SECTION 3:
RESISTIVE CIRCUIT ANALYSIS II
MAE 2055 – Mechetronics I
2
K. Webb
I-V Characteristics
MAE 2055 – Mechetronics I
I-V Characteristics
3




I-V characteristics relate the terminal voltages and
currents for electronic circuit components
Plot terminal current as a function of terminal
voltage
Useful for two-terminal devices, and especially
useful for three-terminal devices, e.g. transistors
Commonly see parameterized plots of I-V
characteristics – e.g. I-V characteristic between two
terminals of a three-terminal device parameterized
by the voltage on the third terminal
K. Webb
MAE 2055 – Mechetronics I
I-V Characteristics
4


The I-V terminal characteristic of a network is a graphical
representation of the voltage across and the current into
the terminals of that network
A graphical answer to one of the following two questions:
Apply a known voltage.
- How much current flows into the
terminals?
K. Webb
Apply a known current.
- How much voltage appears across
the terminals?
MAE 2055 – Mechetronics I
I-V Characteristics – ideal sources
5

An ideal voltage source
supplies constant
voltage regardless of its
terminal current
K. Webb

An ideal current source
supplies constant
current regardless of its
terminal voltages
MAE 2055 – Mechetronics I
I-V Characteristics – resistors
6

Ohm’s Law gives
the I-V relationship
for a resistor
V
I
R

A line whose slope
is the inverse of the
resistance
K. Webb
slope = 1/R
MAE 2055 – Mechetronics I
I-V Characteristics – Example
7
This is in the slope-intercept form:
y  mx  b
The slope is 1/10 A/V, and the current-axis
intercept is -0.5A.
Apply KVL around the loop:
V  I 10  5V  0
Solving for I gives an
equation of a line – the I-V
characteristic:
I 
K. Webb
V
 0.5A
10
MAE 2055 – Mechetronics I
Open-Circuit Voltage/Short-Circuit Current
8
Open Circuit Voltage
• I-V characteristic intercepts
the voltage axis where
terminal current is zero
Open-circuit
voltage
• This is the voltage that would
appear with nothing
connected to the terminals
Short-Circuit Current
• I-V characteristic intercepts
the current axis where
terminal voltage is zero
Short-circuit
current
• This is the current that would
flow with the terminals shortcircuited
K. Webb
MAE 2055 – Mechetronics I
9
K. Webb
Linearity & Superposition
MAE 2055 – Mechetronics I
Linearity
10



In a linear system outputs are linear functions of
the inputs
yi  a1 x1  a2 x2  ...  an xn
Can think of a system as a function that operates
on inputs to produce outputs:
f (x1 )  y1 , f (x 2 )  y 2

A linear system will obey the following:
f (x1   x 2 )    f (x1 )    f (x 2 )    y1    y 2
K. Webb
MAE 2055 – Mechetronics I
Linearity
11




In a linear circuit, outputs may be any circuit
operating condition – node voltages and branch
currents
Inputs may be independent current and voltage
sources
Linear circuits are composed of linear circuit
elements
Components are linear if their I-V characteristics
are linear
K. Webb
MAE 2055 – Mechetronics I
Superposition
12


Consider a circuit with two independent sources
This is a linear circuit, so Vout is a linear function of the
inputs
Vout  a1Vs  a2 I s
where a1 and a2 are constants

The output, Vout , due to both sources is the sum of the outputs due
to each source taken one at a time – this is superposition

Simplifies determining the output of multiple-input linear circuits
and systems
K. Webb
MAE 2055 – Mechetronics I
Superposition
13
The output of a multiple-input system is the sum of
the outputs due to each source acting individually
 Determine the response of a circuit to each
independent source, one at a time, with all other
independent sources set to zero
 Sum the individual responses to get the response
due to all sources
 Setting sources to zero:
 Voltage
sources become short circuits (V = 0)
 Current sources become open circuits (I = 0)
K. Webb
MAE 2055 – Mechetronics I
Superposition – an example
14



Determine the value of Vout in the following circuit
Linear circuit – all components have linear I-V
characteristics
Two independents sources – use superposition
K. Webb
MAE 2055 – Mechetronics I
Superposition Example – step 1
15


Set the current source to zero – open circuit
Determine Vout due to Vs
With Is set to zero, the circuit
becomes a simple voltage divider
𝑉𝑜𝑢𝑡
𝑉𝑜𝑢𝑡
K. Webb
𝑉𝑠
𝑅3
= 𝑉𝑠 ∙
𝑅1 + 𝑅2 + 𝑅3
𝑉𝑠
2𝑘Ω
= 5𝑉 ∙
= 1.67𝑉
6𝑘Ω
MAE 2055 – Mechetronics I
Superposition Example – step 2
16


Set the voltage source to zero – short circuit
Determine Vout due to Is
With Vs set to zero, the circuit
becomes a simple current divider
𝑅1
𝐼3 = 𝐼𝑠 ∙
= 1.67𝑚𝐴
𝑅1 + 𝑅2 + 𝑅3
𝑉𝑜𝑢𝑡
𝑉𝑜𝑢𝑡
K. Webb
𝐼𝑠
𝐼𝑠
= 𝐼3 𝑅3
= 1.67𝑚𝐴 ∙ 2𝑘Ω = 3.33V
MAE 2055 – Mechetronics I
Superposition Example – step 3
17


The total response is the sum of the individual responses
Vout is the sum of Vout due to the voltage source and Vout
due to the current source
Sum the individual values for Vout to
get the total value for Vout
𝑉𝑜𝑢𝑡 = 𝑉𝑜𝑢𝑡
𝑉𝑠
+ 𝑉𝑜𝑢𝑡
𝐼𝑠
𝑉𝑜𝑢𝑡 = 1.67𝑉 + 3.33𝑉
𝑉𝑜𝑢𝑡 = 5𝑉
K. Webb
MAE 2055 – Mechetronics I
18
K. Webb
Thévenin & Norton Equivalents
MAE 2055 – Mechetronics I
Thévenin Equivalent Circuits
19



Any two-terminal linear
network of resistors and
sources can be represented
as single resistor in series
with a single independent
voltage source
The resistor is the Thévenin
equivalent resistance
The voltage source is the
open-circuit voltage
K. Webb
Léon Charles Thévenin, 1857 – 1926
MAE 2055 – Mechetronics I
Thévenin Equivalent Circuits
20


Useful for determining current, voltage, and power
delivered by any complex network to an arbitrary
load
Simplifies the analysis of complex networks
Complex network
K. Webb
Thévenin equivalent network
MAE 2055 – Mechetronics I
Open-Circuit Voltage - Voc
21


Voc, the open-circuit voltage, is the terminal
voltage with no load attached
Determine Voc by using most convenient method –
Ohm’s Law, Kirchhoff’s Laws, mesh or nodal
analysis, etc.
K. Webb
MAE 2055 – Mechetronics I
Thévenin Resistance - Rth
22

Rth, the Thévenin equivalent resistance, is the
resistance seen between the two terminals with all
sources set to zero
sources  short circuits
 Current sources  open circuits
 Voltage
K. Webb
MAE 2055 – Mechetronics I
Thévenin Equivalent – an example
23

Determine the load current and voltage for a 100 Ω
resistor connected to the following network
 Transform
to a Thévenin equivalent circuit, then
connect a 100 Ω load
 IL and VL are then easily determined using Ohm’s Law
K. Webb
MAE 2055 – Mechetronics I
Thévenin Example – find Voc
24

Two independent sources, so use superposition

𝑉𝑜𝑐
K. Webb
First, find Voc due to Vs

R1 is in parallel with a voltage
source, so it can be neglected

No current flows through R5
so it can be neglected

Circuit reduces to a simple
voltage divider
𝑉𝑠
500𝛺
= 10𝑉 ∙
= 5𝑉
1000𝛺
MAE 2055 – Mechetronics I
Thévenin Example – find Voc, cont’d
25

800Ω
𝐼2 = 10𝑚𝐴 ∙
= 8𝑚𝐴
1000Ω
200Ω
𝐼3 = 10𝑚𝐴 ∙
= 2𝑚𝐴
1000Ω
K. Webb
𝑉𝑜𝑐
𝐼𝑠
Next, find Voc due to Is

R1 gets shorted, so it can be
neglected

No current flows through R5
so it can be neglected

Circuit reduces to a simple
current divider
= −𝐼3 𝑅4 = −2𝑚𝐴 ∙ 500Ω = −1𝑉
𝑉𝑜𝑐 = 𝑉𝑜𝑐
𝑉𝑠
+ 𝑉𝑜𝑐
𝐼𝑠
= 5𝑉 − 1𝑉 = 4𝑉
MAE 2055 – Mechetronics I
Thévenin Example – find Rth
26

Set independent sources to zero, then determine resistance
between the two terminals


Voltages sources become short circuits (V = 0)
Current sources become open circuits (I = 0)



R1 gets shorted
R2 and R3 are in series
R4 in parallel with R2 plus R3
Rth  R5  R4 || R2  R3 
Rth  50  500 || 200  300
Rth  300
K. Webb
MAE 2055 – Mechetronics I
Thévenin Example – find IL and VL
27
Find the voltage across the load by
using the voltage divider equation
Thévenin equivalent circuit with
the 100 Ω load resistor connected
VL  Voc 
``
RL
100
 4V 
RL  Rth
400
VL  1V
Ohm’s Law gives the load current
IL 
VL
1V

RL 100
I L  10mA
K. Webb
MAE 2055 – Mechetronics I
Norton Equivalent Circuits
28



Any two-terminal linear
network of resistors and
independent sources can be
represented as single resistor
in parallel with a single
independent current source
The resistor is the Thévenin
equivalent resistance
The current source is the
short-circuit current
K. Webb
Edward Lawry Norton, 1898 – 1983
MAE 2055 – Mechetronics I
Norton Equivalent Circuits
29


An extension of Thévenin’s Theorem
Came about due to the development of vacuum
tubes, which are more appropriately modeled with
current sources
Complex network
K. Webb
Norton equivalent network
MAE 2055 – Mechetronics I
Short-Circuit Current- Isc
30
I , the short-circuit current, is the current that
flows between the short-circuited terminals
Determine Isc by shorting the output terminals,
then using most convenient method – Ohm’s Law,
Kirchhoff’s Laws, mesh or nodal analysis, etc.
 sc

K. Webb
MAE 2055 – Mechetronics I
Thévenin Resistance - Rth
31


Rth, is the same for a Norton equivalent circuit as
for a Thévenin equivalent circuit
The resistance seen between the two terminals
with all sources set to zero
K. Webb
MAE 2055 – Mechetronics I
Thévenin and Norton Equivalents
32

A Thévenin circuit can easily be converted to a
Norton Circuit and vice versa
Voc  I sc Rth
K. Webb
Voc
I sc 
Rth
MAE 2055 – Mechetronics I
33
K. Webb
Dependent Sources
MAE 2055 – Mechetronics I
Dependent Sources
34


Ideal current and voltage sources
Outputs depend on some circuit parameter – branch
current or node voltage
VCVS – voltage-controlled
voltage source
Output voltage is a
function of node voltages
elsewhere in the circuit
VCCS – voltage-controlled
current source
Output current is a
function of node voltages
elsewhere in the circuit
K. Webb
CCVS – current-controlled
voltage source
Output voltage is a function
of a branch current elsewhere
in the circuit
CCCS – voltage-controlled
current source
Output current is a function
of a branch current
elsewhere in the circuit
MAE 2055 – Mechetronics I
Dependent Sources
35



Schematic symbols may vary greatly
May look like an independent source, whose value is
written as a function of a circuit voltages or currents
Dependent source are useful for modeling complex active
devices, such as transistors



K. Webb
Current source is a
dependent source – CCCS
Its output current is the
value of the current into
terminal b, ib, times some
factor, β.
This is a simple model of a
bipolar transistor
MAE 2055 – Mechetronics I