By the end of this chapter you should be able to:
- Distinguish Zener tunnelling from avalanche multiplication by the doping, depletion width and field each requires.
- Explain the opposite temperature coefficients of the two mechanisms and the near-zero crossover at about 5.6 V.
- Replace the ideal Zener with the model \(V_Z = V_{Z0} + I_Zr_z\) and evaluate \(r_z\) from a datasheet.
- Design a shunt regulator: choose \(R_S\) from the worst-case line and load conditions and verify both extremes.
- Derive and compute line regulation, load regulation and ripple attenuation from \(r_z\) and \(R_S\).
- Identify the worst-case power dissipation in both the Zener and the series resistor, and select rated components.
- Use a Zener as a voltage reference and as a symmetrical clipper, and state the limits of the shunt topology.
Every diode has a reverse breakdown voltage, and every chapter so far has treated it as a limit to be respected: Chapter 6 quoted it as a rating, Chapter 8 computed the peak inverse voltage a rectifier must survive, and Chapter 11 warned that a voltage multiplier stresses each diode at \(2V_m\). A Zener diode is built to be operated there deliberately. Its breakdown voltage is manufactured to a tolerance of a few per cent, its characteristic in breakdown is almost vertical, and it is packaged to dissipate the power that flows when a diode conducts hard at 12 V rather than at 0.7 V.
What makes that useful is the shape of the curve. Once past the knee, a Zener holds its terminal voltage nearly constant while its current changes by two orders of magnitude — from a milliampere to a hundred milliamperes for a change of a few hundred millivolts. A device with that property is a voltage reference, and putting one across a load with a resistor feeding it from a larger, dirtier supply gives the shunt regulator, the simplest circuit in electronics that holds an output steady while both the input and the load move.
This chapter derives the two quite different physical mechanisms that produce reverse breakdown, shows why they have opposite temperature coefficients and why the crossover at about 5.6 V matters to anyone building a reference, replaces the ideal Zener with a model containing a dynamic resistance \(r_z\), and then designs a complete shunt regulator end to end for a stated line and load range. Line regulation, load regulation and ripple rejection all fall out of the same small-signal model, and the worst-case power dissipation — which occurs at the moment the regulator is doing the least useful work — decides which device you must buy. Chapter 30 returns to regulation with feedback and IC regulators; everything here is the foundation those circuits are built on.
1 Zener and Avalanche Breakdown
Reverse breakdown is a sudden, reversible collapse of a diode's reverse resistance, and it happens by two entirely different physical routes. Both are triggered by the electric field in the depletion region, and which one dominates is decided by how wide that region is — which in turn is decided by the doping.
Zener breakdown is quantum-mechanical tunnelling. In a very heavily doped junction the depletion region is only tens of nanometres wide, so a modest reverse voltage produces an enormous field. Chapter 5 gave the one-sided abrupt-junction result \(W = \sqrt{2\varepsilon(V_{bi}+V_R)/qN_D}\) and \(E_{\max} = 2(V_{bi}+V_R)/W\). With \(V_{bi} = 0.9\) V and \(V_R = 5\) V, those give:
| \(N_D\) (cm\(^{-3}\)) | Depletion width at 5 V | Peak field | Mechanism |
|---|---|---|---|
| \(10^{19}\) | 27.6 nm | 4.27 × 10\(^6\) V/cm | Zener tunnelling |
| \(10^{18}\) | 87.4 nm | 1.35 × 10\(^6\) V/cm | Zener, mixed |
| \(10^{17}\) | 276 nm | 4.27 × 10\(^5\) V/cm | Avalanche |
| \(10^{16}\) | 874 nm | 1.35 × 10\(^5\) V/cm | Avalanche |
Tunnelling becomes significant above about \(10^6\) V/cm, and the table shows that only the two most heavily doped junctions reach it at a few volts. At that field the valence band on the p side is pulled above the conduction band on the n side, and the barrier separating a filled valence state from an empty conduction state is thin enough that an electron can tunnel straight through it without ever being excited across the gap. No collision is involved, and the process needs no energetic carrier to start it.
Avalanche breakdown is impact ionisation, and it needs the opposite conditions. In a lightly doped junction the depletion region is wide, the field is comparatively low, and a carrier crossing it is accelerated over a long distance. If it acquires more than the band-gap energy between collisions, its next collision with the lattice creates an electron–hole pair; those two carriers are then accelerated in their turn and create two more, and the multiplication runs away. The multiplication factor is usually written
with \(n\) between 3 and 6 for silicon. \(M\) is finite and modest until \(V_R\) approaches \(V_{BR}\), then rises without limit — which is exactly the sharp knee seen on the characteristic.
The temperature behaviour follows directly from the two mechanisms, and it is opposite in the two cases.
- Zener tunnelling has a negative coefficient. Heating the lattice narrows the band gap slightly, which reduces the height of the barrier that must be tunnelled through. Less voltage is then needed to reach the same tunnelling rate, so \(V_Z\) falls as temperature rises.
- Avalanche has a positive coefficient. Heating increases lattice vibration, which shortens the mean free path between collisions. A carrier therefore gains less energy before each collision, so a larger field — a larger voltage — is needed to sustain multiplication, and \(V_{BR}\) rises with temperature.
Because the doping that gives 5 V breakdown is close to the doping at which the two mechanisms overlap, both operate at once in devices between about 4.5 V and 6 V, and their coefficients partly cancel. Figure 12.1 plots the measured coefficient across the range of a standard 1N47xx series.
The practical consequences are worth stating in volts rather than percentages. A 1N4728A (3.3 V) has a coefficient of \(-0.062\ \%/^\circ\text{C}\), so over a 50 °C rise its voltage falls by \(3.3\times0.00062\times50 = 102\) mV, a drift of 3.10 %. A 1N4744A (15 V) has \(+0.073\ \%/^\circ\text{C}\) and drifts \(+548\) mV, or 3.65 %, over the same range. A 1N4734A (5.6 V) has about \(+0.005\ \%/^\circ\text{C}\) and drifts only 14 mV, 0.25 %. Fifteen times better temperature stability, from the same process, for the same money, simply by choosing the voltage at which the two mechanisms cancel.
Where an exact cancellation is needed, the standard trick is to take a 6.2 V device, whose coefficient is a small positive \(+2.2\) mV/°C, and put a forward-biased silicon junction in series with it; the forward drop has a coefficient of about \(-2.0\) mV/°C, so the pair drifts by only \(0.2\) mV/°C, that is 10 mV over 50 °C on a 6.9 V reference. Compensated reference diodes of this kind, sold as the 1N821 to 1N829 series, are exactly a 6.2 V Zener and a diode in one package.
One point of vocabulary: almost every device sold as a "Zener diode" above about 5.5 V actually breaks down by avalanche, not by the Zener mechanism at all. The name has stuck to the whole family because Clarence Zener described the tunnelling process first, in 1934, before avalanche multiplication in junctions was understood.
2 The Zener Model and Dynamic Resistance
Figure 12.2 shows the complete characteristic. In the forward direction a Zener is an ordinary diode, conducting at 0.7 V; the interest is entirely in the third quadrant. Three features of it carry names, and each is a number on the datasheet.
The knee current \(I_{ZK}\), typically 0.25 to 1 mA, is the current below which the characteristic curls and the voltage is no longer well defined. Below the knee a Zener is not a reference; every design in this chapter must guarantee that the current stays above it under all conditions. The test current \(I_{ZT}\) is the current at which the manufacturer measured and binned the device: a 1N4742A is a 12 V \(\pm 5\ \%\) part at 21 mA and at no other current. The maximum current \(I_{ZM}\) is simply \(P_{D(\max)}/V_Z\), the point at which the package overheats.
Between the knee and the maximum the characteristic is very nearly a straight line, so the natural model is a battery in series with a resistance:
\(r_z\) is the dynamic or incremental resistance, quoted on datasheets as \(Z_{ZT}\) at the test current. It is not the static ratio \(V_Z/I_Z\), which for a 12 V device at 20 mA would be 600 \(\Omega\) and means nothing. For the 1N4742A, \(Z_{ZT} = 9\ \Omega\).
Extracting \(V_{Z0}\) from a datasheet is a one-line calculation that is often skipped and should not be. The stated \(V_Z\) applies at \(I_{ZT}\), so
and the model then predicts the voltage at any other current. At 1 mA it gives 11.83 V; at 50 mA, 12.27 V. So a "12 V" Zener is a 12 V device over a 440 mV band, which is 3.7 % — comparable with the \(\pm 5\ \%\) selection tolerance itself. Both effects must be carried into a design, and they are different in kind: the selection tolerance is a fixed unknown offset for a given device, whereas \(r_z\) produces a variation that moves as the operating point moves.
Two further facts about \(r_z\) matter in practice. It is strongly current-dependent, rising sharply as the knee is approached — for the 1N4742A it is 9 \(\Omega\) at 21 mA but 700 \(\Omega\) at 0.25 mA, a factor of 78. Operating a Zener at a comfortable current is therefore not merely about avoiding the knee but about keeping the regulation good. And \(r_z\) has a pronounced minimum around 6 to 8 V for a given power rating: low-voltage Zeners have high \(r_z\) because the tunnelling current depends only weakly on voltage, and high-voltage ones have high \(r_z\) because the die must be larger. A 5.1 V 1N4733A has \(Z_{ZT} = 7\ \Omega\); a 3.3 V 1N4728A has \(10\ \Omega\); a 100 V 1N4764A has \(350\ \Omega\). The best-regulating Zeners are the ones near the temperature crossover, which is a convenient coincidence.
3 The Shunt Regulator and Its Design Procedure
The circuit is a series resistor \(R_S\) from the unregulated input to the output node, and a Zener from that node to ground, reverse biased. The load hangs across the Zener. It works by a simple mechanism: the Zener absorbs whatever current the load does not want, so the total current through \(R_S\) stays nearly constant and the drop across \(R_S\) absorbs the whole of the input variation.
Everything that follows is a consequence of that one node equation. Note what it implies: the Zener current is largest when the load takes least, so the device runs hottest at the moment it is doing the least useful work.
The choice of \(R_S\) is squeezed between two worst cases, and both must be checked.
- Minimum input, maximum load. This is when the Zener has the least current available. If \(I_Z\) falls below \(I_{ZK}\) the device drops out of breakdown and the output collapses. This sets an upper bound: \(R_{S(\max)} = (V_{in(\min)} - V_Z)/(I_{ZK} + I_{L(\max)})\).
- Maximum input, minimum load. This is when the Zener carries the most current and dissipates the most power. It sets a lower bound through the device's power rating, and it is the condition that decides which Zener you must buy.
A design is feasible only if the upper bound exceeds the lower one. When it does not — a wide line range combined with a wide load range — the shunt regulator must be abandoned for the series topologies of Chapter 30.
Specification. Output 12 V. Input 20 V to 26 V (a rectified and filtered supply whose ripple and mains tolerance together give that range). Load current 10 mA to 30 mA. Take \(I_{ZK} = 5\) mA as the minimum acceptable Zener current, which is comfortably above the datasheet knee.
Step 1 — upper bound on \(R_S\). The tightest case is minimum input with maximum load: \(R_{S(\max)} = (20 - 12)/(5 + 30)\ \text{mA} = 8/0.035 = 228.6\ \Omega\).
Step 2 — choose a standard value. Take the next value below, \(R_S = 220\ \Omega\) (E12). Going below the bound gives margin; going above it would starve the Zener.
Step 3 — verify the low-input case. \(I_S = (20-12)/220 = 36.4\) mA, so \(I_Z = 36.4 - 30 = 6.4\) mA. That is above the 5 mA target, so the Zener stays in breakdown.
Step 4 — find the high-input case. \(I_S = (26-12)/220 = 63.6\) mA, and with the load at its minimum 10 mA the Zener takes \(I_{Z(\max)} = 53.6\) mA.
Step 5 — power ratings. \(P_{Z(\max)} = 12 \times 0.0536 = 0.644\) W. A 1 W device (1N4742A) has 36 % margin, which is the least one should accept without derating for ambient temperature. \(P_{RS(\max)} = I_S^2R_S = (0.0636)^2\times220 = 0.891\) W, so specify a 2 W resistor — a 1 W part would run at 89 % of rating.
Step 6 — sanity check on the current range. The Zener current swings from 6.4 mA to 53.6 mA, a range of 8.4:1. Over that range \(r_z\) is not constant, which is why the regulation figures of the next section are estimates rather than guarantees.
Two habits make this procedure reliable. Always compute \(R_{S(\max)}\) first and then round down to a standard value, never up. And always state which two worst cases you checked, because a design verified only at nominal input and nominal load is not verified at all — the failures happen at the corners.
4 Line Regulation, Load Regulation and Ripple
How well does the circuit of Figure 12.3 actually hold 12 V? Replace the Zener by its model and the answer follows from one small-signal circuit. For changes about an operating point, the Zener is just \(r_z\) to ground and \(R_S\) is a resistor to the input, so the output is a potential divider driven by the input change:
This ratio is the whole of the circuit's line performance. It says that 3.93 % of any change at the input arrives at the output, whether that change is a slow drift in the mains or the 100 Hz ripple left by the filter of Chapter 10.
Line regulation is that ratio expressed as a specification. Over the stated 20 V to 26 V range, \(\Delta V_{in} = 6\) V and
which is 1.97 % of the 12 V output over the full line range, or equivalently 0.327 %/V. Different textbooks and datasheets normalise this in different ways — as a bare ratio (3.93 %), as a percentage of output per volt of input (0.327 %/V), or as a percentage of output over the whole range (1.97 %) — so always state which convention is in use. The physical content is the same in all three.
Load regulation comes from the same model with the input held fixed. A change in load current \(\Delta I_L\) must be supplied by the parallel combination of \(r_z\) and \(R_S\), because from the output node those are the two paths to an AC ground:
\(r_z \parallel R_S\) is the output resistance of the regulator, and since \(r_z \ll R_S\) it is essentially \(r_z\). Making \(R_S\) smaller improves nothing here; only a smaller \(r_z\) does.
For the 20 mA load swing of the worked design, \(\Delta V_o = 0.020\times8.646 = 173\) mV. Working the full node equation with \(V_{Z0} = 11.82\) V, \(r_z = 9\ \Omega\), \(R_S = 220\ \Omega\) and \(V_{in} = 23\) V gives the actual outputs: 12.259 V at no load, 12.173 V at 10 mA, and 12.000 V at 30 mA. The conventional figure is
Both regulation figures are a few per cent, and both are dominated by \(r_z\). That is the fundamental limitation of the shunt regulator: it can never be better than the dynamic resistance of a passive device. The feedback regulators of Chapter 30 divide these figures by the loop gain and reach 0.01 % without difficulty.
Ripple is only a fast line variation, so the same divider applies. A capacitor-input filter leaving 2 V peak-to-peak of 100 Hz ripple on the 23 V rail delivers \(2\times0.0393 = 78.6\) mV peak-to-peak at the regulated output — an attenuation of 28.1 dB. This is why a Zener is worth fitting even where the DC accuracy does not matter: it is a ripple filter that needs no large capacitor, and unlike a capacitor its attenuation does not fall off at low frequencies.
One caution about all three numbers. They are computed from a single value of \(r_z\), and \(r_z\) varies over the operating range — in the worked design the Zener current swings by more than eight to one, and \(r_z\) is appreciably larger at 6.4 mA than at 53.6 mA. The figures above are therefore best-case estimates taken at the datasheet test current. A careful design either measures \(r_z\) at the worst-case current or takes the datasheet's \(Z_{ZK}\) value at the knee, which for the 1N4742A is 700 \(\Omega\) and would give a line regulation of 76 % rather than 3.9 %. That enormous difference is the quantitative reason for insisting on \(I_Z > I_{ZK}\) at every operating point.
5 Power Dissipation, Worst Case and Efficiency
The shunt regulator's defining habit is that it keeps the current drawn from the source constant and throws away whatever the load does not use. That makes the power arithmetic simple and the efficiency poor.
The worst-case Zener dissipation occurs at maximum input and minimum load, and is worth having as a formula because it is the number that selects the device:
For the worked design: \(12\times(63.6 - 10)\ \text{mA} = 0.644\) W. If the load can be disconnected entirely — a real possibility with a plug-in load, and one that catches people out — \(I_{L(\min)} = 0\) and the dissipation rises to \(12\times0.0636 = 0.763\) W, so the 1 W device is still adequate but the margin has shrunk from 36 % to 24 %.
Datasheet power ratings are quoted at a stated lead or ambient temperature, usually 50 °C for a 1 W axial device, and must be derated above it. A typical figure is 6.67 mW/°C, so at 75 °C ambient the 1 W part is a 0.83 W part and the 0.763 W worst case leaves only 8 % margin. Junction temperature, not nominal rating, is what destroys Zeners.
Efficiency follows from the same node equation. The input power is \(V_{in}I_S\) and the useful output is \(V_ZI_L\):
| \(V_{in}\) | \(I_L\) | \(I_S\) | \(P_{in}\) | \(P_{out}\) | Efficiency |
|---|---|---|---|---|---|
| 20 V | 30 mA | 36.4 mA | 0.727 W | 0.360 W | 49.5 % |
| 23 V | 20 mA | 50.0 mA | 1.150 W | 0.240 W | 20.9 % |
| 26 V | 30 mA | 63.6 mA | 1.655 W | 0.360 W | 21.8 % |
| 26 V | 10 mA | 63.6 mA | 1.655 W | 0.120 W | 7.3 % |
The best case is 49.5 %, and that occurs only at the one corner where the input is lowest and the load heaviest. At the opposite corner the circuit burns 1.53 W to deliver 0.12 W. The whole of the loss is heat in two components: \(R_S\), which drops the surplus voltage, and the Zener, which absorbs the surplus current.
Three consequences follow, and they define where the shunt regulator belongs.
- It is a low-current circuit. Scaling the design above to a 1 A load would require the Zener to absorb 1 A at no load, dissipating 12 W. Nobody does this; the series regulator exists precisely to avoid it.
- It tolerates a short circuit. With the output shorted, \(R_S\) limits the current to \(V_{in}/R_S = 118\) mA and the Zener sees nothing at all. The shunt regulator is inherently short-circuit proof, which is more than can be said for a naive series regulator.
- Its efficiency is best when it is working hardest. This is the opposite of most circuits and is a direct consequence of the constant total current.
A useful refinement, and the bridge to the next part of the course, is to let the Zener control a transistor instead of carrying the load current itself. Placing an emitter follower between the Zener and the load multiplies the current capability by \(\beta\) while leaving the Zener at a few milliamperes: a 12 V Zener and a transistor with \(\beta = 100\) will hold 11.3 V (one \(V_{BE}\) below the reference) across a 500 mA load while the Zener itself dissipates 60 mW. That circuit is the series regulator, and Chapter 30 develops it with feedback added.
6 The Zener as Reference and as Clipper
Not every Zener is in a regulator. Two other uses appear constantly, and both exploit the same flat characteristic from a different direction.
As a voltage reference, the Zener is not asked to supply current at all — only to define a voltage that some high-impedance circuit will compare against. That changes the design priorities completely. Current capability becomes irrelevant; temperature coefficient and long-term stability become everything. The standard reference circuit therefore feeds the Zener from a constant-current source rather than a resistor, which removes the line-regulation term entirely, since \(\Delta I_Z = 0\) means \(\Delta V_Z = 0\) whatever the input does. A 6.2 V compensated device driven at a fixed 7.5 mA from a current source is a reference good to a few parts per million per degree, and that is the heart of most precision analogue-to-digital converters built before band-gap references took over.
As a clipper, the Zener replaces the battery-and-diode combination of Chapter 11 with a single component. One Zener alone limits at \(V_Z\) in one direction and at \(V_F = 0.7\) V in the other, which is a strongly asymmetric limiter. Two in series, cathode to cathode, give the symmetric case: whichever way the input swings, one device is in breakdown and the other is forward biased, so the limit is
A pair of 5.1 V devices limits at \(\pm 5.8\) V. Two components, no bias supply, and a series resistor chosen by the same \(R = \sqrt{r_fR_L}\) rule as before. This is the standard protection network on the input of an instrumentation amplifier.
Both breakdown mechanisms are shot-noise processes at heart, and avalanche multiplication amplifies the noise along with the signal. A Zener in breakdown generates several microvolts per root hertz of noise — far more than a resistor of the same impedance — and the noise is worst in the avalanche region above 6 V. For a power supply this is irrelevant, because the load does not care. For a reference feeding a 16-bit converter it is fatal, and it is one reason band-gap references, which do not use breakdown at all, replaced Zeners in precision work. Where a Zener reference must be used, a large capacitor across it and a low-pass filter after it are standard practice. The same property is occasionally useful: a reverse-biased Zener is a convenient wideband white-noise source.
Finally, the limits of the topology, since knowing when not to use a circuit is as much a part of design as knowing how. The shunt regulator fails when the line range and the load range together make \(R_{S(\max)}\) smaller than the value the Zener's power rating permits. It fails when the load current is large, because the Zener must be able to absorb all of it. It fails when the output must be adjustable, because \(V_Z\) is fixed at manufacture. And it fails when better than about 1 % regulation is required, because \(r_z\) sets a floor that no choice of \(R_S\) can lower.
What it does supremely well is provide a fixed, moderately accurate voltage at a few tens of milliamperes from two components, with inherent short-circuit protection and 28 dB of ripple rejection thrown in. That is enough for biasing, for reference levels inside a larger circuit, for clamping and for protection, which is why Zeners appear on almost every schematic even though almost no modern equipment uses one as its main regulator. Chapter 13 turns to three more junctions engineered away from ordinary rectification, and Chapter 30 returns to regulation once the transistor has made feedback possible.
7 Summary and Key Results
| Quantity | Expression | Value |
|---|---|---|
| Breakdown mechanism | Set by depletion width and field | Tunnelling below ~5 V; avalanche above ~6 V |
| Field for tunnelling | \(E_{\max} = 2(V_{bi}+V_R)/W\) | \(1.35\times10^6\) V/cm at \(N_D = 10^{18}\), \(W\) = 87.4 nm |
| Temperature coefficient | Negative for Zener, positive for avalanche | −0.062 %/°C at 3.3 V; +0.073 %/°C at 15 V |
| Zero-TC crossover | Where the two mechanisms cancel | ≈ 5.6 V; +0.005 %/°C, 14 mV drift over 50 °C |
| Zener model | \(V_Z = V_{Z0} + I_Zr_z\) | \(V_{Z0}\) = 11.82 V; 11.83 V at 1 mA, 12.27 V at 50 mA |
| Series resistor bound | \(R_{S(\max)} = (V_{in(\min)}-V_Z)/(I_{ZK}+I_{L(\max)})\) | 228.6 \(\Omega\); choose 220 \(\Omega\) |
| Worst-case Zener current | \((V_{in(\max)}-V_Z)/R_S - I_{L(\min)}\) | 53.6 mA; 6.4 mA at the other corner |
| Line regulation | \(\Delta V_o/\Delta V_{in} = r_z/(R_S+r_z)\) | 0.0393; 236 mV over a 6 V line change = 1.97 % |
| Load regulation | \(\Delta V_o = \Delta I_L(r_z \parallel R_S)\) | 8.646 \(\Omega\) output resistance; 173 mV for 20 mA; 2.16 % |
| Ripple attenuation | \(r_z/(R_S+r_z)\) | 2 V pp in → 78.6 mV pp out; 28.1 dB |
| Zener dissipation | \(P_Z = V_ZI_{Z(\max)}\) | 0.644 W loaded, 0.763 W at no load; 1 W device |
| Resistor dissipation | \(P_{R_S} = I_S^2R_S\) | 0.891 W; specify 2 W |
| Efficiency | \(V_ZI_L/(V_{in}I_S)\) | 49.5 % best case, 7.3 % worst case |
| Zener clipper | Two in series, cathode to cathode | \(\pm(V_Z+V_F) = \pm 5.8\) V for a 5.1 V pair |
8 Common Mistakes
\(V_Z/I_Z\) for a 12 V Zener at 20 mA is 600 \(\Omega\), and that number appears in no useful calculation anywhere. What every regulation formula needs is the slope, \(r_z = \partial V_Z/\partial I_Z\), which is 9 \(\Omega\) for the same device — smaller by a factor of 67. Substituting the static value into \(r_z/(R_S+r_z)\) gives a line regulation of 73 % instead of 3.9 %, which would condemn a perfectly good design. The datasheet symbol is \(Z_{ZT}\), and it is always quoted at the test current; if the design operates the device near the knee, use \(Z_{ZK}\) instead, which is typically fifty to a hundred times larger.
A shunt regulator that works perfectly at 23 V input and 20 mA load may do either of two bad things at the corners: starve the Zener below its knee at minimum input and maximum load, so that the output falls out of regulation, or overheat it at maximum input and minimum load. Both corners must be computed explicitly, and the second is the one that selects the device — the Zener runs hottest when the load is lightest, which is the opposite of the intuition carried over from series-pass circuits. Remember also that "minimum load" may be zero if the load can be unplugged.
A 1N4742A is a 12 V \(\pm 5\ \%\) device at its 21 mA test current, and nowhere else. Run it at 5 mA and the model \(V_{Z0}+I_Zr_z\) gives 11.87 V; run it at 55 mA and it gives 12.32 V. That 450 mV spread is on top of the \(\pm 5\ \%\) selection tolerance of \(\pm 600\) mV and the \(\pm 366\) mV of temperature drift over 50 °C. Adding the three worst cases gives an output that could be anywhere between about 10.6 V and 13.4 V. If a design needs better than that, it needs feedback, not a better Zener.
9 Chapter Review
1. Explain why a 3.3 V Zener and a 15 V Zener have temperature coefficients of opposite sign, and compute the drift of each over a 50 °C rise.
The two devices break down by different mechanisms because they are doped differently. The 3.3 V device is very heavily doped, so its depletion region is only tens of nanometres wide and a few volts produce a field above \(10^6\) V/cm — enough for electrons to tunnel directly from the filled valence band on the p side into empty conduction states on the n side. Heating narrows the silicon band gap, which lowers the barrier that must be tunnelled through, so less voltage is needed for the same tunnelling rate and \(V_Z\) falls: the coefficient is negative, about \(-0.062\ \%/^\circ\text{C}\). Over 50 °C the drift is \(3.3\times0.00062\times50 = -0.102\) V, that is \(-3.10\ \%\). The 15 V device is much more lightly doped, its depletion region is wide, the field is lower, and breakdown occurs by impact ionisation: a carrier accelerated across the region gains more than the band-gap energy between collisions and creates a new pair, which multiplies. Heating increases lattice scattering and shortens the mean free path, so a carrier gains less energy per free flight and a higher field is needed to sustain multiplication; \(V_{BR}\) therefore rises. The coefficient is about \(+0.073\ \%/^\circ\text{C}\), giving \(15\times0.00073\times50 = +0.548\) V, that is \(+3.65\ \%\). Between about 5 V and 6 V both mechanisms operate together and the coefficients cancel, which is why a 5.6 V device drifts by only 14 mV over the same range.
2. Design a shunt regulator for a 5.1 V output from an input of 9 V to 12 V, with a load of 0 to 40 mA. Use \(I_{ZK} = 5\) mA. Select \(R_S\), verify both corners, and specify the power ratings.
The tightest condition for the Zener current is minimum input with maximum load: \(R_{S(\max)} = (9 - 5.1)/(5 + 40)\ \text{mA} = 3.9/0.045 = 86.7\ \Omega\). Round down to the standard value \(R_S = 82\ \Omega\). Check the low corner: \(I_S = (9-5.1)/82 = 47.6\) mA, so \(I_Z = 47.6 - 40 = 7.6\) mA, comfortably above the 5 mA target. Check the high corner: \(I_S = (12-5.1)/82 = 84.1\) mA, and with the load at zero the Zener carries all of it, so \(I_{Z(\max)} = 84.1\) mA and \(P_{Z(\max)} = 5.1\times0.0841 = 0.429\) W. A 1 W device such as the 1N4733A (5.1 V, \(Z_{ZT} = 7\ \Omega\)) has 57 % margin, which is ample. The resistor dissipates \(I_S^2R_S = (0.0841)^2\times82 = 0.580\) W, so a 1 W resistor is adequate but a 2 W part is safer in still air. For the regulation figures, \(r_z/(R_S+r_z) = 7/89 = 0.0787\), so a 3 V line change produces 236 mV at the output (4.6 % of 5.1 V) and the output resistance is \(7\parallel82 = 6.45\ \Omega\), giving 258 mV for the 40 mA load swing. The regulation is noticeably worse than the 12 V design of the chapter, because \(R_S\) has had to be made small to keep the Zener alive at the low corner, and a small \(R_S\) makes the divider ratio worse.
3. A Zener regulator has \(R_S = 220\ \Omega\), \(r_z = 9\ \Omega\) and a 12 V output. The unregulated input carries 2 V peak-to-peak of 100 Hz ripple. Find the output ripple, express the attenuation in decibels, and compare the circuit with a capacitor doing the same job.
For any small change, whether slow or fast, the output node sees a divider formed by \(R_S\) from the input and \(r_z\) to ground, so \(\Delta V_o/\Delta V_{in} = r_z/(R_S+r_z) = 9/229 = 0.0393\). The 2 V peak-to-peak input ripple therefore becomes \(2\times0.0393 = 78.6\) mV peak-to-peak at the output. In decibels the attenuation is \(-20\log_{10}(0.0393) = 28.1\) dB. Compared with a capacitor: to achieve the same attenuation at 100 Hz with a simple RC filter using the same 220 \(\Omega\) resistor, the capacitor's reactance would have to be 9 \(\Omega\), requiring \(C = 1/(2\pi\times100\times9) = 177\) µF. The Zener achieves it with a component the size of a grain of rice. More importantly, the Zener's 9 \(\Omega\) is frequency-independent, so it attenuates a slow mains drift and a 100 Hz ripple equally well, whereas the capacitor's reactance rises as the frequency falls and it does nothing at all at DC. That is the real argument for the Zener: it is a low impedance to ground at every frequency down to zero.
4. Why does a shunt regulator dissipate the most power when the load takes the least, and what is the efficiency of the chapter's worked design at each of its four corners?
The series resistor sets the total current entering the output node, \(I_S = (V_{in}-V_Z)/R_S\), and that current is independent of the load. Whatever the load does not take, the Zener must absorb, so \(I_Z = I_S - I_L\) is largest when \(I_L\) is smallest. Since \(V_Z\) is fixed, the Zener dissipation \(V_ZI_Z\) is greatest at maximum input and minimum load — precisely the condition in which the circuit is delivering the least useful power. For the worked design (\(V_Z = 12\) V, \(R_S = 220\ \Omega\)): at 20 V in and 30 mA out, \(I_S = 36.4\) mA, \(P_{in} = 0.727\) W, \(P_{out} = 0.360\) W, efficiency 49.5 %. At 26 V in and 30 mA out, \(I_S = 63.6\) mA, \(P_{in} = 1.655\) W, efficiency 21.8 %. At 20 V in and 10 mA out, \(P_{out} = 0.120\) W and the efficiency is 16.5 %. At 26 V in and 10 mA out, \(P_{in} = 1.655\) W for \(P_{out} = 0.120\) W, an efficiency of 7.3 %. The circuit is at its best at the corner where it is most stressed and at its worst when it is doing nothing, which is the opposite of most engineering. The compensating virtue is that the constant current makes the circuit inherently immune to a short circuit: shorting the output limits the current to \(V_{in}/R_S = 118\) mA and the Zener sees nothing.
5. A 1N4742A (12 V \(\pm 5\ \%\) at 21 mA, \(r_z = 9\ \Omega\), TC \(= +0.061\ \%/^\circ\text{C}\)) is used as a reference. Estimate the total worst-case uncertainty in its voltage over 0 to 70 °C and a 5 to 55 mA current range, and say what could be done to improve it.
There are three independent contributions. Selection tolerance: \(\pm 5\ \%\) of 12 V is \(\pm 600\) mV, and it is a fixed but unknown offset for any given device. Current dependence: using \(V_{Z0} = 12.00 - 0.021\times9 = 11.81\) V, the voltage is \(11.81 + 0.005\times9 = 11.86\) V at 5 mA and \(11.81 + 0.055\times9 = 12.31\) V at 55 mA, a spread of 450 mV, or \(\pm 225\) mV about the mid-point. Temperature: at \(+0.061\ \%/^\circ\text{C}\), a swing from 25 °C to either end of the 0 to 70 °C range gives at worst \(12\times0.00061\times45 = 329\) mV. Adding the three worst cases arithmetically gives \(\pm 1.15\) V, or \(\pm 9.6\ \%\) — an output that could lie anywhere from 10.85 V to 13.15 V. Three improvements are available, in increasing order of effectiveness. Select or measure the individual device, which removes the 600 mV selection term and leaves \(\pm 554\) mV. Feed the Zener from a constant-current source instead of a resistor, which removes the 225 mV current term entirely since \(\Delta I_Z = 0\). And, decisively, change the device: a 5.6 V Zener has a coefficient near zero, or a 6.2 V compensated reference diode such as the 1N829 has a net coefficient of about \(0.2\) mV/°C, which over the same 45 °C is 9 mV instead of 329 mV. Combining a compensated device with a current-source bias gives a reference stable to better than 0.1 %, and beyond that one leaves Zeners behind for a band-gap reference.