Electrical circuits wyklad 6

Electrical Circuits

Dr inż. Agnieszka Wardzińska

Room: 105 Polanka

agnieszka.wardzinska@put.poznan.pl

cygnus.et.put.poznan.pl/~award

Coupling coils

The coupling occurs when two coils are placed near each other (See Fig.3.1). The first

coil current I₁ gives magnetic field B₁. When the distance between two coils are small,

some of the magnetic field will pass through coil 2. Variation of I₁ with time, induce

electromagnetic field associated with the changing magnetic flux in the second coil:

If the current ENTERS the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is POSITIVE at the dotted terminal of the second coil.

If the current LEAVES the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is NEGATIVE at the dotted terminal of the second coil. 1 2 di v M dt  1 2 di v M dt   2 1 di v M dt   2 1 di v M dt 

Dot convention

Series connection

The effect of mutual inductance for

inductors connected together in series so that the magnetic field of one links with the other, changes total inductance. The

increase or decrease of the inductance depends on their orientation to each other. The coils are said to be Cumulatively

Coupled (Fig.3.2) if the magnetic flux

produced by the current flows through the coils in the same direction.

The coils are said to be Differentially Coupled if the current flows through the coils in opposite directions.

Parallel connection

The mutual inductance of the coils connected in parallel, when the currents goes through them in the same way (parallel aiding

inductors, see Fig3.4) can be

calculated as:

If one of the two coils was reversed (see Fig.3.5) the mutual inductance, M will have

a cancelling effect on each coil

instead of an aiding effect (parallel

Time-domain and Frequency-domain

Analysis

Time Domain

Frequency Domain

(

)

(

)

di

di

v

i R

L

M

dt

dt

di

di

v

i R

L

M

dt

dt

V

R

j L I

j MI

V

j MI

R

j L I

































Energy in a Coupled Circuit

1

1

2

2

w



L i



L i



Mi i



_{The total energy w stored in a mutually coupled inductor is:}



_{Positive sign is selected if both currents ENTER or LEAVE the}

dotted terminals.

Methods of analysing the coupligs



_{Kirchoffs laws}



The rules for positive or negative couplings work for

current in branches



_{Mehs current method}



_{The rules for positive or negative couplings work for}

current in loops



_Uncupling



_{The rules for positive or negative couplings work for}

Uncoupling



_{Sometimes it could be useful replace mutual}

coupled inductors by ordinary uncoupled

inductors. If coupled inductors are connected

into same node, then the replacement is

Transformers

 _{On the mutual inductance bases the transformer operation. The}

transformer is constructed of two coils, the flux generated in one of the coils induced voltage across the second coil. The source coil is called primary coil and the coil to which the load is applied is called secondary.

 _{The basic types of transformers:}

 _{the iron-core transformer}  _{the air-core transformer}

 _{the variable-core transformer}

 _{Three basic operations of a transformer are:}

 _{Step up/down}

 _{Impedance matching}  _Isolation

The symbol used for the

Linear Transformers



_{A transformer is generally a four-terminal device comprising two}

or more magnetically coupled coils.



_{The transformer is called LINEAR if the coils are wound on}

magnetically linear material.



_{For a LINEAR TRANSFORMER flux is proportional to current in}

the windings.



Resistances R

and R

account for losses in the coils.

Reflected Impedance for Linear Transformers

(

)

0

(

)

V

R

j L I

j MI

j MI

R

j L

Z I















 







V

M

Z

R

j L

R

j L

Z

I

R

j L

Z



























REFLECTED IMPEDANCE

M

Z

R

j L

Z











•

_{Secondary impedance seen from the primary side is the}

Reflected Impedance

.



Let us obtain the input impedance as seen from the source,

Equivalent T Circuit for Linear Transformers



The coupled transformer can equivalently be represented by an

EQUIVALENT T

circuit using

UNCOUPED INDUCTORS

.

1

,

,

a b c

L

 

L

M

L



L



M

L



M

a) Transformer circuit b) Equivalent T circuit of

the transformer

Equivalent П Circuit for Linear Transformers



The coupled transformer can equivalently be represented by an

EQUIVALENT П circuit using uncoupled inductors.

2 2 2 1 2 1 2 1 2 2 1

,

,

L L

M

L L

M

L L

M

L

L

L

L

M

L

M

M

















a) Transformer circuit b) Equivalent Π circuit of the

transformer

Power – DC circuit

 _{The electric power in watts associated with an electric circuit or a}

circuit component represents the rate at which energy is

converted from the electrical energy of the moving charges to some other form, e.g., heat, mechanical energy, or energy stored in electric fields or magnetic fields.

 _{For a resistor in a DC Circuit the power is given by the product of}

applied voltage and the electric current.

 _{When calculating the power dissipation of resistive components,}

we can also use one of the two other power equations (they are conversions of the above using Ohm’s law):

]

[

]

[

]

[

W

V

A

I

U

P









R

U

I

R

I

U

P

_









power is is additive for any configuration of circuit: series, parallel, series/parallel, or otherwise.

Maximum Power Transfer Theorem



_{Maximum Power Transfer Theorem states}

that the maximum amount of power will be

dissipated by a load if its total resistance R

is

equal to the source total resistance R

of the

network supplying power.

For maximum power:

The Maximum Power Transfer

Theorem does not assume maximum or even high efficiency, what is more important for AC power distribution.

Example



_{Calculate the total power of the load. Check}

the additivity rule. Calculate R

to get the

maximum power transfer.

Power in AC circuits



_{Instantaneous electric power}

The time varying value of the amplitude of the sinusoidally

oscillating magnitude S and doubling the frequency around

the mean value P. It is measured in voltampere (VA).



_{Active power or Real power}

where ' is an phase shift between current and voltage.

The average value of power (for the period) actually consumed by the device, able to be processed into another form (eg.

mechanical, thermal), this power is always non-negative. It is measured in watt (W).



_{Reactive power}

The value a purely contractual linked to periodic changes in

the energy stored in the reactive components (coil,

capacitor), this power can be positive (induction, where ' >

0) or negative (capacitive, when ' < 0). It is measured in

volt-ampere reactive (var).



_{Complex power}

It is proportional to the RMS values of current and voltage, and marked with the

letter S. Complex power is formally defined as a complex number in the form of

a complex product of the RMS voltage U and coupled current I. It is measured

in volt-ampere (VA). The complex power is a complex sum of real and reactive



_{Apparent power}



_{The power resulting from the amplitude of voltage and}

current, including both the active power and reactive

power. The apparent power can be also calculated as the

magnitude of complex power S. It is measured in

volt-ampere (VA). We can easy calculate the apparent power:

reactive (var).

power triangle



_{We can define the power triangle the}

trigonometric form showing the relation

appearant power to true power and reactive

power. It is presented below:



_{The angle between the real and complex power '}

is a phase of voltage relative to current. It

mean the angle of difference (in degrees)

between current and voltage. The ratio between

real power and apparent power in a circuit is

called the power factor. It’s a measure of the

efficiency of a power distribution. The power

factor is the cosine of the phase angle ' between

the current and voltage cos':

The power factor is by definition a dimensionless and its value is between -1 and 1.

When power factor is equal to 0, the energy flow is entirely reactive. When the power factor is 1, all the energy supplied by the source is consumed by the load.