1.2.5 Terminal Characteristics of Junction Diodes
In this section we study the characteristics
of real diodes—specifically, semiconductor
junction diodes made of silicon.
Figure 3.7 shows the i-v characteristic of a
silicon junction diode. The same characteristic is
shown in Fig. 1.18 with some scales expanded
and others compressed to reveal details. Note
that the scale changes have resulted in the
apparent discontinuity at the origin. As
indicated, the characteristic curve consists of
three distinct regions:
1. The forward-bias region, determined by v > 0
2. The reverse-bias region, determined by v < 0
3. The breakdown region, determined by v < -VZK
These three regions of operation are described in
the following sections.
FIGURE 1 . 17 The i-v characteristic of a silicon
junction diode.
The Forward-Bias Region
The forward bias or simply forward region of
operation is entered when the terminal voltage
v is positive. In the forward region the i-v
relationship is closely approximated by
as in eq
(1.18)
In this equation Is is a constant for a given diode
at a given temperature.
The current Is is usually called the saturation
current (for reasons that will become apparent
shortly)- Another name for Is, and one that we
will occasionally use, is the scale current. This
name arises from the fact that Is is directly
proportional to the cross-sectional area of the
diode. Thus doubling of the junction area results
in a diode with double the value of Is and as the
diode equation indicates, double the value of
current i for a given forward voltage v.
FIGURE 1.1 8 The diode i-v relationship with some
scales reveal details. expanded and others
compressed in order to reveal details
Figure 1.19 The basic pn junction diode: (a) simplified
geometry and (b) circuit symbol, and conventional
current direction and voltage polarity
25
Temperature Effects
Since both IS and VT are functions of temperature, the diode
characteristics also vary with temperature. The temperature-related variations
in forward-bias characteristics are illustrated in Figure 1.20. For a given
current, the required forward-bias voltage decreases as temperature increases.
For silicon diodes, the change is approximately 2 mV/°C.
The parameter IS is a function of the intrinsic carrier concentration ni,
which in turn is strongly dependent on temperature. Consequently, the value
of IS approximately doubles for every 5 °C increase in temperature. The actual
reverse-bias diode current, as a general rule, doubles for every 10 °C rise in
temperature. As an example of the importance of this effect, the relative value
of ni in germanium, is large, resulting in a large reverse-saturation current in
germanium-based diodes. Increases in this reverse current with increases in
the temperature make the germanium diode highly impractical for most circuit
applications.
Figure 1.20 Forward-biased pn junction characteristics versus temperature. The required diode voltage to
produce a given current decreases with an increase in temperature.
Breakdown Voltage
When a reverse-bias voltage is applied to a pn junction, the electric field in the
space-charge region increases. The electric field may become large enough that
covalent bonds are broken and electron–hole pairs are created. Electrons are
swept into the n-region and holes are swept into the p-region by the electric
field, generating a large reverse bias current. This phenomenon is called
breakdown. The reverse-bias current created by the breakdown mechanism is
limited only by the external circuit. If the current is not sufficiently limited, a
large power can be dissipated in the junctionthat may damage the device and
cause burnout. The current–voltage characteristic of a diode in breakdown is
shown in Figure 1.21.
26
Figure 1.21 Reverse-biased diode characteristics showing breakdown for a low-doped pn junction and a highdoped pn junction. The reverse-bias current increases rapidly once breakdown has occurred.
The most common breakdown
mechanism
is
called
avalanche
breakdown, which occurs when carriers
crossing the space charge region gain
sufficient kinetic energy from the high
electric field to be able to break covalent
bonds during a collision process. The
basic avalanche multiplication process is
demonstrated in Figure 1.22. The
generated
electron–hole
pairs
can
themselves be involved in a collision
process generating additional electron–
hole pairs, thus the avalanche process.
The breakdown voltage is a function of
the doping concentrations in then- and pregions of the junction. Larger doping
concentrations
result
in
smaller
breakdown voltages.
Figure 1.22 The avalanche multiplication process in the space
charge region. Shown are the collisions of electrons creating
additional electron–hole pairs. Holes can also be involved in
collisions creating additional electron–hole pairs.
A second breakdown mechanism is called Zener breakdown and is a
avoided.
result of tunneling of carriers across the junction. This effect is prominent at
very high doping concentrations and results in breakdown voltages less than
5V.The voltage at which breakdown occurs depends on fabrication parameters
of the pn junction, but is usually in the range of 50 to 200 V for discrete
devices, although breakdown voltages outside this range are possible—in
excess of 1000 V, for example. A pn junction is usually rated in terms of its
peak inverse voltage or PIV. The PIV of a diode must never be exceeded in
circuit operation if reverse breakdown is to be avoided. Diodes can be fabricated
with a specifically designed breakdown voltage and are designed to operate in
the breakdown region. These diodes are called Zener diodes.
27
Switching Transient
Since the pn junction diode can be
used as an electrical switch, an
important parameter is its transient
response, that is, its speed and
characteristics, as it is switched from
one state to the other. Assume, for
example, that the diode is switched
from the forward-bias “on” state to the
reverse-bias “off” state. Figure 1.23
shows a simple circuit that will switch
the applied voltage at time t = 0. For t <
0, the forward-bias current iD is
(1.19)
The
minority
carrier
concentrations
for
an
applied
forward-bias voltage and an applied
reverse-bias voltage are shown in
Figure 1.24. Here, we neglect the
change in the space charge region
width. When a forward-bias voltage
is applied, excess minority carrier
charge is stored in both the p- and n
regions. The excess charge is the
difference between the minority
carrier concentrations for a forwardbias voltage and those for a reversebias voltage as indicated in the
figure. This charge must be removed
when the diode is switched from the
forward to the reverse bias.
Figure 1.23 Simple circuit for switching a diode
from forward to reverse bias
Figure 1.24 Stored excess minority carrier charge under forward
bias compared to reverse bias. This charge must be removed as
the diode is switched from forward to reverse bias.
As the forward-bias voltage is removed, relatively large diffusion
currents are created in the reverse-bias direction. This happens because the
excess minority carrier electrons flow back across the junction into the nregion, and the excess minority carrier holes flow back across the junction into
the p-region.
The large reverse-bias current is initially limited by resistor RR to approximately
28
(1.20)
The junction capacitances do not allow
the
junction
voltage
to
change
instantaneously. The reverse current IR is
approximately constant for 0+ < t < ts ,
where ts is the storage time, which is the
length of time required for the minority
carrier concentrations at the space-charge
region edges to reach the thermal
equilibrium values. After this time, the
voltage across the junction begins to change.
The fall time tf is typically defined as the
time required for the current to fall to 10
percent of its initial value. The total turn-off
time is the sum of the storage time and the
fall time. Figure1.25 shows the current
characteristics as this entire process takes
place.
Figure 1.25 Current characteristics versus time during
diode switching
In order to switch a diode quickly, the diode must have a small excess
minority carrier lifetime, and we must be able to produce a large reverse
current pulse. Therefore, in the design of diode circuits, we must provide a
path for the transient reverse-bias current pulse. These same transient effects
impact the switching of transistors. For example, the switching speed of
transistors in digital circuits will affect the speed of computers.
The turn-on transient occurs when the diode is switched from the “off”
state to the forward-bias “on” state, which can be initiated by applying a
forward-bias current pulse. The transient turn-on time is the time required to
establish the forward-bias minority carrier distributions. During this time, the
voltage across the junction gradually increases toward its steady-state value.
Although the turn-on time for the pn junction diode is not zero, it is usually
less than the transient turn-off time.
1.1
Modeling the Diode forward characteristics
Having studied the diode terminal characteristics we are now ready to
consider the analysis of circuits employing forward-conducting diodes. Figure
1.26 shows such a circuit. It consists of a dc source VDD, a resistor R, and a
diode. We wish to analyze this circuit to determine the diode voltage VD and
current ID. We already know of two such models: the ideal diode model, and the
exponential model. In the following discussion we shall assess the suitability of
these two models in various analysis situations.
29
1.3.1 The Exponential Model
The most accurate description of the diode operation in the forward
region is provided by the exponential model. Unfortunately, however, its
severely nonlinear nature makes this model the most difficult to use. To
illustrate, let ' s analyze the circuit in Fig.1 .26 using the exponential diode
model.
Assuming that VDD is greater than 0.5 V
or so, the diode current will be much greater
than Is, and we can represent the diode i-v
characteristic by the exponential relationship,
resulting in
(1.21)
The other equation that governs circuit
FIGURE 1.26 A simple circuit used to
illustrate the analysis of circuits in
which the diode is forward conducting.
operation is obtained by writing a Kirchhoff loop equation, resulting in
(1.22)
Assuming that the diode parameters Is and n are known, Eqs. (1.21) and (1.22)
are two equations in the two unknown quantities ID and VD. Two alternative
ways for obtaining the solution are graphical analysis and iterative analysis.
Graphical Analysis Using the Exponential Model
Graphical
analysis
is
performed
by
plotting
the
relationships of Eqs. (1.21) and
(1.22) on the i-v plane. The solution
can then be obtained as the
coordinates
of
the
point
of
intersection of the two graphs. A
sketch of the graphical construction
is shown in Fig. 1.27. The curve
represents the exponential diode
equation (Eq.1.21), and the straight
line represents Eq. (1.22). Such a
straight line is known as the l o a d
FIGURE 1.27 Graphical analysis of the circuit in Fig 1.26
using the exponential diode model.
line a name that will become more meaningful in later chapters. The load line
intersects the diode curve at point Q, which represents the operating point of
the circuit. Its coordinates give the values of ID and VD.
30
Iterative Analysis Using the Exponential Model
Equations (1.21) and (1.22) can be solved using a simple iterative procedure, as
illustrated in the following example.
Example 1.8
Determine the current ID and the diode voltage VD for the circuit in Fig. 1.26
with VDD=5V and R=1 K . Assume that the diode has a current of 1 mA at a
voltage of 0.7V and that its voltage drop changes by 0.1V for every decade
change in current.
To begin the iteration, we assume that VD = 0.7 V and use Eq. (1.22) to
determine the current
ID =
We then use the diode equation to obtain a better estimate for VD. This can be
done by employing Eq. as
V2-V1=23 η
For our case, 23nVT= 0.1 V. Thus,
Substituting V1 = 0.7 V, I1 = 1 mA, and I2 = 4.3 mA results in V2 = 0.763 V.
Thus the results of the first iteration are ID = 4.3 mA and VD = 0.763 V. The
second iteration proceeds in a similar manner:
[
]
= 0.762 V.
Thus the second iteration yields ID = 4.237 mA and VD = 0.762 V. Since these
values are not much different from the values obtained after the first iteration,
no further iterations are necessary, and the solution is ID = 4.237 mA and VD =
0.762 V.
1.3.2 The Need for Rapid Analysis
The iterative analysis procedure utilized in the example above is simple and
yields accurate results after two or three iterations. Nevertheless, there are
situations in which the effort and time required are still greater than can be
justified. Specifically, if one is doing a pencil-and paper design of a relatively
complex circuit, rapid circuit analysis is a necessity. Through quick analysis,
the designer is able to evaluate various possibilities before deciding on a
suitable circuit design. To speed up the analysis process one must be content
with less precise results. This, however, is seldom a problem, because the more
accurate analysis can be postponed until a final or almost-final design is
obtained. Accurate analysis of the almost-final design can be performed with
31
the aid of a computer circuit-analysis program such as SPICE (see Section 3.9).
The results of such an analysis can then be used to further refine or "fine tune"
the design. To speed up the analysis process, we must find simpler models for
the diode forward characteristic.
The Piecewise-Linear Model
The
analysis
can
be
greatly
simplified if we can find linear
relationships to describe the diode
terminal characteristics. An attempt
in this direction is illustrated in Fig.
1.28, where the exponential curve is
approximated by two straight lines,
line A with zero slope and line B with
a slope of 1/rD. It can be seen that
for the particular case shown in Fig.
1.28, over the current range of 0.1
mA to 10 mA the voltages predicted
by the straight-lines model shown
differ from those predicted by the
exponential model by less than 50
mV.
Fig. 1.28 Approximating the diode forward characteristics
with two straight lines: piecewise linear model
Obviously the choice of these two straight lines is not unique; one can
obtain a closer approximation by restricting the current range over which the
approximation is required.
The straight-lines (or piecewise-linear) model of Fig. 1.28 can be described
by
(1.23)
where VD0 is the intercept of line B on the voltage axis and rD is the inverse of
the slope of line B.
The piecewise-linear model described by Eqs. (1.23) can be represented by the
equivalent circuit shown in Fig. 3.13. Note that an ideal diode is included in
this model to constrain iD to flow in the forward direction only. This model is
also known as the battery-plus resistance model.
32
Fig. 1.29 piecewise linear model of the diode
forward characteristic and its equivalent circuit
Example 1.9
Repeat the problem in Example 3.4 utilizing the piecewise-linear model whose
parameters are given in Fig.1.28 (Vm = 0.65 V, rD = 20 Ω). Note that the
characteristics depicted in this figure are those of the diode described in
Example 1.8 (1 mA at 0.7 V and 0.1 V/decade).
Replacing the diode in the circuit of Fig. 1.26 with the equivalent circuit model
of Fig. 1.29 results in the circuit in Fig. 1.30, from which we can write for the
current ID,
where the model parameters VD0 and rD are seen
from Fig. 1.28 to be VD0 = 0.65 V and rD = 20 Ω.
Thus,
The diode voltage VD can now be computed:
FIGURE 1.30 The circuit of Fig.1.26
with the diode replaced with its
piecewise-linear model of Fig. 1.29.
.
The Constant-Voltage-Drop Model
An even simpler model of the diode forward characteristics can be
obtained if we use a vertical straight line to approximate the fast-rising part of
the exponential curve, as shown in Fig. 1.31. The resulting model simply says
that a forward-conducting diode exhibits a constant voltage drop VD. The value
of VD is usually taken to be 0.7 V. Note that for the particular diode whose
characteristics are depicted in Fig. 1.31, this model predicts the diode voltage
to within ±0.1 V over the current range of 0.1 mA to 10 mA. The constantvoltage drop model can be represented by the equivalent circuit shown in Fig.
1.32. The constant-voltage-drop model is the one most frequently employed in
the initial phases of analysis and design. This is especially true if at these
33
stages one does not have detailed information about the diode characteristics,
which is often the case. Finally, note that if w e employ the constant-voltagedrop model to solve the problem in Examples 1.8 and 1.9, w e obtain
VD= 0.7 V and
FIGURE 1.31 Development of the constant
voltage drop model of the diode forward
characteristics. A vertical straight line (B) is used
to approximate the fast-rising exponential.
Observe that this simple model predicts VD to
within +0.1V over the current range of 0.1 mA to
10 mA.
FIGURE 1.32.constant-voltage-drop model of the diode forward
characteristics and its equivalent circuit representation
The Ideal-Diode Model
In applications that involve voltages much greater than the diode
voltage drop (0.6 - 0.8V), we may neglect the diode voltage drop altogether while
calculating the diode current. For the circuit in 1.8 and 1.9(i.e., Fig1.26 with
VDD - 5 V and R = 1 kΩ), utilization of the ideal-diode model leads to
which for a very quick analysis would not be bad as a gross estimate. However,
with almost no additional work, the 0.7-V-drop model yields much more
realistic results. We note, however, that the greatest utility of the ideal-diode
model is in determining which diodes are on and which are off in a multi diode
circuit.
The Small-Signal Model
There are applications in which a diode is biased to operate at a point
on the forward i-v characteristic and a small ac signal is superimposed on the
dc quantities. For this situation, we first have to determine the dc operating
point (VD and ID) of the diode using one of the models discussed above. Most
34
frequently, the 0.7-V-drop model is utilized. Then, for small-signal operation
around the dc bias point, the diode is best modeled by a resistance equal to the
inverse of the slope of the tangent to the exponential i-v characteristic at the
bias point. In the following, we develop such a small-signal model for the
junction diode and illustrate its application.
FIGURE 1.33 Development of the diode small-signal model. Note that
the numerical values shown are for a diode with n = 2.
Consider the conceptual circuit in Fig. 1.33(a) and the corresponding
graphical representation in Fig. 1.33(b). A dc voltage VD, represented by a
battery, is applied to the diode, and a time-varying signal vd(t), assumed
(arbitrarily) to have a triangular waveform, is superimposed on the dc voltage
VD. In the absence of the signal vd(t) the diode voltage is equal to VD, and
correspondingly, the diode will conduct a dc current ID given by
(1.24)
When the signal vd(t) is applied, the total instantaneous diode voltage vD(t) will
be given
(1.25)
Correspondingly, the total instantaneous diode current iD(t) will be
(1.26)
Substituting for vD from Eq. (1.25) gives
which can be written as
Using Eq. (1.24) we obtain
(1.27)
35
Now if the amplitude of the signal vd(t) is kept sufficiently small such that
(1.28)
then we may expand the exponential of Eq. (1.27) in a series and truncate the
series after the first two terms to obtain the approximate expression
(1.29)
This is the small-signal approximation. It is valid for signals whose
amplitudes are smaller than about 10 mV for the case n = 2 and 5 mV for n =1
(see Eq. 1.28 and recall that VT = 25 mV).
From Eq. (1.29) we have
(1.30)
Thus, superimposed on the dc current ID, we have a signal current component
directly proportional to t he signal voltage vd. That is,
(1.31)
Where
(1.32)
The quantity relating the signal current id to the signal voltage vd has the
dimensions of conductance, mhos, and is called the diode small-signal
conductance. The inverse of this parameter is the diode small-signal
resistance, or incremental resistance, rd,
(1.33)
Note that the value of rd is inversely proportional to the bias current ID.
Let us return to the graphical representation in Fig. 1.33(b). It is easy
to see that using the small-signal approximation is equivalent to assuming that
the signal amplitude is sufficiently small such that the excursion along the i-v
curve is limited to a short almost-linear segment. The slope of this segment,
which is equal to the slope of the tangent to the i-v curve at the operating point
Q, is equal to the small-signal conductance. The reader is encouraged to prove
that the slope of the i-v curve at i = ID is equal to ID/nVT, which is 1/rd; that is,
iD=ID
(1.34)
From the preceding we conclude that superimposed on the quantities VD
and ID that define the dc bias point, or quiescent point, of the diode will be the
small-signal quantities vd(t) and id(t), which are related by the diode smallsignal resistance rd evaluated at the bias point (Eq. 1.33). Thus the smallsignal analysis can be performed separately from the dc bias analysis, a great
36
convenience that results from the linearization of the diode characteristics
inherent in the small-signal approximation. Specifically, after the dc analysis is
performed, the small-signal equivalent circuit is obtained by eliminating all dc
sources (i.e., short circuiting dc voltage sources and open-circuiting dc current
sources) and replacing the diode by its small-signal resistance.
Example 1.10
Consider the circuit shown in Fig. 3.18(a) for the case in which R = 10 kΩ. The
power supply V+ has dc value of 10V on which is superimposed a 60-Hz
sinusoid of 1-V peak amplitude. (This “signal" component of the power-supply
voltage is an imperfection in the power-supply design. Ii is known as the
power-supply ripple. Calculate both the dc voltage of the diode and the
amplitude of the sine-wave signal appearing across it. Assume the diode to
have a 0.7V drop at 1 mA current and n = 2.
Considering dc quantities only, we assume VD = 0.7 V
and calculate the diode dc current
Since this value is very close to1mA, the diode
voltage will be very close to the assumed value of 0.7
V.
At this operating point, the diode incremental
resistance rd is
.
Figure 1.34 (a) Circuit for Example 1.34 (b) Circuit
for calculating the dc operating point.
The signal voltage across the diode can be
found from the small-signal equivalent circuit in Fig
1.34(c). Here vs denotes the 60-Hz 1-V peak
sinusoidal component of V+, and vd is the
corresponding signal across the diode. Using the
voltage-divider rule provides the peak amplitude of vd
as follows:
Figure 1.34.(c)small-signal equivalent circuit.
Finally we note that since this value is quite small, our use of the smallsignal model of the diode is justified.
37