, also called the gain, is not a constant, since the amplifier is imperfect (as shown by the curve in the amplification graph above). In other words, the value of A changes a bit as the input signal level varies.

Now let’s consider the negative feedback amplifier, as conceived by Harold Black:

What we have done here seems strange; a fraction b of the amplifier’s output if being subtracted from the input. The signal going into the amplifier is then v_in – bv_out. This is then increased by A to produce v_out. The equation that describes this is then:

v_out = A(v_in – bv_out)

The overall gain (v_out/v_in) is no longer just A; let’s find out what it is:

v_out = Av_in - Abv_out, so

v_out +Abv_out = Av_in,

v_out(1 + Ab) = Av_in,

Finally,

Since the denominator is larger than 1, we have clearly reduced the gain from the original value of A. But that’s not the most important fact; A is fairly large, say 30, and we can arrange to have bA in the denominator be considerably larger than 1. In that case, we will have

Therefore, the gain, though smaller than before, is no longer dependant on A. Now, as it turns out in practice, it is easy to keep the value of b constant. Normally, this value is determined by the values of two components called resistors, and resistors are linear, stable, and cheap.

What Black’s invention does is trade gain for accurate amplification, and gain tends to be cheap. If we need more gain, a second amplifier can be placed in series with the first, and the gain before feedback becomes A². And the benefits don’t end with solving the distortion problem – anything that causes the gain A to vary will have its effect reduced by negative feedback. For example, temperature can cause A to change, but feedback can make the effective amplification insensitive to temperature.

So, the next time you are enjoying crystal clear music from your sound system, remember that Harold Black’s simple equation makes it possible.

Afterword

It took Black 10 years to be granted a patent for negative feedback. Apparently, the people at the patent office couldn’t believe at first that it would really work.

Black spent the rest of his career with Bell Laboratories, producing over 100 other patents, none of them as important as the first one. Before his death in 1983, he had begun writing an autobiography, to be called “Before the Ferry Docked”.

Black’s copy of the New York Times, show here, is on display at Bell Laboratories in Mountainside, NJ.

This entry was posted on Saturday, October 3rd, 2009 at 11:48 pm and is filed under Mathematics History, Thrilling Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.