Section 3 resistive circuit analysis ii

SECTION 3:RESISTIVE CIRCUIT ANALYSIS II

MAE 2055 – Mechetronics I

I-V Characteristics2

K. Webb MAE 2055 – Mechetronics I

I-V Characteristics

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


3

I-V Characteristics

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:


4

Apply a known voltage.- How much current flows into the terminals?

Apply a known current.- How much voltage appears across the terminals?

I-V Characteristics – ideal sources

An ideal voltage source supplies constant voltage regardless of its terminal current


5

An ideal current source supplies constant current regardless of its terminal voltages

I-V Characteristics – resistors

Ohm’s Law gives the I-V relationship for a resistor

A line whose slope is the inverse of the resistance


6

R

VI

slope = 1/R

I-V Characteristics – Example


7

5 010I VV Apply KVL around the loop:

Solving for I gives an equation of a line – the I-V characteristic:

100.5

VAI

y mx b This is in the slope-intercept form:

The slope is 1/10 A/V, and the current-axis intercept is -0.5A.

Open-Circuit Voltage/Short-Circuit Current


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Open Circuit Voltage•I-V characteristic intercepts the voltage axis where terminal current is zero•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•This is the current that would flow with the terminals short-circuited

Short-circuit current

Open-circuit voltage

Linearity & Superposition9


Linearity

In a linear system outputs are linear functions of the inputs

Can think of a system as a function that operates on inputs to produce outputs:

A linear system will obey the following:


10

nni xaxaxay ...2211

1 2 21( ) , ( )f x y f x y

2 1 2 11 2(( ) ) )(f x f x f xx y y

Linearity

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


11

Superposition

Consider a circuit with two independent sources This is a linear circuit, so Vout is a linear function of the

inputs


12

ssout IaVaV 21 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

Superposition

The output of a multiple-input system is the sum of the outputs due to each source acting individuallyDetermine the response of a circuit to each independent source, one at a time, with all other independent sources set to zeroSum the individual responses to get the response due to all sourcesSetting sources to zero:

Voltage sources become short circuits (V = 0) Current sources become open circuits (I = 0)


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Superposition – an example

Determine the value of Vout in the following circuit Linear circuit – all components have linear I-V

characteristics Two independents sources – use superposition


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Superposition Example – step 1

Set the current source to zero – open circuit Determine Vout due to Vs


15

321

30 RRR

RVV sIout

s

VK

KVV

sIout 67.16

25

0

With Is set to zero, the circuit becomes a simple voltage divider


Set the voltage source to zero – short circuit Determine Vout due to Is


16

mARRR

RII s 67.1

321

13

330RIV

sVout

With Vs set to zero, the circuit becomes a simple current divider

VKmAVsVout 33.3267.10


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


17

00

ss VoutIoutout VVV

Sum the individual values for Vout to get the total value for Vout

VVVout 33.367.1

VVout 5

Thévenin & Norton Equivalents18


Thévenin Equivalent Circuits

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


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Léon Charles Thévenin, 1857 – 1926

Thévenin Equivalent Circuits

Useful for determining current, voltage, and power delivered by any complex network to an arbitrary load

Simplifies the analysis of complex networks


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Complex network Thévenin equivalent network

Open-Circuit Voltage - Voc

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.


21

Thévenin Resistance - Rth

Rth, the Thévenin equivalent resistance, is the resistance seen between the two terminals with all sources set to zero Voltage sources short circuits Current sources open circuits


22

Thévenin Equivalent – an example

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


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Thévenin Example – find Voc

Two independent sources, so use superposition


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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

VVVsIoc 5

1000

50010

0

Thévenin Example – find Voc, cont’d


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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 VmARIV

sVoc 15002430

mAmAI

mAmAI

21000

20010

81000

80010

3

2

VVVVVVss VocIococ 41500

Thévenin Example – find Rth

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)


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300200||50050

|| 3245

th

th

R

RRRRR

R1 gets shorted R2 and R3 are in series R4 in parallel with R2 plus R3

300thR

``

Thévenin Example – find IL and VL


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400

1004V

RR

RVV

thL

LocL

Thévenin equivalent circuit with the 100 Ω load resistor connected

Find the voltage across the load by using the voltage divider equation

VVL 1

Ohm’s Law gives the load current

100

1V

R

VI

L

LL

mAIL 10

Norton Equivalent Circuits

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


28

Edward Lawry Norton, 1898 – 1983

Norton Equivalent Circuits

An extension of Thévenin’s Theorem Came about due to the development of vacuum

tubes, which are more appropriately modeled with current sources


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Complex network Norton equivalent network

Short-Circuit Current- Isc

Isc, 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.


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Thévenin Resistance - Rth

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


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Thévenin and Norton Equivalents

A Thévenin circuit can easily be converted to a Norton Circuit and vice versa


32

thscoc RIV th

ocsc R

VI

Dependent Sources33


Dependent Sources

Ideal current and voltage sources Outputs depend on some circuit parameter – branch

current or node voltage


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VCVS – voltage-controlled voltage source

VCCS – voltage-controlled current source

VCVS – voltage-controlled voltage source

VCCS – voltage-controlled current source

Output voltage is a function of node voltages elsewhere in the circuit

Output current is a function of a branch current elsewhere in the circuit

Output voltage is a function of a branch current elsewhere in the circuit

Output current is a function of node voltages elsewhere in the circuit

Dependent Sources

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


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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

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Section 3 resistive circuit analysis ii

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