Panning for nuggets on a rainy day. All the following items are fair-use exerpts for educational purposes from "Pulse, Digital, and Switching Waveforms" by Millman and Taub, McGraw-Hill, 1965.


The static resistance R of a diode is defined as the ratio V/I of the voltage to the current. At any point on the volt-ampere characteristic of the diode ... the resistance R is equal to the reciprocal of the slope of a line joining the operating point to the origin. The static resistance varies widely with V and I and is not a useful parameter.


For small-signal operation the dynamic or incremental resistance r is an important parameter and is defined as the reciprocal of the slope of the volt-ampere characteristic, r -= dVIdl. The dynamic resistance is not a constant but depends upon the operating voltage.

For a reverse bias greater than a few tenths of a volt ... g is extremely small and r is very large. On the other hand, for a forward bias greater than a few tenths of a volt ... r is given approximately by r = nVt/I [where Vt= 26mV, I is the operating current, and n is the recombination coefficient, = 1]

The dynamic resistance varies inversely with current; at room temperature, and for n=1, r = 26/I, where I is in mA and r is in ohms. For a forward current of 26 mA, the dynamic resistance is 1 ohm. The ohmic body resistance of the semiconductor may be of the same order of magnitude or even much higher than this value. Although r varies with current, in a small-signal model it is reasonable to use the parameter r as a constant.

Cutoff in a transistor is defined by the condition Ie=0. ... we see that the cutoff condition Ie = 0 implies that Ic = Ico [the collector base leakage current].It is important to note that in the common-emitter configuration the transistor will not be at cutoff if the base is open-circuited. ...

In germanium, even near cutoff, [reverse beta] may be as large as 0.9 and Ic = 10 Ico. Therefore the transistor is not at cutoff. In Sec. 6-17 we find that Ib = 0 corresponds to a small forward bias and that to bring the germanium transistor to cutoff we need to establish a reverse-biasing voltage between base and emitter of about 0.1 V. In silicon, at collector currents of the order of Ico, it is found that [reverse beta] is very nearly zero because of recombination in the junction transition region. Hence, even with Ib = 0, we find from Eq. (6-14) that Ic= Ico = -Ie, so that the transistor is still very close to cutoff. We verify in Sec. 6-17 that, in silicon, cutoff (Ie = 0) occurs at Vbe= 0 V, corresponding to a short-circuited base.

The Reverse Collector Saturation Current Icbo
The collector current when the emitter current is zero is designated by the symbol Icbo. Two factors cooperate to make Icbo larger than Ico. First, there exists a leakage current which flows not through the junction but around it and across the surfaces. The leakage current is proportional to the voltage across the junction. The second reason why Icbo exceeds Ico is that new carriers may be generated by collision in the junction transition region, leading to avalanche multiplication of current and eventual breakdown, as discussed in Sec. 6-9. But even before breakdown is approached, this multiplication component current may attain considerable proportions.

At 25C, Icbo for a germanium transistor, whose power dissipation is the range of some hundreds of milliwatts, is of the order of microamperes. Under similar conditions a silicon transistor has an Icbo in the range of nanoamperes.

The Break Region
The piecewise linear approximation... indicates an abrupt discontinuity in slope at [the diode cutin voltage]. Actually, the transition of the diode from the OFF condition to the ON condition is not abrupt. Therefore, a waveform which is transmitted through a clipper will not show an abrupt onset of clipping at a break point but will instead exhibit a break region of transition from unattenuated to attenuated transmission.

Since the limiting circuit clips or does not clip depending on whether the diode incremental resistance r is very large or very small in comparison with the circuit resistance R, let us arbitrarily define the break region as the range over which the diode resistance is multiplied by some large factor, say 100. The incremental diode resistance is
r = nVt/Id
Note that r varies inversely with the quiescent current and directly with the absolute temperature.

The resistance will be multiplied by a factor of 100 over the voltage range deltaV provided that {big math jumble}=100. We have then deltaV

... If the signal is only of the order of magnitude of the extent of the break region, the output will not display sharp limiting.


In many vacuum tubes there develops with use a cathode interface layer between the base metal of the cathode and the active emitting surface of the cathode ... The interface compound is a semiconductor compound formed as a result of the chemical interaction between the oxide-emitting material and the base metal or with some reducing constituent of the base metal. The resistance of the interface layer may lie in the range from several ohms to several hundred ohms and may therefore have an appreciable influence on tube operation. Additionally, the emitting surface and the cathode base metal serve as the electrodes of a capacitor, the cathode interface layer acting as a leaky dielectric between these electrodes. The overall effect of the interface layer is to introduce into the cathode a parallel resistance-capacitance combination whose time constant, it is found experimentally, normally lies in the approximate range 0.2 to 2.0 uSec.

In video amplifiers the effect of cathode interface resistance may well be serious. For a signal whose period is very large in comparison with the interface time constant, the principal effect is a loss in gain since the effective transconductance of the tube will be reduced...

An abrupt discontinuity applied to the tube grid will appear at the output similarly reduced in amplitude but accompanied by an overshoot at the leading edge of the pulse.

Interface resistance is present to some extent in all tubes with oxide coated cathodes but is usually particularly pronounced in tubes whose cathode base material contains a large amount of silicon. Interface resistance is inversely proportional to cathode area and is therefore more serious in tubes with small cathode areas. Also, since the effect of interface resistance is to reduce the effective transconductance by the factor 1 + gmRi, high-gm tubes are particularly sensitive to interface effects. Interface resistance increases with the total number of hours that the cathode has been heated, and the end of the useful life of a tube may be the result of interface resistance rather than loss in cathode emission.

A second disease which is often characteristic of video amplifier tubes has the popular designation "slump." The term is applied to a tube which behaves as though there were present in the cathode a parallel resistance-capacitance combination with a time constant in the range of several seconds. The response of such a tube to an input negative step is an output positive step which gradually slumps to a lower voltage level. The origin of "slump" is not well understood. The effect is often a source of difficulty in the design of d-c amplifiers for cathode-ray oscilloscopes.


Typical triodes used in pulse applications, as well as in other types of circuits, are the 6CG7, the 12AU7 (or its equivalent the 5963), the 12AT7, the 12AX7, and the 5965. These are miniature tubes and each contains two triode sections in one envelope. The 6SN7 is a nonminiaturized tube similar to the type 6CG7 and was the tube most commonly used in pulse-type equipment during World War II. The 5963 and 5965 were designed for use in high-speed digital computers. ... The curves for the 5965 are given in Figs. 6-30 and 6-31. In these latter characteristics, curves for positive grid voltages have been included because, as we shall see, the grid of a tube is often driven positive in pulse circuits. If the region near small plate voltages is ignored, then the positive-grid curves are very similar in shape and spacing to those for negative-grid values. Hence, if the grid signal is supplied from a source of low impedance, so that the loading effect on the source due to the flow of grid current may be ignored, the tube will continue to operate linearly even if the grid signal makes an excursion into the positive-grid region. This linearity will continue so long as the grid current is a small fraction of the total cathode current.

In pulse applications, large voltage swings are often encountered, and the small-signal equivalent circuit of Sec. 1-5 is meaningless because the tube parameters ii, r,, and g_ are not constant. The variation of these parameters with plate current is given in Fig. 6-32.

The grid volt-ampere characteristics of the 5965 tube are given in Fig. 6-33. At a given plate voltage the grid circuit behaves as a diode. By analogy with the definition of the dynamic plate resistance, the dynamic grid resistance r is given by dV/dIg, where V, and IG are the instantaneous values of grid voltage and current, respectively. The static grid resistance rg is defined as the ratio VG/IG. From Fig. 6-33 it appears that the difference in values between the static and dynamic resistances is not great, except possibly for small grid voltages. Furthermore, the value of the grid resistance ra is not a sensitive function of plate voltage. From Fig. 6-33 we find that for the 5965 tube, 250 ohms is a reasonable value for rG. For other tubes, the grid resistance may be much more variable than indicated above. For example, for a 12AU7 the static rG has values ranging from about 500 to 1,500 ohms, depending upon the values of grid and plate voltages

A Clamped Grid
If, as in Fig. 6-34, the grid leak is tied to the Vpp supply instead of to the cathode, then the grid-to-cathode voltage will approach nominal zero for values of R which are large compared with rG. For example, if R, = 1 M and Vpp = 300 V, then the grid current will be approximately 300 uA. From Fig. 6-31, we find that the grid voltage corresponding to this grid current is about -0.05 V. (If we assume that the value of rG = 250 ohms is valid at low grid voltages, then the calculated value of VG is 0.3 X 0.25 +0.075 V.) In many pulse circuits it is common to use this connection of the grid leak to a high positive voltage. Under such circumstances, where the grid is held at the cathode voltage because of the flow of grid current, we shall refer to the grid as being clamped to the cathode. Alternatively, the tube is said to be in clamp.

Variability of Characteristics If the grid voltage is made a few volts negative, then the grid current reverses. This negative current is caused by the positive ions which are attracted to the grid. Since the positive-ion current comes from the residual gas in the "vacuum" tube, it is very variable from tube to tube, and is usually a small fraction of a microampere. Negative grid current can also result from thermionic or photoelectric emission from the grid.

Clipper using Grid-Cathode "Diode"
We observe in Fig. 6-33 that the grid-voltage grid current characteristic of a multielectrode tube has much the same form as the volt-ampere characteristic of a simple thermionic diode. In the case of a triode, the characteristic depends somewhat on plate voltage but this dependence is small enough to be neglected in our present considerations. It then appears that a triode may be viewed as a combination of a diode and an ideal triode that draws no grid current. A clipping circuit using this grid-cathode "diode" is called a grid-current limiter. [figure of a triode with a resistor in series with the grid] ... a sinusoidal input is indicated and as appears, the grid-to-cathode signal Vg displays a clipped positive peak. ... under these circumstances the grid is said to be clamped to the cathode.

Since the grid signal does not ... cross any breakpoints of the ... transfer characteristic, the output current waveform has the same form as the grid signal. It is important to recognize that the clipping which appears in the output current waveform is NOT due to the failure of the plate current to respond to the grid voltage but rather to the failure of the grid voltage to respond to the applied signal. ... the plate current will respond almost linearly to grid voltage even for positive grid voltages up to the point where the grid current becomes an appreciable part of the total cathode current.

... the [grid] diode break is sharper than the cutoff break in a triode. Therefore if the series [with the grid] resistor R ... is large enough, the clipping which takes place in a triode at the occurrence of grid current may well be sharper than the clipping at cutoff.

Limiting by Bottoming
There is a third type of limiting possible with a triode [other than cutoff and grid limiting]. Consider [a triode circuit] without the series grid resistor necessary for grid current limiting. The largest possible plate current is Vpp/R;. If we apply to the grid from a low impedance source, a signal large enough to make the plate current nearly equal to Vpp/Rp, limiting will take place. ... Such limiting is sometimes referred to as plate current saturation but is not to be confused with any effect associated with maximum cathode emission. This type of limiting is also referred to as bottoming.