Equation (9.6), a curve of
v
versus
ln( )
i
will have a slope given by
1
nV
T
and y-intercept of
ln( )
I
S
. The following example illustrates how to find
n
and
I
S
from an experimental data. Since the example requires curve fitting,
the MATLAB function polyfit will be covered before doing the example.
9.1.2 MATLAB function polyfit
The polyfit function is used to compute the best fit of a set of data points to a
polynomial with a specified degree. The general form of the function is
coeff xy polyfit x y n
_ ( , , )
= (9.7)
where
x
and
y
are the data points.
n
is the
n
th
degree polynomial that will fit the vectors
x
and
y
.
coeff xy
_
is a polynomial that fits the data in vector
y
to
x
in the
least square sense.
coeff xy
_
returns n+1 coeffi-
cients in descending powers of
x
.
Thus, if the polynomial fit to data in vectors
x
and
y
is given as
coeff xy x c x c x c
n n
m
_ ( )
= + + +
−
1 2
1
The degree of the polynomial is n and the number of coefficients
m n
= +
1
and the coefficients
( , , , )
c c c
m
1 2
are returned by the MATLAB polyfit
function.
Example 9.1
A forward-biased diode has the following corresponding voltage and current.
Use MATLAB to determine the reverse saturation current,
I
S
and diode pa-
rameter
n
.
© 1999 CRC Press LLC© 1999 CRC Press LLC
0.1 0.133e-12
0.2 1.79e-12
0.3 24.02e-12
0.4 0.321e-9
0.5 4.31e-9
0.6 57.69e-9
0.7 7.726e-7
Solution
diary ex9_1.dat
% Diode parameters
vt = 25.67e-3;
v = [0.1 0.2 0.3 0.4 0.5 0.6 0.7];
i = [0.133e-12 1.79e-12 24.02e-12 321.66e-12 4.31e-9 57.69e-9
772.58e-9];
%
lni = log(i); % Natural log of current
% Coefficients of Best fit linear model is obtained
p_fit = polyfit(v,lni,1);
% linear equation is y = m*x + b
b = p_fit(2);
m = p_fit(1);
ifit = m*v + b;
% Calculate Is and n
Is = exp(b)
n = 1/(m*vt)
% Plot v versus ln(i), and best fit linear model
plot(v,ifit,'w', v, lni,'ow')
axis([0,0.8,-35,-10])
Forward Voltage, V Forward Current, A
© 1999 CRC Press LLC© 1999 CRC Press LLC
xlabel('Voltage (V)')
ylabel('ln(i)')
title('Best fit linear model')
diary
The results obtained from MATLAB are
Is = 9.9525e-015
n = 1.5009
Figure 9.3 shows the best fit linear model used to determine the reverse satura-
tion current,
I
S
,
and diode parameter,
n
.
Figure 9.3 Best Fit Linear Model of Voltage versus Natural
Logarithm of Current
© 1999 CRC Press LLC© 1999 CRC Press LLC
9.1.3 Temperature effects
From the diode equation (9.1), the thermal voltage and the reverse saturation
current are temperature dependent. The thermal voltage is directly propor-
tional to temperature. This is expressed in Equation (9.2). The reverse satura-
tion current
I
S
increases approximately 7.2% /
o
C for both silicon and germa-
nium diodes. The expression for the reverse saturation current as a function of
temperature is
I T I T e
S S
k T T
S
( ) ( )
[ ( )]
2 1
2 1
=
−
(9.8)
where
k
S
= 0.072 /
o
C.
T
1
and
T
2
are two different temperatures.
Since
e
072.
is approximately equal to 2, Equation (9.8) can be simplified and
rewritten as
I T I T
S S
T T
( ) ( )
( )/
2 1
10
2
2 1
=
−
(9.9)
Example 9.2
The saturation current of a diode at 25
o
C is 10
-12
A. Assuming that the
emission constant of the diode is 1.9, (a) Plot the i-v characteristic of the di-
ode at the following temperatures:
T
1
= 0
o
C,
T
2
= 100
o
C.
Solution
MATLAB Script
% Temperature effects on diode characteristics
%
k = 1.38e-23; q = 1.6e-19;
t1 = 273 + 0;
t2 = 273 + 100;
ls1 = 1.0e-12;
ks = 0.072;
ls2 = ls1*exp(ks*(t2-t1));
v = 0.45:0.01:0.7;
© 1999 CRC Press LLC© 1999 CRC Press LLC
l1 = ls1*exp(q*v/(k*t1));
l2 = ls2*exp(q*v/(k*t2));
plot(v,l1,'wo',v,l2,'w+')
axis([0.45,0.75,0,10])
title('Diode I-V Curve at two Temperatures')
xlabel('Voltage (V)')
ylabel('Current (A)')
text(0.5,8,'o is for 100 degrees C')
text(0.5,7, '+ is for 0 degree C')
Figure 9.4 shows the temperature effects of the diode forward characteristics.
Figure 9.4 Temperature Effects on the Diode Forward
Characteristics
© 1999 CRC Press LLC© 1999 CRC Press LLC
9.2 ANALYSIS OF DIODE CIRCUITS
Figure 9.5 shows a diode circuit consisting of a dc source
V
DC
,
resistance
R
,
and a diode. We want to determine the diode current
I
D
and the diode volt-
age
V
D
.
V
DC
I
D
V
D
R
+
-
+
-
Figure 9.5 Basic Diode Circuit
Using Kirchoff Voltage Law, we can write the loadline equation
V RI V
DC D D
= + (9.10)
The diode current and voltage will be related by the diode equation
i I e
D S
vnV
D T
=
( / )
(9.11)
Equations (9.10) and (9.11) can be used to solve for the current
I
D
and volt-
age
V
D
.
There are several approaches for solving
I
D
and
V
D
.
In one approach,
Equations (9.10) and (9.11) are plotted and the intersection of the linear curve
of Equation (9.10) and the nonlinear curve of Equation (9.11) will be the op-
erating point of the diode. This is illustrated by the following example.
© 1999 CRC Press LLC© 1999 CRC Press LLC
Example 9.3
For the circuit shown in Figure 9.5,if
R
= 10 kΩ ,
V
DC
= 10V, and the
reverse saturation current of the diode is 10
-12
A and
n
= 2.0. (Assume a
temperature of 25
o
C.)
(a) Use MATLAB to plot the diode forward characteristic curve and the
loadline.
(b) From the plot estimate the operating point of the diode.
Solution
MATLAB Script
% Determination of operating point using
% graphical technique
%
% diode equation
k = 1.38e-23;q = 1.6e-19;
t1 = 273 + 25; vt = k*t1/q;
v1 = 0.25:0.05:1.1;
i1 = 1.0e-12*exp(v1/(2.0*vt));
% load line 10=(1.0e4)i2 + v2
vdc = 10;
r = 1.0e4;
v2 = 0:2:10;
i2 = (vdc - v2)/r;
% plot
plot(v1,i1,'w', v2,i2,'w')
axis([0,2, 0, 0.0015])
title('Graphical method - operating point')
xlabel('Voltage (V)')
ylabel('Current (A)')
text(0.4,1.05e-3,'Loadline')
text(1.08,0.3e-3,'Diode curve')
Figure 9.6 shows the intersection of the diode forward characteristics and the
loadline.
© 1999 CRC Press LLC© 1999 CRC Press LLC
Figure 9.6 Loadline and Diode Forward Characteristics
From Figure 9.6,the operating point of the diode is the intersection of the
loadline and the diode forward characteristic curve. The operating point is ap-
proximately
I
D
=
0 9.
mA
V
D
=
0 7.
V
The second approach for obtaining the diode current
I
D
and diode voltage
V
D
of Figure 9.5 is to use iteration. Assume that
( )
I V
D D
1 1
,
and
( )
I V
D D
2 2
,
are two corresponding points on the diode forward characteris-
tics. Then, from Equation (9.3), we have
i I e
D S
vnV
D T
1
1
=
( / )
(9.12)
i I e
D S
vnV
D T
2
2
=
( / )
(9.13)
© 1999 CRC Press LLC© 1999 CRC Press LLC
Dividing Equation (9.13) by (9.12), we have
I
I
e
D
D
V V nV
D D T
2
1
2 1
=
−
( / )
(9.14)
Simplifying Equation (9.14), we have
v v nV
I
I
D D T
D
D
2 1
2
1
= +
ln
(9.15)
Using iteration, Equation (9.15) and the loadline Equation (9.10) can be used
to obtain the operating point of the diode.
To show how the iterative technique is used, we assume that
I
D
1
= 1mA and
V
D
1
= 0.7 V. Using Equation (9.10),
I
D
2
is calculated by
I
V V
R
D
DC D
2
1
=
−
(9.16)
Using Equation (9.15),
V
D
2
is calculated by
V V nV
I
I
D D T
D
D
2 1
2
1
= +
ln
(9.17)
Using Equation (9.10),
I
D
3
is calculated by
I
V V
R
D
DC D
3
2
=
−
(9.18)
Using Equation (9.15) , V
D3
is calculated by
V V nV
I
I
D D T
D
D
3 1
3
1
= +
ln
(9.19)
Similarly,
I
D
4
and
V
D
4
are calculated by
© 1999 CRC Press LLC© 1999 CRC Press LLC
I
V V
R
D
DC D
4
3
=
−
(9.20)
V V nV
I
I
D D T
D
D
4 1
4
1
= +
ln( )
(9.21)
The iteration is stopped when
V
Dn
is approximately equal to
V
Dn
−
1
or
I
Dn
is approximately equal to
I
Dn
−
1
to the desired decimal points. The iteration
technique is particularly facilitated by using computers. Example 9.4 illus-
trates the use of MATLAB for doing the iteration technique.
Example 9.4
Redo Example 9.3 using the iterative technique. The iteration can be stopped
when the current and previous value of the diode voltage are different by
10
7
−
volts.
Solution
MATLAB Script
% Determination of diode operating point using
% iterative method
k = 1.38e-23;q = 1.6e-19;
t1 = 273 + 25; vt = k*t1/q;
vdc = 10;
r = 1.0e4;
n = 2;
id(1) = 1.0e-3; vd(1) = 0.7;
reltol = 1.0e-7;
i = 1;
vdiff = 1;
while vdiff > reltol
id(i+1) = (vdc - vd(i))/r;
vd(i+1) = vd(i) + n*vt*log(id(i+1)/id(i));
vdiff = abs(vd(i+1) - vd(i));
i = i+1;
end
k = 0:i-1;
% operating point of diode is (vdiode, idiode)
idiode = id(i)
© 1999 CRC Press LLC© 1999 CRC Press LLC
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