No Slide Title · 2015-09-12 · Thévenin's theorem –Example K O N E CKLIK2 Norton's theorem =...

2. lecture

• basic electronic circuit concepts

• resistors, capacitors, inductors

Jiří Jakovenko – Electronics and Microelectronics - Department of Microelectronics – CTU

Electronics and Microelectronics AE4B34EM


Electronics and Microelectronics AE4B34EM

Studing materials: server MOODLE

http://moodle.kme.fel.cvut.cz

AE4B34EM – Electronics and Microelectronics

Book: Sedra, Smith: Microelectronic Circuits

Basic concept

Electric current

Amount of charge that passes through a certain cross section per unit time

SI Unit Ampere

Ampere is a constant electrical current which passes through two direct parallel infinitely long wires of negligible circular cross section placed in a vacuum in the distance of one meter that produces a constant force of 2.10-7 Newtons per metre of wire length


André Marie Ampére French mathematician and physicist who became famous especially for his work in the field of magnetism and electrodynamics.

Voltage

Voltage The difference of electrical potential between two points in space

Unit - Volt

The volt is defined as the value of the voltage across a conductor when a current of one ampere dissipates one watt of power in the conductor.

It can be written in terms of SI base units as: m2 · kg · s−3 · A−1. It is also equal to one joule of energy per coulomb of charge, J/C


Alessandro Giuseppe Antonio Anastasio Volta Italian physicist famous for his discoveries in electricity. Invented such as frictional electricity, electrical cell or a capacitor.

Voltage

DC voltage is a voltage, which does not change polarity over time , the value may vary.

AC voltage is the voltage that changes over time with a certain period, while the median value may be zero. Waveform (shape) can be any shape, most often we can meet with sinusoidal track.

RMS (root mean square) AC voltage is a DC equivalent voltage at which the same conductors create the same amount of heat.


Electric power

Electric power

Electric power is the rate at which electrical energy is transferred by an electric circuit.

In direct current resistive circuits, electrical power is calculated using Joule's law:

where P is the electric power, V the potential difference, and I the electric current.

In the case of resistive (Ohmic, or linear) loads, Joule's law can be combined with Ohm's law (I = V/R) to produce alternative expressions for the dissipated power:


IVP .

R

VRIP

22.

http://moodle.kme.fel.cvut.cz/

http://moodle.kme.fel.cvut.cz/

Kirchhoff’s laws

Deals with the conservation of charge and energy in electrical circuits

First Kirchhoff’s law (Kirchhoff's current law (KCL))

The algebraic sum of currents in a network of conductors meeting at a point is zero.


I1

I2 I3

0I

Kirchhoff’s laws

Second Kirchhoff’s law (Kirchhoff's voltage law (KVL))

The directed sum of the electrical potential differences (voltage) around any closed circuit is zero.


Kirchhoff’s laws

+

+

+

+ +

+

+

+

+

+

+

-

- - -

-

-

- -

-

-

- V1

V2

U4

V3

V12

V11 V9

V8

V6

V5

V7

v10

+

-

“a” •

Blue loop from “a”

- V7 + V10 – V9 + V8 = 0

• “b”

Red loop from “b”

+V2 – V5 – V6 – V8 + V9 – V11

– V12 + V1 = 0

Yellow loop from “b”

+ V2 – V5 – V6 – V7 + V10 – V11

- V12 + V1 = 0

The sum of voltage drop in a closed loop = 0


Ohm’s law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference or voltage across the two points, and inversely proportional to the resistance between them.

R is the resistance of the conductor (units of ohms) between two points on conductor


U = R . I

Georg Simon Ohm (16. března 1789, Erlangen, Bavorsko – 7.

července 1854)

Ohm's and Kirchhoff's laws – Example


Calculate the voltage drop across the resistor R1, R2 and R3 if you know it, the power supply Ucc = 5 V UD1 = 1.7 V – red LED diode UD2 = 2.2 V – green LED diode UD3 = 3.0 V – blue LED diode

Solution: -Ucc + UR1 + UD1 = 0 UR1 = Ucc - UD1 = 5 – 1.7 = 3.3 V -Ucc + UR2 + UD2 = 0 UR2 = Ucc – UD2 = 5 – 2.2 = 2.8 V -Ucc + UR3 + UD3 = 0 UR3 = Ucc – UD3 = 5 – 3.0 = 2.0 V

Ohm's and Kirchhoff's laws – Example


Calculate the size of the resistors R1, R2 and R3 and total current when (according to the catalog values) currents flowing through diodes I1 = 12 mA – red LED diode I2 = 8 mA – green LED diode I3 = 20 mA – blue LED diode

Solution: R1 = UR1 / I1 = 257 R2 = UR2 / I2 = 350 R3 = UR3 / I3 = 100 Total current: Icc - I1 – I2 – I3 = 0 Icc = I1 + I2 + I3 = 40 mA

Voltage and current sources

Ideal voltage source is a circuit element where the voltage across it is independent of the current through it

Ideal current source is a circuit element where the current through it is independent of the voltage across it

Nonideal voltage or current source is a combination of ideal source U0 or Ik and its internal resistance Ri


Unloaded voltage source Unloaded current source Voltage across the ideal current

source approaches infinity as the load resistance approaches infinity

(an open circuit)

Voltage and current sources

Loaded voltage or current source

The voltage of across load resistance Rz is smaller then reference voltage U0 (internal resistance Ri - voltage divider)

The current of a load resistance is smaller then reference current Ik (internal resistance Ri)


Loaded voltage source – voltage drop on internal

resistance result a voltage decrease at the output terminals of the source

Loaded current source – Desired current should flow

through load resistance. Requirement Rz << Ri

The duality of voltage and current source

In practice, we often uses the duality. Connection of voltage source and its internal resistance can be converted to the current source and its internal resistance. This can be applied vice versa.


0.URR

RU

Zi

Z

0

..

.U

RR

RI

RR

RRU

Zi

Zk

Zi

iZ

Thévenin's theorem

Any combination of voltage sources, current sources, and resistors with two terminals is for linear electrical networks electrically equivalent to a single voltage source U and a single series resistor R.

Calculating procedure:

1. Calculate the output voltage U, in open circuit condition (no load resistor—meaning infinite resistance).

2. Now replace voltage sources with short circuits and current sources with open circuits

3. Replace the load circuit with an imaginary ohmmeter and measure the total resistance R, "looking back" into the circuit.


Thévenin's theorem - Example

Simplify the circuit in Figure. Voltage U1 = 15 V, resistance R1 = 1KW and R2 = 2.2 KW

Voltage U0 is given as the open circuit voltage

Ri is determined by parallel combination of R1 and R2, voltage source is replaced by a short circuit

Output voltage:


VURR

RU 3.10. 1

21

2

W

5.685.

21

21

RR

RRRi

IIRUU i .5.6 8 73.1 0.0

R3R1

R2 R4U0

I2

KLIK KLIK

Thévenin's theorem – Example 2

Calculate current I2 through resistance R2, when is given:

U0 = 15 V

R1 = 5 kΩ

R2 = 15 kΩ

R3 = 10 kΩ

R4 = 5 kΩ


R3R1

R2 R4U0

I2 2

2´

Uv

2

2´

Rv

KLIK KLIK KLIK

Every linear system can be replaced by the voltage source at the output terminals.

First, we have to indicate the output terminals.

We wont to calculate current through resistance R2 (I2).

At these terminals we can replace the remainder of the circuit by voltage source.



Circuit is divided into two parts:

Linear system with its output terminals (2 - 2´) and

Load (R2)

R3

R4

R1

U0

2

2´

R2

Circuit will be converted to a equivalent circuit with one

alternative voltage source its internal resistance Rv.

Uv

2

2´

Rv

R2

I2

For the equivalent circuit is necessary to determine the characteristic parameters of the alternative power source - Uv and Rv.

KLIK KLIK KLIK KLIK KLIK



R3

R4

R1

U0

2

2´

Uv

2

2´

Rv

R2

I2

Voltage across alternative voltage source is determined as the open circuit voltage at the output

terminals (Uv = U22´).

U22´

Uv

Total current I0 creates a voltage drop U1, U3 and U4 on resistors.

I0 U1 U3

U4

Voltage U22´ is given by:

U22´ = U0 - U1 = U3 + U4

Individual voltage drops are calculated from Ohm's Law :U1 = I0.R1; U3 = I0.R3; U4 = I0.R4

First we calculate the total current I0:

Substituting into the equation for voltage U22´

mA 75,05105

15

431

00

RRR

UI

Now we calculate the individual voltage drops :

V 75,35.75,0.

V 5,710.75,0.U

V 75,35.75,0.

404

303

101

RIU

RI

RIU

By substituting we get :

U22´ = 15 - 3,75 = 7,5 + 3,75 = 11,25 V = Uv

KLIK KLIK KLIK KLIK KLIK KLIK



The second equivalent parameter of the alternative source is internal resistance Rv.

R3

R4

R1

U0

2

2´

Uv

2

2´

Rv

R2

I2

The internal resistance of alternative source: is determined as total resistance at the output terminal when the voltage sources of the system is removed (Rv = R22´).

The system has only one ideal voltage source (U0), that will be shorted.

Rv

R22´

The circuit can be adjusted:

R 3

R 4

R 1

2

2´

R22´

For searched resistance R22´is given:

W

k 75,3)510//(5

)//(R

,

,

22

43122

R

RRR

KLIK KLIK KLIK KLIK KLIK KLIK



Uv

2

2´

Rv

R2

I2 Searched current I2 is given by:

Uv = 11.25 V Rv = 3,75 kW

The calculation of I2 for specified values of R2

W

kV,mA; 2

2RR

UI

v

v

KLIK KLIK KLIK K O N E C


= 0,6 mA


Norton's theorem

Is an extension of Thévenin's theorem

Any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R.


Passive electronic components

Resistor

Capacitor

Induktor


Passive components - Resistors

In practical terms, resistors convert the current to voltage and vice versa

For the resistance of electrically conductive material R can be described by:

r – specific electrical resistance

l – Wire length

S – Wire cross sectional area


S

lR r

Resistors parameters

Characterize the functional characteristics of components and are listed in the catalog

Eg. nominal resistance value, load, tolerance...

Ranges of nominal values

Resistors (capacitors, inductors) are manufactured only for certain values according to the tolerance

Typical ranges: E6, E12, E24 (divide resistor values decades to a number of sections)


Range

Resistors parameters

Power rating (Power dissipation)

Temperature coefficient of resistance

Voltage coefficient of resistance (dependence on the supplied voltage)

Dependence on the ambient humidity

Noise properties

Parasitic inductance and capacitance

maximum working voltage


R

VIRIVP

22..

Resistors marking

Numerical Marking

Color coding


Capacitors

Passive electronic component consisting of a pair of conductors separated by a dielectric (insulator)

Component able to accumulate an electric charge Q

The relationship between voltage and charge is:

Capacitor current is directly proportional to the voltage change at the terminals :

Capacitor charging


UCQ .

dt

dUCI

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50 55 60

uC

(t)

[V]

t [ms]

Capacitor charging

U0

R

C

PR1

2

Switch PR is off.

For the circuit values conditions:

uC(t) = 0 - capacitor is discharged;

i(t) = 0 - the circuit is disconnected (PR in possition 0);

uR(t) = 0 - due to the fact that i(t) = 0, there will be no voltage drop at resistance R.

uC

uR i

Initial conditions: ie the conditions at the time t = 0.

uC(0) = i(0) = uR(0) = 0.

Capacitor charging


U0

R

C

PR1

2

uC

uR i(t)

Now we switch the switch PR from a position 1 to position 2.

Capacitor begins to charge through resistor R and voltage uc will vary according to the relation

Since the capacitor C is discharged and Rv is equal to zero, the initial circuit current will be limited only by resistor R.

i(t) = U0/R.

)1.(0

t

c eUtu

For the current is valid:

The voltage across the resistor R is given by :

tt

R eUeR

URtiRtu

00..

ttt

C eR

U

R

eUUU

R

eUU

R

tuUti

0000000 .)1.(

Capacitor charging


the voltage on the capacitor uC(t)

the voltage across the resistor uR(t)

charging current i(t)

ss,V,V; )1.(0

t

c eUtu

ss,V,V; .0

t

R eUtu

ss,,V,A; .0 W

t

eR

Uti

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50 55 60

uC

(t)

[V]

t [ms]

CAPACITOR CHARGING uc(t)

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50 55 60

uR

(t) [V

]

t [ms]

CAPACITOR CHARGING uR(t)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

0 5 10 15 20 25 30 35 40 45 50 55 60

i(t)

[m

A]

t [ms]

CAPACITOR CHARGING i(t)

Capacitor charging


The lines in one graph ...

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50 55 60

t [ms]

Capacitor charging

Capacitor charging

uR(t)

i(t)

uC(t)


Curves for different values of resistance R

R = 5 kΩ; C = 1 μF; τ = 5 ms

R = 10 kΩ; C = 1 μF; τ = 10 ms

R = 15 kΩ; C = 1 μF; τ = 15 ms

0

2

4

6

8

10

12

14

16

0 5 10 15 20 25 30 35 40 45 50 55 60

uc

(t)

[V]

t [ms]

Capacitor charging - Time constant

Capacitor charging


Capacitors - properties

Capacity of plate capacitor :

Capacitor parameters:

Accuracy, Capacitance instability

Temperature coefficient

leakage current

Breakdown voltage


d

SC r 0

Capacitor types

Polyester film capacitors

Big dimensions, accurate, good RF properties

Ceramic capacitors Small size, very inaccurate, non-linear

Electrolytic Capacitors Very large capacitance to volume ratio, inexpensive, polarized. Primary

applications are as smoothing and reservoir capacitors in power supplies

Monolithic integrated capacitors

Very limited size


Inductors

Passive electrical component that can store energy in a magnetic field created by the electric current passing through it.

If electric current passes through the closed circuit creates a magnetic flux:

Inductance of 1 henry produces an EMF of 1 volt when the current through the inductor changes at the rate of 1 ampere per second

L = inductance (H)

μ0 = permeability of free space = 4π × 10−7 H/m

μr = relative permeability of core material

N = number of turns

l = length of coil (m)

S = crossection area


WbIL.

2

0 .Nl

SL r

Inductance - Example

Calculate the inductance of the air inductor in Figure of length l = 50 mm, diameter d = 8 mm, the number of turns N = 100


7.12..4

.. 2

2

0

2

0 Nl

dN

l

SL rr

Inductors

Inductor parameters

Nominal value and power load

DC resistance

quality factor Q (ratio of reactance and resistance of unwanted dc)

dielectric strength (break down voltage)

Operating temperature range

Applications

Choke

Transformer

The electric motor, speaker, relay ...

Filters, tuned circuits

Types of inductors

Air core coil - Radio frequency inductors

Ferromagnetic core coil - A magnetic core can increase the inductance of a coil by a factor of several thousand

Variable inductor


No Slide Title · 2015-09-12 · Thévenin's theorem –Example K O N E CKLIK2 Norton's theorem =...

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Transcript of No Slide Title · 2015-09-12 · Thévenin's theorem –Example K O N E CKLIK2 Norton's theorem =...