Negative approach to positive thinking

There is often felt to be something odd about negative components, such as negative resistance or inductance, an arcane aura setting them apart from the real world of practical circuit design. The circuit designer in the development labs of a large firm can go along to stores and draw a dozen 100 kΩ resistors or half a dozen 10 μF tantalums for example, but however handy it would be, it is not possible to go and draw a −4.7 kΩ resistor. Yet negative resistors would be so useful in a number of applications; for example when using mismatch pads to bridge the interfaces between two systems with different characteristic impedances. Even when the difference is not very great, for example testing a 75 Ω bandpass filter using a 50 Ω network analyser, the loss associated with each pad is round 6 dB, immediately cutting 12 dB off how far down you can measure in the stopband. With a few negative resistors in the junk box, you could make a pair of mismatch pads with 0 dB insertion loss each.

But in circuit design, negative component values do turn up from time to time and the experienced designer knows when to accommodate them, and when to redesign in order to avoid them. For example, in a filter design it may turn out that a −3 pF capacitor, say, must be added between nodes X and Y. Provided that an earlier stage of the computation has resulted in a capacitance of more than this value appearing between those nodes, there is no problem; it is simply reduced by 3 pF to give the Final value. In the case where the final value is still negative, it may be necessary to redesign to avoid the problem, particularly at UHF and above. At lower frequencies, there is always the option of using a ‘real’ negative capacitator (or something that behaves exactly like one); this is easily implemented with an ‘ordinary’ (positive) capacitor and an opamp or two, as are negative resistors and inductors. However, before looking at negative components using active devices, note that they can be implemented in entirely passive circuits if you know how (Roddam, 1959). Figure 1.1(a) shows a parallel tuned circuit placed in series with a signal path, to act as a trap, notch or rejector circuit. Clearly it only works well if the load resistance Rl is low compared with the tuned circuit’s dynamic impedance Rd. If Rl is near infinite, the trap makes no difference, so Rd should be much greater than Rl; indeed, ideally we would make Rd infinite by using an inductor (and capacitor) with infinite Q. An equally effective ploy would be to connect a resistance of −Rd in parallel with the capacitor, cancelling out the coil’s loss exactly and effectively raising Q to infinity. This is quite easily done, as in Figure 1.1(b), where the capacitor has been split in two, and the tuned circuit’s dynamic resistance Rd (Rd = QωL, assuming the capacitor is perfect) replaced by an equivalent series loss component r associated with the coil (r = ωL/Q). From the junction of the two capacitors, a resistor R has been connected to ground. This forms a star network with the two capacitors, and the next step is to transform it to a delta network, using the star-delta equivalence formulae. The result is as in Figure 1.1(c) and the circuit can now provide a deep notch even Rl is infinite, owing to the presence of the shunt impedance Zp across the output, if the right value for R is chosen. So, let R′ = –r, making the resistive component of Zs (in parallel form) equal to –Rd. Now R′ turns out to be −1/(4ω2C2R) and equating this to –r gives R = Rd/4.

Negative inductor

Now for a negative inductor, and all entirely passive – not an opamp in sight. Figure 1.2(a) shows a section of constant-K lowpass filter acting as a lumped passive delay line. It provides a group delay dB/dω of √(LC) seconds per section. Figure 1.2(b) at dc and low frequencies, maintained fairly constant over much of the passband of the filter. A constant group delay (also known as envelope delay) means that all frequency components passing through the delay line (or through a filter of any sort) emerge at the same time as each other at the far end, implying that the phase delay B = ω √(LC) radians per section is proportional to frequency. (Thus a complex waveform such as an AM signal with 100% modulation will emerge unscathed, with its envelope delayed but otherwise preserved unchanged. Similarly, a squarewave will be undistorted provided all the significant harmonics lie within the range of frequencies for which a filter exhibits a constant group delay. Constant group delay is thus particularly important for an IF bandpass filter handling phase modulated signals.) If you connect an inductance L′ (of suitable value) in series with each of the shunt capacitors, the line becomes an ‘m-derived’ lowpass filter instead of a constant-K filter, with the result that the increase of attenuation beyond the cut-off frequency is much more rapid. However, that is no great benefit in this application, a delay line is desired above all to provide a constant group delay over a given bandwidth and the variation in group delay of an m-derived filter is much worse even than that of a constant-K type. Note that L′ may not be a separate physical component at all, but due to mutual coupling between adjacent sections of series inductance, often wound one after the other, between tapping points on a cylindrical former in one long continuous winding. If the presence of shunt inductive components L′ makes matters worse than the constant-K case, the addition of negative L′ improves matters. This is easily arranged (Figure 1.2(c)) by winding each series section of inductance in the opposite sense to the previous one.

Real pictures

Now for some negative components that may, in a sense, seem more real, implemented using active circuitry. Imagine connecting the output of an adjustable power supply to a 1 Ω resistor whose other end, like that of the supply’s return lead, is connected to ground. Then for every volt positive (or negative) that you apply to the resistor, 1 A will flow into (or out of) it. Now imagine that, without changing the supply’s connections, you arrange that the previously earthy end of the resistor is automatically jacked up to twice the power supply output voltage, whatever that happens to be. Now, the voltage across the resistor is always equal to the power supply output voltage, but of the opposite polarity. So when, previously, current flowed into the resistor, it now supplies an output current, and vice versa. With the current always of the wrong sign, Ohm’s law will still hold if we label the value of the resistor −1 Ω. Figure 1.3(a) shows the scheme, this time put to use to provide a capacitance of −C μF, and clearly substituting L for C will give a negative inductance. For a...