Chapter 3 – Diodes
Introduction
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Introduction
The Ideal Diode
The ideal diode: (a) diode circuit symbol; (b) i-v characteristic; (c) equivalent circuit in the reverse direction; (d) equivalent circuit in
the forward direction.
Rectifier Circuit
Input waveform.
Rectifier circuit
Equivalent circuit when v1 > 0
Equivalent circuit when v1  0
Output waveform.
Rectifier Circuit
Example 3.1
Rectifier Circuit
Example 3.2
Terminal Characteristics of Junction Diodes – Forward Region
Example 3.3
Terminal Characteristics of Junction Diodes – Reverse-Bias Region
Exercise 3.9
Rectifier Circuit
Exercises 3.4 and 3.5
Diode – i-v Characteristic
The i-v characteristic of a silicon
junction diode.
Diode – i-v Characteristic
 v

n VT


IS  e
 1
i
VT
k T
Thermal Voltage
q
25mV at room temp.
for i>> Is
i
v
 v 
n VT


IS  e

n  VT ln 


IS 
 
i
e is the base for the natural log
The diode i-v relationship with some scales expanded and others compressed in order to reveal details.
ln = 2.3 log
Diode – i-v Characteristic
Exercise 3.6
Consider a silicon diode with n=1.5. Find the change in voltage if current changes
from 0.1 mA to 10 mA.
v
i
IS e
n VT
v1
I1
IS e
n  1.5
I1  0.0001
n VT
v2
IS e
n VT
I2  0.01
( v1 v2)
I1
I2
VT  0.025
I2
e
n VT
v1_v2  n  VT  ln 
I1 

I2
 
v1_v2  0.173
V
Diode – i-v Characteristic
A diode for which the forward voltage drop is 0.7 V at 1 mA and for which n=1 is
operated at 0.5 V. What is the value of the current?
v
i
IS e
n VT
0.7
0.001
IS e
1 0.025
0.5
I
IS e
1 0.025
 0.2
I  0.001 e
I  0.335
0.025
uA
6
 10
Diode – Simplified Physical Structure
Simplified physical structure of the junction diode.
(Actual geometries are given on Appendix A.)
Diode – Semiconductor Physics
The semiconductor diode is what is called a pn junction and is shown in the figure on the right
Both the p and the n sections are part of the same crystal of silicon. At room temp., some
of the covalent bonds in silicon break and electrons are attracted to other atoms. These moving
electrons leave a hole behind that is filled by another electron, thus continuing the cycle. In
thermal equilibrium the concentration of holes (p) and the concentration of free electrons (n) are
equal to each other and to ni which is the number of holes or free electrons in silicon at a given
temp. Study of semiconductor physics yields the following equation for the free electrons.
In this eqaution, B is a material dependant parameter (5.4x10^31) for silicon, E g
is a parameter known as the bandgap energy (1.12electron volts (eV) for silicon),
and k is the Boltsmann's constant (8.62x10^-5 eV/K). Thus, at room temp
(300K), the number of holes or free electrons is 1.5x10^22.
Diode – Semiconductor Physics
The two methods by which holes and electrons move are called diffusion and drift. In diffusion, the flow
moves from areas of higher concentration of p or n to areas of lower concentration. This flow gives rise to
a flow of charge or diffusion current. This is given by the following
Here, Jp is the current density (in Amps/cm^2), q is the magnitude of electron
charge (1.6x10^-19C) and Dp is called the diffusivity of the holes. the differential
dP is the rate of change for hole concentration.
Jp
d
q  D p  P
dx
Jn
Here, Jn is also the current density. The only difference is the differential dn.
d
q  D n  n Here, it stands for the rate of change of free electron concentration.
dx
Diode – Semiconductor Physics
The other method is called drift. Free holes and e- are moved by an electric field (E) and have a velocity
pE
v drift
p is the mobility of holes and has the units of (cm^2)/V*s.
This method also gives rise to diffusion currents.
q  p  p  E
Jp_drift
Jn_drift
q  n  n  E
The total drift current is written as follows.
Jdrift


q pp  nn E
Diode – Semiconductor Physics
A relationship, known as the Einstein relationship can be noted below.
Dn
Dp
n
p
The Vt term is what is called the thermal voltage. At room temperature, the
thermal voltage is approximately equal to 25mV
VT
Silicon is often doped to give it better conductivity. Doping is the process by which impurity atoms are
added to the silicon to provide more holes (p-type) or more free electrons (n-type). One thing to note is
the way the various terms are defined. For n-type silicon, the hole concentration is n p and the electron
concentration is nn . The concentration of donor (n-type) atoms or accpetor (p-type) atoms is denoted N d
and NA respectively. Study of doped silicon revealed the following.
2
Pn
ni
ND
nn
Nd
pp
NA
2
np
ni
NA
Diode – Semiconductor Physics
The pn junction under open-circuit conditions
When a diode (like the one shown above) is existing in a open circuit form two currents will naturally
occur.
Because the concentration of holes is high in the p-region and low in the n-region, holes will diffuse
from the p-region to the n-region. Likewise, free electrons will diffuse from the n-region to the p-region.
These two components add together to form the diffusion current Id whose direction is from the p-side
to the n-side. When the free electrons diffuse to the p-side, many are paired up with holes very close
to the border of the two regions thus depleting the number of free electrons. This uncovers a positive
charge at the edge of the n-region. On the other side, due to the same (but exact opposite) situation, a
negative charge is uncovered at the boundary of the p-region. This results in what is called the
carrier-depletion region (shown in the figure below) because the two extremes create a potential
difference to result. This is in the form of a barrier that holes and electrons must cross in order to
create a current. Thus, it can be seen that the current ID depends on this barrier voltage (which is
denoted V 0 ).
Diode – Semiconductor Physics
The voltage across this region can be found by the following equation:
Vo
 NA  ND 
VT 
 ni2 


In this equation, Na, Nd, and Vt are the same as they are previously
defined above. In open circuit conditions, V0 is 0 because the voltages
existing at the metal contacts of the diode (in the diagram above)
counteract the voltage at the barrier. If this were not so, energy could be
drawn from the isolated diode which is clearly incorrect.
In addition to the diffusion current, there is also a drift current. When some of the thermally generated
holes (holes created by a temp. increase which releases some outer electrons from their bonds) reach
the edge of the depletion region, they are swept accross the area because of the electric field present
in that region. This also happens to the free electrons. The addition of these two currents is the drift
current Is which flows in the opposite direction of ID. Under open circuit conditions, no external current
exists and the two currents are equal to each other. Under these conditions, if one current for some
reason is not equal to the other, they will shift and change until the equilibrium is once again attained.
Diode – Semiconductor Physics
The carrier depletion region has a width and if the doping was equal for both the n and p regions, the
depletion region would be symetrical. However, this is not the case so the widths will be different on
either side. Because the depletion region has a balanced amount of charge, it will have to extend
deeper into the lighter doped region so that the holes and electrons will all have a match. If the width in
the p side is denoted x p and the width in the n side is x n , the charge-eqaulity condition is this:
q  xp  A  NA
xn
NA
xp
ND
q  xn  A  ND
Here, A is the cross-sectional area of the junction.
This equation can be rewritten to yield the following, more clear
equality.
In actual practice, one side is usually more heavily doped than the
other causing the depletion region to exist almost entirely on one
side (the lightly doped side).
The width of the depletion region can be given (based on the above assumption that it exists
mainly on one side) by the following equation.
W depletion_region
xn  xp
 1  1 V

 o
q  NA
ND 
2s
In this equation, s is the electrical permittivity of silicon (1.04x10^-12 Farrads/cm). The width of
the region is usually 0.1 to 1 m.
Diode –Physical Structure
The pn Junction under reverse-bias conditions
To best describe the pn junction diode under reverse bias conditions, the diode is modelled to
have a current source exciting it. This current is kept lower than I s to keep the diode from
experiencing breakdown. The current (I) will be carried by electrons that move (in the opposite
direction of I) from the n-region to the p-region. This process creates a voltage (denoted V r) for
which the voltage tries to flow into the n-region of the diode. This voltage is, like the current,
less than the breakdown voltage (the voltage at which current will flow in the negative direction).
Because the positive end of the voltage is trying to enter the diode through the positive wall of
the barrier, the voltage actually increases the barrier's size. No current, therefore, is allowed to
pass through the diode. From this reasoning, it is relatively easy to see that as the current (and
thus Vr) changes, the charge in the depletion region is going to change. In this way, the diode
has some capacitance. The charge in the depletion region q j can be found by finding the charge
in either of the two regions (because the charge is equal in both). Using the n-side, the following
equation is derived.
qj
qn
qj
q
q  ND  xn  A
NA  ND
NA  ND
For this equation, A is the cross-sectional area of the junction.
Using the above eq. for the width of the depletion region, the
following can be written
 A  W depletion_region
Here, Wdep is slightly different from the above
equation and can be written as follows.
Diode –Physical Structure
2s 1
1 
This
adjustment is made
W depletion_region
xn  xpbecause

Vo  VR


ND 
it is no longer an open circuit. q  NA
There is now an external voltage source acting on the diode so the voltage is the addition of the
open circuit voltage and the reverse voltage. The capacitance can then be written as the
derivative of the charge in the depletion region. WIth a little bit of algebra and the combining of
previous equations, the value for the capacitance (C j ) can be found to be as follows:
Cj0
Cj
VR 

1  V 
o

Cj0
A
m
Here, Cjo is the value of the capacitance of the open circuit diode
and is defined below. m is a value that depends on the manner in
which the concentration changes from the p to the n side of the
juncion. It is called the grading coefficient (for the case of the
diode we are using, it is .5 but it ranges from .5 to about .333)
q  s  NA  ND
  1  This is the value of the open circuit capacitance. This is

 
2  NA  ND   Vo  readily made apparant by the fact that it does not depend
on Vr.
Diode –Physical Structure
The pn Junction in the Breakdown Region
As was explained earlier, the voltage for reverse bias conditions cannot exceed the breakdown voltage.
If it does, the barrier will no longer hinder the flow of current. As the voltage, in reverse, accross the
diode increases, the voltage across the depletion layer also increases. When this voltage is sufficiently
high enough, one of two mechanisms comes into play that allows current to flow through the diode.
The first of these mechanisms is called the zener effect. If the breakdown voltage of the diode is less
than 5V, the zener effect is usually what happens. Zener breakdown occurs when the electric field in
the depletion layer increases to the point where it can break covalent bonds and generate electron-hole
pairs. These holes and pairs are then swept to their respective sides causing a reverse current to form
which supports the outside current source. While this happens, the voltage across the diode will
remain relatively close to the breakdown voltage while the current will be largely determined by the
outside source.
The other mechanism is called the avalanche effect. This usually happens when the breakdown
voltage is greater than 7V. In this effect, the voltage or current basically forces its way through the
barrier. The covalent bonds are broken by the incoming voltage/current and as the first few bonds
break, more carriers are freed up to break more bonds causing an avalanche of current to flow. This
causes large current changes for small voltage changes.
A couple of things to note:
1. In the 5V to 7V range, the breakdown could be either zener or avalanche or a combination fo the
two.
2. pn junction breakdown is not a destructive process. It only causes problems when the maximum
dissapated power is exceeded. This max. value, in turn, implies a max. value for the reverse current.
Diode –Physical Structure
The pn Junction Under Forward-Bias Conditions
When a diode is under forward bias condtions, an external voltage or current source is applied with the
positive side of the voltage entering the p-side of the diode. When this happens, the electrons from the
incoming current (which enter at the n-side) and the holes from the positive voltage (entering from the
p-side) will nuetralize and diminish the depletion region barrier. A result of this is that the concentration
of minority carriers at the edge of the depletion region (denoted pn(xn) is related to the forward voltage
V by the following equation.
V
p n xn
p n0  e
VT
In this equation, pn0 is the concentration of minority carriers when no external
source is added. VT is the thermal voltage (ususlly 25mV). This is known as
the law of the junction.
The concentration of excess holes in the n-region can be expressed as the following.


 x xn
p n( x)
p n0   p n xn  p n0  e
Lp
Here, Lp is a canstant that determines the steepness of
the exponential decay of excess holes. It is called the
diffusion length of holes in the n-type silicon. The smaller
the value of Lp, the faster the injected holes will
recombined with the majority electrons.
The constant Lp is also related to another parameter called the excess-minority-carrier lifetime p.
This is the average time it takes for an injected hole to recombine. Lp is also related to the diffusion
constant Dp.
Lp
Dp  p
Typical values for Lp range from 1 to 100um and p usually ranges from 1 to
10,000ns.
Diode –Physical Structure
The holes diffusing in the n-region will give rise to a holes current. The density (which is greatest at the
edge of the depletion region - x = xn) is given by the following.
Jp
  V 1
  VT 
Dp
q
 p n0  e

Lp
In a similar manner, the analysis can extendd to the electrons injected into the p-region
Jn
  V 1
  VT 
Dn

q
 n p0  e

Ln is the diffusion length of the electrons in the p-region.
Ln
Because both of the densities are in the same direction, the total current (I) can be found. Substituting
for pn0 = ni^2/ND and similarly for np0, the current can be expressed as follows.
 V 1 
Dn   VT 
2  Dp
I A  q  ni  


 e
L

N
L

N
p
D
n
A


Minority-carrier distribution in a forward-biased pn junction. It
is assumed that the p region is more heavily doped than the n
region; NA  ND.
Diode –Physical Structure
Thus, the diode saturation current is defined to be
IS
2

Dp
A  q  ni  
 Lp  ND



Ln  NA 
Dn
Because of the excess carriers in forward bias, when the voltage is changed, the charge of the diode
will have to change to achieve steady state. This causes a form of charge storing in the depletion
region. This charge can be calculated by adding up the charge in the p and n regions. This charge
turns out to be as follows.
Q
p  Ip  n  In
Q T  I
Because it was earlier defined that Ip + In is equal to I, the equation can be rewritten as
where T is related to n and p and is called the mean tranit time. With this, it can be shown that
the capacitance is defined as follows.
Cd
 T 
 I
 VT 
As can be seen, the diffusion capacitance is directly proportional to the current.
This means that the capacitance for reverse bias is almost negligably small.
The capacitance over the depletion layer under forward bias is written as Cj
2  Cj0
This eqation is actually a fairly poor model so it is used as a rule of thumb rather than a solid fact.
Diode – Characteristic
Lessons In Electric Circuits copyright (C) 2000-2002 Tony R. Kuphaldt
Diode – Characteristic
Lessons In Electric Circuits copyright (C) 2000-2002 Tony R. Kuphaldt
Diode – Characteristic
Lessons In Electric Circuits copyright (C) 2000-2002 Tony R. Kuphaldt
Diode – Characteristic
Lessons In Electric Circuits copyright (C) 2000-2002 Tony R. Kuphaldt
Diode – Applications
Lessons In Electric Circuits copyright (C) 2000-2002 Tony R. Kuphaldt
Diode – Applications
Lessons In Electric Circuits copyright (C) 2000-2002 Tony R. Kuphaldt
Diode – Applications
Diode – Applications
Diode – Applications
Diode – Applications
Analysis of Diode Circuits
A simple diode circuit.
Graphical Analysis
v1
ID
ID
Graphical analysis of the circuit above
IS e
n VT
VDD  VD
R
Iterative Analysis
Example 3.4
Determine ID and VD for this circuit with VDD = 5 V and R1 = 1 K ohm.
Assume diode current 1 mA at voltage 0.7 V, and that its voltage drop changes by 0.1 V for
every decade change in current.
VDD  5
ID 
R1  1000
VDD  VD
VD  0.7
3
ID  4.3  10
1000
We then use the diode equation to obtain a better estimate for VD
V2  V1
2.3 n  VT  log 


I2 

I1 
V1  0.7 V
I2  4.3
mA
I1  1
mA
For our case 2.3.n.VT = 0.1 V (This results from the condition of 0.1 V change for every
decade change in current
V2  V1  0.1 log 


ID2 
I2 

I1 
5  0.763
V2  0.763
3
ID2  4.237  10
1000
V2  0.763  0.1 log 


ID2 
ID


V2  0.762
Simplified Diode Models
Approximating the diode forward characteristic with two straight lines.
Simplified Diode Models
Example 3.5
Piecewise-linear model of the diode forward characteristic and its equivalent circuit representation.
The Constant-Voltage Drop Model
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.
The Constant-Voltage Drop Model
The constant-voltage-drop model of the diode forward characteristic and its equivalent circuit representation.
The Small-Signal Model
Development of the diode small-signal model. Note that the numerical values shown are for a diode with n = 2.
The Small-Signal Model
Example 3.6
Equivalent circuit model for the diode for small changes around bias point Q. The incremental resistance rd is the inverse of the slope
of the tangent at Q, and VD0 is the intercept of the tangent on the vD axis.
The Small-Signal Model
The analysis of the circuit in (a), which contains both dc and signal quantities, can be performed by replacing the diode with the model
of previous figure, as shown in (b). This allows separating the dc analysis [the circuit in (c)] from the signal analysis [the circuit in
(d)].
Zener Diode - Characteristics
6.8 –V, 10mA
0.5W, 6.8-V, 70mA
Vz = Vzo + r2Iz
Vz > Vzo
Circuit symbol for a zener diode.
The diode i-v characteristic with the breakdown region shown in some detail.
Model for the zener diode.
Rectifier Circuits
Block diagram of a dc power supply.
Rectifier Circuits
v s  VDO
vo 0
vo
(a) Half-wave rectifier. (b) Equivalent
circuit of the half-wave rectifier with the
diode replaced with its battery-plusresistance model. (c) transfer
characteristic of the rectifier circuit. (d)
Input and output waveforms, assuming
that rD  R.
R
R  rD
rD  R
PIV Vs
 v s  VDO
R
R  rD
v o v s  VDO
v s  VDO
Rectifier Circuits
PIV
Full-wave rectifier utilizing a transformer with a center-tapped
secondary winding. (a) Circuit. (b) Transfer characteristic
assuming a constant-voltage-drop model for the diodes. (c) Input
and output waveforms.
2 V s  V DO
Rectifier Circuits
The bridge rectifier: (a) circuit and (b) input and output
waveforms.
PIV
Vs  VDO
Rectifier Circuits
With A Filter Capacitor
Voltage and current waveforms in the peak rectifier circuit with CR  T. The diode is assumed ideal.
Rectifier Circuits
With A Filter Capacitor
iL
iD
vo
R
iC  IL
d
C v I  iL
dt
Vp
IL
R
t
CR >> T
vo
Vp  e
C R
at the end of the discharge intervall
T
Vp  Vr
Vp  e
C R
T
SInce CR >> T
Vr
Vp 
and
T
CR
Vr
Vp
f  C R
e
CR
1
T
CR
Vr << Vp
Rectifier Circuits
With A Filter Capacitor
Vp  cos  t 
Vp  Vr
for small angles (t)
cos  t 
Vp  
1 

1
1
2
1
2
  t 
1
2
Vp    t 
2
  t 
2


2
Vp  Vr
Vr
t
2
Vr
Vp
to determine the average diode current during
conduction we equate the charge that the diode
supplies the capacitor
iCav t
Qsup plied
to the charge the capacitor losses during the discha
Qlost
iDav
iDmax
C Vr

IL  1   2




Vr 
Vp
IL  1  2  2



Vr 
Vp
Rectifier Circuits
With A Filter Capacitor
Vp  100
R  10000
Vr  5
C 
If Vp = 100 V
R = 10 K
Calculate the value of the
capacitance C that will result
in a peak-to-peak ripple
Vr of 5 V, the conduction
angle and the average and
peak values of the diode
current.
IL 
Vp
Vp
IL  0.01
R
mA
5
C  3.333  10
Vr f  R
t 
f  60
2
Vr
Vp
t  0.316
rad
Vp 

iDav  IL  1   2

V
r 

iDav  0.209
Vp 

iDmax  IL  1  2  2

V
r 

iDmax  0.407
The Spice Diode Model and Simulation Examples
The dc characteristics of the diode are determined by the parameters IS and N. An ohmic resistance, RS, is included.
Charge storage effects are modeled by a transit time, TT, and a nonlinear depletion layer capacitance which is
determined by the parameters CJO, VJ, and M. The temperature dependence of the saturation current is defined by the
parameters EG, the energy and XTI, the saturation current temperature exponent. Reverse breakdown is modeled by an
exponential increase in the reverse diode current and is determined by the parameters BV and IBV (both of which are
positive numbers).
name
parameter
units
default example
----
---------
1
IS
saturation current
A
1.0E-14 1.0E-14 *
2
RS
*
ohmic resistance
Ohm
0
10
3
N
emission coefficient
-
1
1.0
4
TT
transit-time
sec
0
0.1Ns
5
CJO
zero-bias junction capacitance
F
0
2PF
6
VJ
junction potential
V
1
0.6
7
M
grading coefficient
-
0.5
0.5
8
EG
activation energy
eV
1.11
1.11 Si
-----
-------
0.69 Sbd
0.67 Ge
*
The Spice Diode Model and Simulation Examples
The dc characteristics of the diode are determined by the parameters IS and N. An ohmic resistance, RS, is included.
Charge storage effects are modeled by a transit time, TT, and a nonlinear depletion layer capacitance which is
determined by the parameters CJO, VJ, and M. The temperature dependence of the saturation current is defined by the
parameters EG, the energy and XTI, the saturation current temperature exponent. Reverse breakdown is modeled by an
exponential increase in the reverse diode current and is determined by the parameters BV and IBV (both of which are
positive numbers).
name
parameter
----
---------
XTI
saturation-current temp. exp
units
default example
a
-----
------- --
3.0
3.0 jn
----9
-
2.0 Sbd
10
KF
flicker noise coefficient
-
0
11
AF
flicker noise exponent
-
1
12
FC
coefficient for forward-bias
-
0.5
depletion capacitance formula
13
BV
reverse breakdown voltage V
14
IBV
current at breakdown voltage
infinite
A
1.0E-3
40.0
The Spice Diode Model and Simulation Examples
PN Junction Diodes
Name
Parameter
Units
Default
IS
saturation current
A
1.0E-14
N
emission coefficient
-
1
BV
reverse breakdown voltage
V
infinite
RS
diode series resistance

0
CJO
zero-bias junction capacitance
F
0
VJ
junction potential V
1
M
grading coefficient
-
0.5
Limiting and Clamping Circuits
A variety of basic limiting circuits.