02 Powerand Energy Kirchhoffs Laws 2011

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Lecture Quiz 2Topic Questions and Answers

Energy Unit? conversion between electrical and mechanicalenergy? how much energy produced measured incents when you carry 25 boxes of 35kg each tothe top level of your college for your collegemaster? 1 cent 5 cents 50 cents 250 cents?

Power unit? conversion between electrical and mechanicalpower

Thermal Energy unit? Conversion between electrical and thermalenergy

Kirchhoffs Laws and

application to circuitanalysis

what have a round trip Servery common room Servery in College, the River Junction LesumWeser and Kirchhoffs Laws in common?

direction of counting arrows in sources ?

Prof. W. Bergholz Gen. EE 1 (Fall 2011) 1


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Case Study #4a: Tour de France Cycling

Uphill: assume a gradient of 20%, and 18km/h speed, andtotal weight of 70kg (including the bike).

What is the power the cyclist has to deliver?

What is the equivalent in electrical power?

02_Power&Energy, Kirchhoffs laws


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Objectives: Understand how power and energy are related to U and I

Understand the electric, mechanical and heat energy

equivalent and being able to convert it

Understand Kirchhoffs laws and their first applications



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Electric power: We know:

Voltage U is a potential difference between 2 points per unit ofcharge potential difference = energy difference per unit of charge

Current I is the amount of charge transported between the 2 points /through a cross section A per second

So U x I is energy / second = Power Hence we define:

Definition 8:Electric Power P = U x I VA

The unit is also called Watt = W = VA

1 2A



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Electric power: Practical Examples:

Mobile phone talking 100400mW (e.g. 3.6V x 100mA)

Light bulb 40 - 100W ( e.g 230V x 300mA)

Washing machine 3.450 kW (e.g. 230V x 15A)

Large power plant 1100 MW

Eye and ear sensitivity 1 pW

Racing cyclist 5001000 W

Audio Amplifier 10 500 W



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Electric energy: Power is the energy delivered per second

If you have to pay your electricity bill you do notpay for power (which varies all the time!) but forthe total electric energy consumed

How is this calculated: You add up the powersconsumed in every second, the mathematicalprocess for this operation is integration, hence:

Definition 9:Electric energy Wel = P(t) dt

Unit = Watt x seconds = Ws = Joule

(since during integration you multiply P by dt,which has the unit of seconds!)

t

P

Area under the

curve is theelectric energy

Wel



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Electric energy: Definition 9a:forconstant power P, the electric energy = P x t

Definition 9b:the commercial unit for electric energy is

1 kWh = 103 Wx 3600s

= 3.6 x 106 Ws

The price for 1 kWh in Germany for households isabout 24 cents

and for energy from PV: 22 - 24 cents, so it is cheaperto generate your own electricity!

This is dirt cheap if you e.g. consider the amount ofmechanical energy you can get out of it by an electricmotor

t

Area under the

curve is the

electric energyWelP

t



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Electric and mechanical energy equivalent: We know from physics the law of conservation of energy, i.e. if you

convert electrical to mechnical energy, no energy can be lost

(this neglects [for a moment] the losses by ohmic heating in the

coils of an electric motor, for now we assume efficiency = 100%)

When introducing the unit for the current A, it was linked to the

number of electrons and to a force between 2 straight wires .

What is the link to mechanical quantities for the voltage V?



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Electric and mechanical energy equivalent: This is link is simple and useful at the same time (made that way

intentionally!).

The unit 1 Volt has been chosen so that the units for electricalenergy and mechanical energy are the same, i.e. the conversion

factor is 1!

1 Ws = 1 Nm


great ... and what is one Nm?????


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Answer: It is the work done if you move against a force of 1 Newton over a

distance of 1m (of course)

The weight of a mass of about 102 g is about 1 Newton, becausethe weight is:

Weight = mg m= mass in kg

g = 9.81 ms-2

Example: If you lift 1kg 1m height you have done the mechanicalwork of

1kg x 9.81 ms-2 x 1m 10 Nm = 10 Ws


great ... and what is one Nm?????

Electric and mechanical energy equivalent:


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Case Study #4a: Tour de France Cycling

Uphill: assume a gradient of 20%, and 18km/h speed (5ms-1),and total weight of 70kg (including the bike).

What is the power the cyclist has to deliver?

What is the equivalent in electrical power?

Mechanical Power

Emech = m x g x h = 70kg x 10ms-2 x 0.2 x 5ms-1 x 1s

= 700 kgm2s-2

= 700 Nm .. every second

Hence:

Pmech = 700 Nm/s

Pel = 700 W



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Case study 4b:Electric and mechanical energyequivalent:

You have to lift 40 boxes of 25kg each 15m (4 stories)

Mechanical Work:


02 & i hh ff l


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Mechanical Work:

Force F approx. F = mg = 40 x 25kg x 10 ms-2

Distance = heigh: h = 15m

Mechanical work: Wmech = 104 x 15 kgm2s-2 = 1.5 x 105 kgm2s-2

= 1.5 x 105 Nm

How much electric energy, and how much does this cost?


02 P &E Ki hh ff l


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Mechanical Work:

Force F approx. F = mg = 40 x 25kg x 10 ms-2

Distance = heigh: h = 15m

Mechanical work: Wmech = 104 x 15 kgm2s-2 = 1.5 x 105 kgm2s-2

= 1.5 x 105 Nm

How much electric energy, and how much does this cost?

Remember: 1 Nm = 1 Ws

So: Wel = 1.5 x 105 Ws which is about 0.04 kWh

So the cost is approx. 1 cent

For a healthy average person, that is one hour of strenuous

work!


02 P &E Ki hh ff l


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Electric and thermal energy equivalent: If thermal energy is measured in Ws, the conversion is as straightforward

BUT

For applications this is rarely possible, since most thermal energies and heat

capacitances are stated in calories (cal) or Kilocalories (Kcal)

WARNING: In food science, the unit calory is in reality 1 Kcal!!!!!


Definition 10:1 cal is the energy needed to heat 1g of clean water

from 14.5 0C to 15.5 0C

02 P &E Ki hh ff l


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Electric and thermal energy equivalent:

Conversion Factor:1 cal = 4.186 4.2 Ws

Case Study #5: Heating water in an electric kettlewater kettle with 2100 W electric power

0,5 liter (= 500g) water to be heated from 20 to 100 0C

How long does it take to boiling ?


Definition 10:1 cal is the energy needed to heat 1g of clean water

from 14.5 0C to 15.5 0C

02 P &E Ki hh ff l


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water kettle with 2100 W electric power

0,5 liter (= 500g) water to be heated from 20 to 100 0C

How long does it take to boiling ?

Solution:Energy needed for heating the water:

500g x 1cal/g.K x T = 500 x 80 cal = 40 000 cal = 168 000 Ws

Time: t = energy / power

t = 168 000 Ws / 2100 W = 80s

In reality: about 100s .... Why ?


02 Power&Energy Kirchhoffs laws


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Relation between power and resistance: Given: ideal Voltage Source of voltage U0 and a resistor R Question: for drawing more power, does R have to be bigger or

smaller?




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Relation between power and resistance: Given: ideal Voltage Source of voltage U0 and an ohmic resistor R Question: for drawing more power, does R have to be bigger or

smaller (i.e. for constant voltage) ?


Answer:Remember the relation: P = U x I

Insert the Ohms law: P = U x U/R = U2 / RTherefore: to make P larger, R has to be made smaller!

Extreme examples:

Short circuit: R = 0 infinite power, therefore we have fuses!

Open circuit: R = infinity zero power



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Relation between power and resistance:

P = U x I = I x R x I = I2 R

Application: Trams and trains (U=const) cause most power

losses in lines when the need for power is high, i.e. double

power double current 4 times as much losses!

therefore this should be avoided to save energy!

How?

02_Power&Energy, Kirchhoff s laws

Answer:Remember the relation: P = U x IInsert the linear relation: P = U x U/R = U2 / RTherefore: to make P larger, R has to be made smaller!

Extreme examples:

Short circuit: R = 0 infinite power, therefore we have fuses!

Open circuit: R = infinity zero power



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Energy and Laws which govern the analysis of

Electrical networks:02_Power&Energy, Kirchhoff s laws

Next Task: analyze electrical networks (rather than simple 1

loop circuits)

- voltage between all the network nodes

- currents through all the network branches

The Route:

- current: we need an idea about direction of current. i.e. what is

positive and what is negative current

- voltage: energy difference positive or negative

Kirchhoffs 2 laws & a Standard (convention) will help us



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Kirchhoffs first law (Kirchhoffs current law KCL)02_Power&Energy, Kirchhoff s laws

I2I3

I4I5

I1



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Kirchhoffs first law (Kirchhoffs current law KCL)The sum of all currents into and out of a node is always zero

Ii = 0 Rationale: you cannot have a creation

of annihilation of charge in one point

(at least not in classical physics)

Convention:

Current into node is counted positive (charge is added) Current out of node is counted negative (charge is withdrawn)

(as it is with your bank account!)


I2I3

I4I5

I1



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Kirchhoffs second law: (Kirchhoffs Voltage Law KVL)

The sum of all voltages around a loop (mesh) is always zero

Ui = 0Rationale: if you move around a closed loop you must

come to the same potential when returning to your

starting point (the potential cannot be multivalued. Ifthis were the case, you would have invented the

perpetuum mobile)

Count voltage over resistors and sources (opposite signs !) - Why?

Over resistors energy is lost, in a source energy is gained




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Ui = 0A counter example by Max Escher?




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Standard - Conventions: (boring but important!)

Current goes from higher to lower potential (i.e. from + to - )

conventional current direction (water flows down, not up)

Voltage over a resistor is counted positive via a counting arrow in the

direction of the conventional current, i.e. from + to -

(in this direction the charge carriers lose energy (and heat the resistor))

In a voltage or current source, the charge carriers gain energy, therefore

the counting arrow has the opposite direction, going in the opposite

direction of the current flow

Note: counting arrows always from higher to lower potential

Each loop of a network is assigned a sense of direction (e.g. counter

clockwise or clock-wise, you can do what you prefer as long as you keep

it constant)




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Application of KVL: Resistance of 2 resistors in series:

By KVL:

0 = - U0 + U1 + U2

U0 = I R1 + IR2

U0= I (R

1+ R

2)

R tot = R1 + R2 (R = Ri )

Negative because

counting arrow and

sense of direction areopposite


U1

U2

+

+

-

-

+

-

U0

LOOP


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Lecture Quiz 2Topic Questions and Answers

Energy Unit? conversion between electrical and mechanicalenergy? how much energy produced measured incents when you carry 25 boxes of 35kg each tothe top level of your college for your collegemaster? 1 cent 5 cents 50 cents 250 cents?

Power unit? conversion between electrical and mechanicalpower

Thermal Energy unit? Conversion between electrical and thermalenergy

Kirchhoffs Laws and

application to circuitanalysis

what have a round trip Servery common room Servery in College, the River Junction LesumWeser and Kirchhoffs Laws in common?

direction of counting arrows in sources ?


02 Powerand Energy Kirchhoffs Laws 2011

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Transcript of 02 Powerand Energy Kirchhoffs Laws 2011