İzmir University of Economics
EEE201 Electrical Circuits I Lab
EXPERIMENT 2:
Linear and Nonlinear Resistors
Objective:
The students will be able to learn

how to find v-i characteristics of linear and nonlinear resistors by dc voltage and current
measurements and

how to obtain a simple yet realistic model for a given resistor.
A. Pre-lab Tasks
•
•
•
Read Chapter 2 of the book by Dorf & Svoboda and also Chapter 2 of the book by L. O.
Chua, C. Desoer and E. Kuh.
Read the background material provided below.
Complete the pre-lab assignment. (You should submit it at the beginning of the lab
session.)
Pre-lab assignment:
1. Calculate the resistances of resistors which are given below and write down each steps
Resistance Value
Tolerance
red - green - brown – gold
red- black- red- red
Figure 1. Reading resistor value from color codes
2. Color the resistance color bands of the resistors given in Figure 2.
270 Ω (±5%)
47 k Ω (±20%)
Figure 2. Coloring resistors according to resistance values
3. Assume that R1=1.2 K and R2=2.4k with %10 tolerances for the circuit in Figure 3.
What would be the minimum value for the voltage difference across R1?
R1=1.2K
E=12V
R2=2.4K
Figure 3. The effect of resistor tolerances
4. For a resistor with 1 K (linear) resistance, plot the V-I characteristic on Figure 4 by
determining the current values I corresponding to the voltage values -5, -4, -3, -2, -1,
0, 1, 2, 3, 4, 5 (Volts). (Indicate the units of the axis!)
1-1
i [?]
-5 -4 -3 -2 -1
01 2 3 4 5
v [?]
Figure 4. v-i characteristic of a linear resistor with 1K resistance
5. Draw v-i characteristic of an ideal switch which corresponds to an open-circuit element
when is off and a short-circuit element when is on. Then,
o Explain why a switch is a resistive element.
o Classify the switch as

linear or nonlinear,

time-varying or time-invariant
6. Draw the v-i characteristic of the nonlinear resistor defined by I=Io(eV/VT-1). Where, e
symbol denotes the exponential function, the negative saturation current Io is 4x10-10
Ampers and the thermal voltage VT is 50 mili volts. Then, determine whether or not the
given resistor is linear and/or time-invariant.
References:





R. C. Dorf & J. A. Svoboda, Introduction to Electric Circuits, 8th Edition, Wiley, 2011.
L. O. Chua, C. Desoer and E. Kuh, Linear and Nonliear Circuits, McGraw Hill, 1987
P. M. Jansson, Networks-I, Lecture Slides on http://users.rowan.edu/~jansson/ (15:00,
27.09.2011)
Physics 111 Lab, University of California at Berkeley,
http://socrates.berkeley.edu/~phylabs/bsc/PDFFiles/bsc1.pdf (15:00, 27.09.2011)
Diodes and rectifiers, http://www.allaboutcircuits.com/vol_3/chpt_3/1.html (16:00,
3.10.2011)
B. Background
B.1. Definition and classification of resistive circuit elements
Resistors are circuit elements whose constitutive relations are algebraic equations. Their driving
point characteristics which can be determined by voltage and current measurements realized at
their terminals define V-I surfaces in the most general multi-terminal (and also multi port) case
and V-I curves in the 2-terminal case. Resistive circuit elements can classified as time-varying
or time-invariant and also linear or nonlinear depending on their constitutive relations.
A 2-terminal resistor is linear if and only if its constitutive relation f(v(t),i(t); t)=0 is defined by
a function f(.,.;t) which is linear in v and i. A 2-terminal resistor is time-invariant if and only if
its constitutive relation f(v(t),i(t); t)=0 does not depend on time t explicitly so constituting a
single v-i curve for all time t. In other words, for time-invariant resistors, f(v(t),i(t); t)=0 can be
rewritten as f(v(t), i(t))=0 meaning the constitutive relation f(v(t), i(t); t) does not depend on
the third variable t.
Checking linearity: Driving point characteristics of linear resistors satisfy both of the additivity
and multiplicativity properties whereas both (or one) of these properties fail(s) to be satisfied
for nonlinear resistors. These two properties, which are actually integrated as a single property,
called as superposition in engineering domain and linearity in mathematics, can be checked
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whether or not any linear combination [c1 v1 (t) + c2 v2 (t), c1 i1 (t) + c2 i2 (t)] of any two
different v-i measurements [v1 (t), i1 (t)] and [v2 (t), i2 (t)] (called also as admissible v(t)-i(t)
signal pairs) satisfies the constitutive relation of the resistor:
Do f(v1(t), i1(t);t)=0 & f(v2(t), i2(t);t)=0 imply f(c1 v1 (t) + c2 v2 (t), c1 i1 (t) + c2 i2 (t);t)=0 for
all real c1 and c2 and admissible v-i measurements [v1 (t), i1 (t)] and [v2 (t), i2 (t)] ?
Checking time-invariance: Time-invariance can also be checked in terms of the admissible
(v(t),i(t)) signal pairs as:
Does f(v(t), i(t);t)=0 imply f(v(t-τ), i(t- τ) ;t)=0 for all τ delays and admissible (v(t), i(t)) signal
pairs ?
B.2. Ohm’s law for linear time-invariant resistors
Georg Simon Ohm (1789-1854), a German scientist, discovered that there
is a linear relationship between the voltage over a kind of resistor and the
current through, i.e., v = R i as in Fig. 2.1. He published his work in the
book entitled 'Die galvanische Kette, mathematische bearbeitet' in 1827.
The SI unit of the electrical resistance of Linear Time-Invariant (LTI)
resistors, the Ohm, is named after him.
i(t)
+
v(t)
-
v(t) = R i(t)
R
Figure 5. Symbol of the LTI resistors
The Ohm’s law for 2-terminal LTI resistors may also be written as:
v(t)
=R
i(t)
or neglecting the time time dependence simply
v
i
= R.
The units of the quantities in Ohm’s law are given in Table 2.1.
Table 1
Quantity
Symbol
Unit
Unit Symbol
Current
i, I
Ampere
A
Voltage
v, V or e, E
Volt
V
Resistance
r, R
Ohm
Ω
For an electrical component, the graphics of voltage (v) as a function of current (i) is called v-i
charactestics of the component. For a 2-terminal LTI resistor, the v-i characteristics is a linear
curve passing through origin. The slope of the line is R, as shown in Figure 6.
1-3
v
i
+
v
-
slope=R
R
i
Figure 6. The v-i characterictics of a 2-terminal LTI resistor
To obtain the v-i characteristics of a resistor of 1 k Ω = 1 x 103 Ω, different values to i are
assigned and the values of corresponding voltages are measured (or calculated) as in Table 2.
Table 2
Current (A)
Voltage (V) (V = R I)
0
1 x 10
0
1 x 10
-3
-3
x 1 x 103 = 1
2 x 10-3
2 x 10-3 x 1 x 103 = 2
3 x 10-3
3 x 10-3 x 1 x 103 = 3
4 x 10-3
4 x 10-3 x 1 x 103 = 4
5 x 10-3
5 x 10-3 x 1 x 103 = 5
The values listed in Table 2 are plotted in Figure 7.
v, volts
5
4
3
2
1
i, 10-3 A
1 2 3 4 5
i, mA
Figure 7. The v-i characteristics of 1 k Ω LTI resistor
B.3. Color Codes for LTI Resistors
LTI resistors have been developed to meet different circuit specifications. For different power
dissipation ranges, resistors of different materials are produced as shown in Figure 8. When an
application requires a resistor of 1 Watt power rating, possibly a wire wound resistor shown in
Figure 8.a is chosen. Figure 8.b shows 1W metal film resistor. In Figure 8.c and 8.d, carbon
composition resistors of ½ W and ¼ W are shown. For low power applications on the printed
circuit boards, surface mount resistors may be used during production as shown in Figure 8.e.
When a circuit requires resistors of equal resistances, single-in-line package (SIP) resistors of
Figure 8.f may be used as shown in Figure 8.f.
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(b) 1W Metal Film Resistor
(a) 10 W Wire Wound Resitor
(d) ¼ W Carbon Composition Resistor
(c) ½ W Carbon Composition Resistor
Surface Mount
Resistor
(e) Surface Mount Resistors
(f) Single-in-Line Package (SIP) Resistors
Figure 8. LTI resistors of different type
The value of an LTI resistor is usually indicated by 4 color band over the resistor as shown in
Figure 9.a.
Gold (5%)
Silver (10%)
Tolerance
Multiplier
2nd Digit
1stDigit
(a)
(c)
No color
(20%)
(b)
Figure 9. LTI resistor color codes (4 band)
The first band indicates 1st digit, second band is for 2nd digit and the third band indicates the
multiplier. The numerical values corresponding to the color bands are listed in Figure 9.b.
Since the first color band is brown (1) and the second band is black (0), the first two digits
constitute 10. The multiplier color band being red (2) means the number of zeros added will be
2. So the value of the resistor given in Fig. 1.2(a) is determined as:
Color
Value
1st Digit
Brown
1
2nd Digit
Black
0
Multiplier
2
00
Code
Resistance
1000
1kΩ
Hence, the above LTI resistor has a value of 1 k Ω. The above LTI resistor has a tolerance color
of gold; which means error (tolerance) in the resistance value is ±5%. If the tolerance band
color is silver, than it means the tolerance in the resistance value is ±10%. No color means
±20% tolerance.
1-5
Gold (5%)
Silver (10%)
Tolerance
Multiplier
3rd Digit
(a)
(c)
No color
(20%)
2nd Digit
1st Digit
(b)
Figure 10. LTI resistor color codes (5 band)
The color band scheme for 5-color bands is given in Figure 10. For the resistor given in Figure
10.a, the first color band is brown (1), the second band is black (0), and the third band is black
(0), so the first three digits constitute 100. The multiplier color band being red (2) means the
number of zeros added will be 2. So, the value of the resistor given in Figure 10.a is determined
as:
Color
Value
1st Digit
Brown
1
2nd Digit
Black
0
Multiplier
2
00
Black
0
Code
Resistance
10000
10 k Ω
The above resistance has a tolerance color of gold; which means tolerance is ±5%.
The resistance values for each tolerance band are fixed. The nominal values for the first two
digits used in each tolerance band are listed in Table 3.
Table 3 Nominal Resistance Values
±5%
10
11
12
13
15
16
18
20
22
24
27
30
33
36
39
43
47
51
56
62
68
75
82
91
±10%
±20%
10
12
15
18
22
27
33
39
47
56
68
82
10
15
22
33
47
68
Therefore a resistance of value of 1.1 k can be found in 5% tolerance range. If such a resistor is
required and if ±10% resistors are available, the suitable choices may be 1k or 1.2k
resistances.
B.4. Diodes
Diodes are two terminal resistive elements which are made from semiconducting materials,
typically crystalline silicon. They are isolated p-n junctions. Where, p and n types regions are
obtained by doping the silicon with two different types of dopant. This process give an n-type
(doped) semiconductor with having more mobile electrons than holes and a p-type (doped)
semiconductor with more holes than mobile electrons.
The current passing through a p-n junction diode can be as an exponential function of the
voltage across its terminals: I=Io (eV/VT-1). Where, VT= nkT/e with n varies between 1 and 2
depending on the particular diode, typically equal to 2, e = 1.6 × 10-19 Coulomb is the charge of
an electron, k = 1.38 × 10-23 J/K is the Boltzmann’s constant and T is the temperature in
1-6
Kelvin. The constant Io is the reverse saturation current which depends on the temperature, the
particular geometry and the material used for the p-n junction.
Due to the exponential dependence of the current on the voltage, the current passing through
the diode is high when the voltage difference across the diode is positive. In the reverse biased
case (the voltage difference is negative), the current gets saturated at a very small negative
value –Io. So, the diodes can be said as one-way valves so that they enable the current only in
one direction. As shown in Figure 11, the positive terminal of the diode is called as the anode
and the negative terminal as the cathode.
Figure 11. p-n junction diode: a) Physical representation, b) Circuit symbol and c) Appearance of a real
diode (http://www.allaboutcircuits.com/vol_3/chpt_3/1.html)
C. In-Lab Experimental Work
C.1. Derivation of driving point characteristics of LTI resistors
Task C.1.1.: Build the below circuit on the protoboard available in Yıldırım Electronics Basic
Training Set Y-0016.
i
R=2.2K
V
+
V
-
Figure 12. Obtaining v-i characteristics of an LTI resistor
Using the variable power supply, apply the voltages given in Table 4 to the resistor,
respectively, and measure the current values for every step. Write your measurements onto the
Table 4. Calculate V/A ratio for every step and fill in write to table again. (Indicate the unit of
V/A!)
Table 4
v (Volts)
I (Ampere)
V
(………)
-5
-4
-3
-2
-1
0
1
2
3
4
5
I
Are the ratios invariant? Compare your finding to the nominal resistance value by using i) color
codes and ii) ohmmeter mode of the multi meter Fluke 177.
Average V/I ratio obtained by the v-i measurements : ……………………………..
Ohmmeter Result
: ……………………………..
Color Code Result
: ……………………………..
Task C.1.2.: Draw the v-i curve of this resistive element below, by using voltage-current
measurements in Table 4. If your measurements do not fit to a line in an exact way, then
propose a way for obtaining a line which best fits to your measurements.
1-7
I[]
-5 -4 -3 -2 -1
V[]
01 2 3 4 5
Figure 13. Driving point characteristics of the resistor
Task C.1.3.: What do you think about the linearity of the resistor?
Task C.1.4.: Now, build the below circuit on the protoboard to plot the v-i characteristics of a
1N4003 diode.
R
V
iD
1N4003
+
vD
-
Figure 14. Obtaining v-i characteristics of a diode
Using the variable power supply and the R resistor in series to the diode, apply the voltages
given in Table 5 to the diode, respectively, and measure the iD current values for every step.
Use R=1K resistor for the forward biased and 220 K for the backward biased. Write your
measurements onto the Table 5 and then draw the diode characteristics in Figure 15.
Table 5
v (Volts)
I (Amp.)
V
I
-20
-16
-12
-8
-4
0
0.2
0.4
0.5
0.6
0.7
0.8
1.0
1.4
1.8
2.0
(Ohm)
i[]
0
v[]
Figure 13. p-n junction diode characteristics
1-8
Note that the diode current values are about mili Ampers when it is forward biased and about
micro Ampers when it is reverse biased. So, the current values especially in the reverse biased
may not be measured due to the resolution of the digital multi meter. Find a way for measuring
small currents in an indirect way.
Task C.1.4.: Specify whether or not the diode having the v-i characteristics obtained in Figure
13 is linear and/or time-invariant?
D. Post-Lab Tasks: Lab Report
The lab report should have the information contained in the pre-lab assignment and also the
answers of the following questions. The lab report should be submitted at the beginning of the
succeeding lab session.



Explain
o What you have done in this lab.
o How do you measure the quantities related to the the experiment.
Reply the questions in the Tasks C.1.1-C.1.4.
Find the best line fitting to your measurements in Task C.1.2 as finding the slope of this
line by using minimizing the total fitting error.
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