source of electromotive force

A device that, by doing work on charge carriers, maintains a potential difference between its terminals is called a source of electromotive force (emf).

Other form of energy is converted into electricity in a source of electromotive force:

battery - chemical energyelectric generator - mechanical energysolar cell - electromagnetic radiationthermopile - internal energyliving cell - chemical energy

electromotive force

The maximum electric potential difference that can exist between the terminals of the voltage source is called the electromotive force of that source.

I

r+ - IrVV

Voltage produced by a real source of electromotive force:

direct and alternating current

If the charge moves in a circuit in the same direction at all times, the current is said to be direct current (DC). Constant current (independent of time) is a special case of direct current.

If the charges move (across a surface) changing their direction of motion, the current is said to be alternating current (AC).

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circuit analysis Kirchhoff's Junction Rule:

The sum of all the currents entering a junction is zero.

Kirchhoff's Loop Rule:Around any closed circuit loop

the sum of potential differences is zero.

I1

I2

In

i

i 0I V1

V2

Vn

i

i 0V

electrical measurements Current is measured with an ammeter, which must be inserted into the circuit in series with the element in which the current is measured.

Voltage is measured with a voltmeter, which must be inserted into the circuit in parallel to the elements across which the voltage is measured.

The resistance of passive elements can be measured with an ohmmeter.

V

+

-

A

V

?

electric current & the human body

Currents of 200 mA can be fatal. A current that strong can affect the proper operation of the heart.

A current above 100 mA can cause muscle spasm.

A person can sense an AC with a current of 1mA.

NEVER TOUCH AN OPERATING CIRCUIT WITH BOTH HANDS !!!

inductors

An inductor is an element of a circuit with two sides for which (at any instant) the potential difference V between its terminals is proportional to the rate of change in current I passing through this element.

I

VaVb dt

dILVV ba

The proportionality coefficient L is called the inductance of the inductor.

In SI the henry is the unit of inductance HVsA

sinusoidal alternating current

For a sinusoidal alternating current, both the voltage V(t) across an element and the current I(t) through this element are sinusoidal functions of time.

V t V tm v( ) sin( )

I t I tm I( ) sin( )

Va(t) Vb(t)

I (t)

Vm, Im - the peak value

(t+) - the phase

= 2f - the angular frequency

- the initial phase

t

VI

electric power

t

V

I

P

tVtI)t(P

Electric power delivered to an electrical element is a sinusoidal function of time.

)tsin()tsin(VI VImm

VIIVmm t2coscosVI2

1

cosVIP rmsrmsav

Pav

where IV

21

av2

rms tff

AC in the US standard one phase power line

The voltage oscillates with frequency f = 60 Hz.

0 V

0 V

0 V

120 V

“ground”

“zero”“hot”

breaker

AC in the US three phase power line

The rms voltage between any "hot" wire and the zero wire is 127 V.

Three "hot" wires with phases differing by , the "zero" and the "ground" wires are connected to the outlet.

3

2

The voltage between any two "hot" wires is 220 V.

t

V

complex voltage

The complex function V(t) such that the voltage across the element is

V(t) = Im V(t) is called the complex voltage.

Sinusoidal voltage: V V0t V e e emi i t i tV

tIm V VVm tsinitcosVIm Vm tsinV tV

t

VIm V

Re V

V (t)Vm

t

-Vm

t+V

complex current

The complex function I(t) such that the current through the element is

I(t) = Im I(t) is called the complex current.

Sinusoidal current: titiim eeeIt V 0II

tImI VVm tsinitcosIIm Vm tsinI tI

t

IIm I

Re I

I (t)Im

t+I

t

-Im

relation between voltage and current

The coefficient Z relating the peak values of the voltage across the system with the peak value of the current through the system is called the impedance of the system.

Vm = Im·ZIV

Vm

Im

t

note that: Vrms = Irms·Z

The number relating the phase of the voltage across the system with the phase of the current through the system is called the phase angle between the current and the voltage

IV tt

and V = I +

i

m

m eI

V

Z

complex impedance

The complex coefficient Z, relating the complex voltage across an element with the complex current through this element, is called the complex impedance of this element at frequency :

Im

Re

V I ZIV tt

Complex impedance Z includes information about both the impedance Z as well as about the phase angle

Z

ZZZ Z

Re

Imtan

tiim eeV VV IZ

ZIV tt ZIV ,0,0

tiim eeI I

AC in a resistor

R

a

b

Ireal analysis

tI Itsin mI RIm RtV

0 RZR ,

impedance and phase angle

average power:

t

V

I

0cosVIP rmsrmsav

RI2rms

R

V2rms

AC in a resistorcomplex analysis

R

a

b

I

IRV

0Re

Imtan

ZZ

RZR Z ,

complex impedance

tIm V tRIm I tRt IV

ZR() = R

t

V

I

Im

ReZR

VI

L

a

b

I

AC in an inductorreal analysis

tI Im tsinI dt

dLtV

2

LZL ,

impedance and phase angle

average power:

t

VI0

2cosVIP rmsrmsav

dt

dL =

Im tcosIL

2

tsinIL Im

L

a

b

I

complex analysis

t

VI

dt

dILV tIm V

dt

dLIm

I dt

dLt

IV tie

dt

dL 0I tLi I

complex impedanceZL() = iL

Im

Re

ZLI

V

ZZ

Re

ImtanLZL Z ,

AC in an capacitorreal analysis

Vm tsinVtV

2

C

1ZC

,

impedance and phase angle

average power:

t02

cosVIP rmsrmsav

Vm tcosVC

2

tsinCV Vm

VI

C

a

b

I

Q

-Q

CtQ dt

dtI