Chapter 20: Circuits Current and EMF Ohm’s Law and Resistance Electrical Power Alternating...

Chapter 20: Circuits

Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits

Current and EMF

A few definitions: Circuit: A continuous path made of conducting materials EMF: an alternative term for “potential difference” or “voltage,”

especially when applied to something that acts as a source of electrical power in a circuit (such as a battery)

Current: the rate of motion of charge in a circuit:

Symbol: I (or sometimes i).

SI units: C/s = ampere (A)

t

qI

Current and EMF

“Conventional current:” assumed to consist of the motion of positive charges.

Conventional current flows from higher to lower potential.

+ -

12 V

I

Current and EMF

Direct current (DC) flows in one direction around the circuit

Alternating current (AC) “sloshes” back and forth, due to a time-varying EMF that changes its sign periodically

Ohm’s Law and Resistance

The current that flows through an object is directly proportional to the voltage applied across the object:

A constant of proportionality makes this an equation:

IV

IRV


The constant of proportionality, R, is called the resistance of the object.

SI unit of resistance: ohm ()

IRV


Resistance depends on the geometry of the object, and a property, resistivity, of the material from which it is made:

Resistivity symbol: SI units of resistivity: ohm m ( m)

L

cross-sectional area A

A

LR


In most materials, resistivity increases with temperature, according to the material’s temperature coefficient of resistivity, :

(resistivity is 0, and R = R0, at temperature T0)

SI units of : (C°)-1

00

00

1

1

TTRR

TT

Electrical Power

Power is the time rate of doing work:

Voltage is the work done per unit charge:

Current is the time rate at which charge goes by:

Combining:

t

WP

q

WV

t

qI

VIt

q

q

W

t

WP

Electrical Power

Ohm’s Law substitutions allow us to write several equivalent expressions for power:

Regardless of how specified, power always has SI units of watts (W)

R

VRIVIP

22

Alternating Current

EMF can be produced by rotating a coil of wire in a magnetic field.

This results in a time-varying EMF:

ftVVt 2sin0

peak voltage frequency (Hz)

time, s

Alternating Current tVt Hz 602sin78

-80

-60

-40

-20

0

20

40

60

80

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

time, s

volt

age,

V

V0

T

Alternating Current

The time-varying voltage produces a time-varying current, according to Ohm’s Law:

ftIftR

V

R

VI t

t 2sin2sin 00

peak current frequency (Hz)

time, s

Alternating Current tI t

Hz 602sin

100

V 78

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

time, s

curr

ent,

A

Alternating Current

Calculate the power:

ftIVIVP ttt 2sin 200

Alternating Current

0

10

20

30

40

50

60

70

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

time, s

po

wer

, W

ftIVPt 2sin 200

Alternating Current

Calculate the power:

Average power:

ftIVIVP ttt 2sin 200

rmsrms VIVI

P

VIVIPP

22

2

1

2

1

2

1

2

1

00

00000

Series Connection

A circuit, or a set of circuit elements, are said to be connected “in series” if there is only one electrical path through them.

+ -

R1 R2

R3

V

I

Series Connection


The same current flows through all series-connected elements. (Equation of continuity)

Series Connection


The same current flows through all series-connected elements. (Equation of continuity)

A set of series-connected resistors is equivalent to a single resistor having the sum of the resistance values in the set.

Series Connection

+ -

R1 R2

R3

V

I

+ -

Req = R1 + R2 + R3

V

I

Series Connection

Potential drops add in series.

+ -

R1 R2

R3

V

I

V1 V2

V3

321

332211

RRRIIRV

IRVIRVIRV

eq

Parallel Connection

A circuit, or a set of circuit elements, are said to be connected “in parallel” if the circuit current is divided among them.

The same potential difference exists across all parallel-connected elements.

Parallel Connection

R2

R3

R1

+ -

V

I1

I2

I3

I = I

1 +

I 2 +

I 3

Parallel Connection

What is the equivalent resistance?

The equation of continuity requires that: I = I1 + I2 + I3

R2

R3

R1

+ -

V

I1

I2

I3

I = I

1 +

I 2 +

I 3

Req

+ -

V

I

I

Parallel Connection

Applying Ohm’s Law:

R2

R3

R1

+ -

V

I1

I2

I3

I = I

1 +

I 2 +

I 3

321

321321

321

1111

1111

1

RRRR

RV

RRRV

R

V

R

V

R

VI

IIIR

VR

VI

eq

eq

eqeq

Series - Parallel Networks

Resistive loads may be so connected that both series and parallel connections are present.

+ -

V

R1

R2

R4R3

R5

R6


Simplify this network by small steps:

+ -

V

R1

R2

R4R3

R5

R6

+ -

V

R1

R2

R7

R8

437 RRR

65

8658

111

111

RR

RRRR


Continue the simplification:

+ -

V

R1

R2

R7

R8

721

97219

1111

1111

RRR

RRRRR

+ -

V

R9 R8


Finally:

9810 RRR

+ -

V

R9 R8

+ -

V

R10

Kirchoff’s Rules and Circuit Analysis

The Loop Rule:

Around any closed loop in a circuit, the sum of the potential drops and the potential rises are equal and opposite; or …

Around any closed loop in a circuit, the sum of the potential changes must equal zero.

(Energy conservation)

Kirchoff’s Rules and Circuit Analysis

The Junction Rule:

At any point in a circuit, the total of the currents flowing into that point must be equal to the total of the currents flowing out of that point.

(Charge conservation; equation of continuity)

Internal Resistance

An ideal battery has a constant potential difference between its terminals, no matter what current flows through it.

This is not true of a real battery. The voltage of a real battery decreases as more current is drawn from it.

Internal Resistance

A real battery can be modeled as ideal one, connected in series with a small resistor (representing the internal resistance of the battery). The voltage drop with increased current is due to Ohm’s Law in the internal resistance.

Capacitors in Circuits

Like resistors, capacitors in circuits can be connected in series, in parallel, or in more-complex networks containing both series and parallel connections.

+ -

C3

V

C2C1

+ -

V

C2

C3

C1

Capacitors in Parallel

Parallel-connected capacitors all have the same potential difference across their terminals.

+ -

V

C2

C3

C1

321

321

321

332211

CCCC

VCVCVCVC

VCqqqQ

VCqVCqVCq

eq

eq

eq

Capacitors in Series

Capacitors in series all have the same charge, but different potential differences.

+ -

C3

V

C2C1

V1 V2 V3

321

321

321

1111

CCCC

C

q

C

q

C

q

C

q

C

qVVVV

eq

eq

eq

RC Circuits

A capacitor connected in series with a resistor is part of an RC circuit.

Resistance limits charging current

Capacitance determines ultimate charge

RC

+ -

V

RC Circuits

At the instant when the circuit is first completed, there is no potential difference across the capacitor.

At that time, the current charging the capacitor is determined by Ohm’s Law at the resistor.

RC

+ -

V

0

0

0

00

CV

qR

VI

RC Circuits

In the final steady state, the capacitor is fully charged.

The full potential difference appears across the capacitor.

There is no charging current.

There is no potential difference across the resistor.

RC

+ -

V

VV

CVqI

fC

ff

0

RC Circuits

Between the initial state and the final state, the charge approaches its final value according to:

The product RC is the “time constant” of the circuit.

RC

+ -

V

RC

t

t eqq 10

sV

C

C

sV

V

C

sCV

V

C

A

VΩF

RC Circuits RC Charging Curve

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

time, s

char

ge,

C

R = 100 K

C = 10 F

(RC = 1 s)

V = 12 V

RC

t

t eqq 10

RC Circuits

During discharge, the time dependence of the capacitor charge is:

RCt

t eqq

0

R

C

RC Circuits RC Discharge Curve

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

time, s

char

ge,

C

R = 100 K

C = 10 F

(RC = 1 s)

V = 12 V

RCt

t eqq

0

Chapter 20: Circuits Current and EMF Ohm’s Law and Resistance Electrical Power Alternating...

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Transcript of Chapter 20: Circuits Current and EMF Ohm’s Law and Resistance Electrical Power Alternating...