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Coupling coils The coupling occurs when two coils are placed near each other (See Fig.3.1). The first coil current I 1 gives magnetic field B 1. When the distance between two coils are small, some of the magnetic field will pass through coil 2. Variation of I 1 with time, induce electromagnetic field associated with the changing magnetic flux in the second coil:

Mutual inductance

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. 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 opposing inductors).

Time-domain and Frequency-domain Analysis V1V1 V2V2 I1I1 I2I2 jL1jL1 jL2jL2 jMjM a) Time-domain circuit b) Frequency-domain circuit

Energy in a Coupled Circuit  The total energy w stored in a mutually coupled inductor is:  Positive sign is selected if both currents ENTER or LEAVE the dotted terminals.  Otherwise we use Negative sign.

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 the node joining the branches with coupled coils.

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

EXAMPLES

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 transformer in circuit theory

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 1 and R 2 account for losses in the coils.  The coils are named as PRIMARY and SECONDARY.

Reflected Impedance for Linear Transformers Secondary impedance seen from the primary side is the Reflected Impedance.  Let us obtain the input impedance as seen from the source, ZRZR

Equivalent T Circuit for Linear Transformers  The coupled transformer can equivalently be represented by an EQUIVALENT T circuit using UNCOUPED INDUCTORS. 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. 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): 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 l is equal to the source total resistance R s 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 w 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 power:

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). or

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.