2Topics Covered in Chapter 19 19-7: Transformer Ratings19-8: Impedance Transformation19-9: Core Losses19-10: Types of Cores19-11: Variable Inductance19-12: Inductances in Series or Parallel19-13: Energy in Magnetic Field of Inductance19-14: Stray Capacitive and Inductive Effects19-15: Measuring and Testing InductorsMcGraw-Hill
319-1: Induction by Alternating Current Induced voltage is the result of flux cutting across a conductor.This action can be produced by physical motion of either the magnetic field or the conductor.Variations in current level (or amplitude) induces voltage in a conductor because the variations of current and its magnetic field are equivalent to the motion of the flux.Thus, the varying current can produce induced voltage without the need for motion of the conductor.The ability of a conductor to induce voltage in itself when the current changes is called self-inductance, or simply inductance.
419-1: Induction by Alternating Current Induction by a varying current results from the change in current, not the current value itself. The current must change to provide motion of the flux.The faster the current changes, the higher the induced voltage.
619-1: Induction by Alternating Current Point C has maximum current and maximum counterclockwise flux.At point D there is less flux than at C. Now the field is collapsing because of reduced current.
719-1: Induction by Alternating Current Point E with zero current, there is no magnetic flux. The field can be considered collapsed into the wire.The next half-cycle of current allows the field to expand and collapse again, but the directions are reversed.When the flux expands at points F and G, the field lines are clockwise.From G to H and I, this clockwise field collapses into the wire.
819-1: Induction by Alternating Current Characteristics of inductance are important in:AC circuits: In these circuits, the current is continuously changing and producing induced voltage.DC circuits in which the current changes in value: DC circuits that are turned off and on (changing between zero and its steady value) can produce induced voltage.
919-2: Self-Inductance LThe symbol for inductance is L, for linkages of magnetic flux.VL is in volts, di/dt is the current change in amperes per second.The henry (H) is the basic unit of inductance.One henry causes 1 V to be induced when the current is changing at the rate of 1 A per second.L =VLdi / dt
10ExamplesThe current in an inductor changes from 12 to 16 A in 1 s. How much is the di/dt rate of current change in amperes per second?The current in an inductor changes by 50 mA in 2 µs. How much is the di/dt rate of current change in amperes per second?How much is the inductance of a coil that induces 40 V when its current changes at the rate of 4 A/s?How much is the inductance of a coil that induces 1000 V when its current changes at the rate of 50 mA in 2 µs?
1119-2: Self-Inductance L Inductance of Coils The inductance of a coil depends on how it is wound.A greater number of turns (N) increases L because more voltage can be induced (L increases in proportion to N).More area enclosed by each turn increases L.The L increases with the permeability of the core.The L decreases with more length for the same number of turns, as the magnetic field is less concentrated.
1319-2: Self-Inductance L Typical Coil Inductance Values Air-core coils for RF applications have L values in millihenrys (mH) and microhenrys (μH).Practical inductor values are in these ranges:1 H to 10 H (for iron-core inductors)1 mH (millihenry) = 1 × 10-3 H1 µH (microhenry) = 1 × 10-6 H
1419-3: Self-Induced Voltage vL ( )didtvLL=Formula:Induced voltage is proportional to inductance (L).Induced voltage is proportional to the rate of current change:
16ExamplesHow much is the self-induced voltage across a 4-H inductance produced by a current change of 12 A/s?The current through a 200-mH L changes from 0 to 100 mA in 2 µs. How much is vL ?
1719-4: How vL Opposes a Change in Current Lenz’ Law states that the induced voltage produces current that opposes the changes in the current causing the induction.The polarity of vL depends on the direction of the current variation di.When di increases, vL has polarity that opposes the increase in current.When di decreases, vL has opposite polarity to oppose the decrease in current.In both cases, the change in current is opposed by the induced voltage.
1819-5: Mutual Inductance LM Mutual inductance (LM) occurs when current flowing through one conductor creates a magnetic field which induces a voltage in a nearby conductor.Two coils have a mutual inductance of 1 H when a current change of 1A/s induces 1 V in the other coil.Unit: Henrys (H)Formula:LkM=12
1919-5: Mutual Inductance LM Coefficient of coupling, k, is the fraction of total flux from one coil linking another coil nearby.Specifically, the coefficient of coupling isk = flux linkages between L1 and L2 divided byflux produced by L1There are no units for k, because it is a ratio of two values of magnetic flux. The value of k is generally stated as a decimal fraction.
2019-5: Mutual Inductance LM The coefficient of coupling is increased by placing the coils close together, possibly with one wound on top of the other, by placing them parallel, or by winding the coils on a common core.A high value of k, called tight coupling, allows the current in one coil to induce more voltage in the other.Loose coupling, with a low value of k, has the opposite effect.Two coils may be placed perpendicular to each other and far apart for essentially zero coupling to minimize interaction between the coils.
2219-5: Mutual Inductance LM Calculating LMMutual inductance increases with higher values for primary and secondary inductances.LMwhere L1 and L2 are the self-inductance values of the two coils, k is the coefficient of coupling, and LM is the mutual inductance.
2619-6: Transformers Turns Ratio The ratio of the number of turns in the primary to the number in the secondary is the turns ratio of the transformer.Turns ratio equals NP/NS.where NP equals the number of turns in the primary and NS equals the number of turns in the secondary.The turns ratio NP/NS is sometimes represented by the lowercase letter a.
2719-6: Transformers The voltage ratio is the same as the turns ratio: VP / VS = NP / NSVP = primary voltage, VS = secondary voltageNP = number of turns of wire in the primaryNS = number of turns of wire in the secondaryWhen transformer efficiency is 100%, the power at the primary equals the power at the secondary.Power ratings refer to the secondary winding in real transformers (efficiency < 100%).
2919-6: TransformersCurrent Ratio is the inverse of the voltage ratio. (That is voltage step-up in the secondary means current step-down, and vice versa.)The secondary does not generate power but takes it from the primary.The current step-up or step-down is terms of the secondary current IS, which is determined by the load resistance across the secondary voltage.
3119-6: TransformersTransformer efficiency is the ratio of power out to power in.Stated as a formula% Efficiency = Pout/Pin x 100Assuming zero losses in the transformer, power out equals power in and the efficiency is 100%.Actual power transformers have an efficiency of approximately 80 to 90%.
3319-6: Transformers Loaded Power Transformer 1:6 Calculate VS from the turns ratio and VP.Use VS to calculate IS:IS = VS/RLUse IS to calculate PS:PS = VS x ISUse PS to find PP:PP = PSFinally, IP can be calculated:IP = PP/VP20:1
3419-6: Transformers Autotransformers An autotransformer is a transformer made of one continuous coil with a tapped connection between the end terminals.An autotransformer has only three leads and provides no isolation between the primary and secondary.
3519-7: Transformer Ratings Transformer voltage, current, and power ratings must not be exceeded; doing so will destroy the transformer.Typical Ratings:Voltage values are specified for primary and secondary windings.CurrentPower (apparent power – VA)Frequency
3619-7: Transformer Ratings Voltage RatingsManufacturers always specify the voltage rating of the primary and secondary windings.Under no circumstances should the primary voltage rating be exceeded.In many cases, the rated primary and secondary voltages are printed on the transformer.Regardless of how the secondary voltage is specified, the rated value is always specified under full load conditions with the rated primary voltage applied.
3719-7: Transformer Ratings Current RatingsManufacturers usually specify current ratings only for secondary windings.If the secondary current is not exceeded, there is no possible way the primary current can be exceeded.If the secondary current exceeds its rated value, excessive I2R losses will result in the secondary winding.
3819-7: Transformer Ratings Power RatingsThe power rating is the amount of power the transformer can deliver to a resistive load.The power rating is specified in volt-amperes (VA).The product VA is called apparent power, since it is the power that is apparently used by the transformer.The unit of apparent power is VA because the watt is reserved for the dissipation of power in a resistance.
3919-7: Transformer Ratings Frequency RatingsTypical ratings for a power transformer are 50, 60, and 400 Hz.A power transformer with a frequency rating of 400 Hz cannot be used at 50 or 60 Hz because it will overheat.Many power transformers are designed to operate at either 50 or 60 Hz.Power transformers with a 400-Hz rating are often used in aircraft because these transformers are much smaller and lighter that 50- or 60-Hz transformers.
4019-8: Impedance Transformation Transformers are often used to change or transform a secondary load impedance to a new value as seen by the primary.The secondary load impedance is said to be reflected back into the primary and is called a reflected impedance.The reflected impedance of the secondary may be stepped up or down.An equation for the reflected impedance is:ZNPS=æèçöø÷2secondaryprimary
4119-8: Impedance Transformation If the turns ratio NP/NS is less than 1, ZS will be stepped down in value.Transformer impedance matching is related to the turns ratio:NZPS=
4319-8: Impedance Transformation Impedance Matching for Maximum Power TransferTransformers are used when it is necessary to achieve maximum transfer of power from a generator to a load when the generator and load impedances are not the same.This application of a transformer is called impedance matching.
4819-9: Core Losses Eddy Currents Eddy currents are induced in the iron core of an inductor or transformer.Eddy currents raise the temperature of the core. Wasted power is dissipated as heat.Losses increase with frequency.
5019-9: Core Losses Hysteresis losses The hysteresis losses result from the additional power needed to reverse the magnetic field in magnetic materials in the presence of alternating current.The greater the frequency, the more hysteresis loss.Air-core coilsAir has practically no losses from eddy currents or hysteresis.The inductance for small coils with an air core is, however, limited to low values (e.g. mH or μH).
5119-10: Types of CoresLosses can be reduced by using a laminated core or a powered-iron core.The type of steel used can reduce hysteresis losses.The most common types of insulation are:Laminated corePowdered iron coreFerrite core
5219-10: Types of Cores Laminated core A shell-type core formed with a group of individual laminations.Each laminated section is insulated by a very thin coating of iron oxide.Powdered ironConsists of individual insulated granules pressed into one solid form called a slug.Ferrite coreSynthetic ceramic materials that are ferromagnetic.
5719-12: Inductances in Series or Parallel With no mutual coupling:For series circuits, inductances add just like resistances.For parallel circuits, inductances combine according to a reciprocal formula as with resistances.LT = L1 + L2 + L etc.LEQ =1etc.+L3L2L1
5819-12: Inductances in Series or Parallel Series Coils for LMLM depends on the amount of mutual coupling and on whether the coils are connected series-aiding or series-opposing.Series-aiding means that the common current produces the same direction of magnetic field for the two coils.The series-opposing connection results in opposite fields.
6019-12: Inductances in Series or Parallel To calculate the total inductance of two coils that are series-connected and have mutual inductance,LT = L1 + L2 ± 2LMThe mutual inductance LM is plus, increasing the total inductance, when the coils are series-aiding, or minus when they are series-opposing to reduce the total inductance.
6119-13: Energy in Magnetic Field of Inductance The magnetic flux of current in an inductance has electric energy supplied by the voltage source producing the current.The energy is stored in the field, since it can do the work of producing induced voltage when the flux moves.The amount of electric energy stored isEnergy = ε = ½ LI2The factor of ½ gives the average result of I in producing energy.
6219-14: Stray Capacitive and Inductive Effects Stray capacitive and inductive effects can occur in all circuits with all types of components.A capacitor has a small amount of inductance in the conductors.A coil has some capacitance between windings.A resistor has a small amount of inductance and capacitance.
6319-14: Stray Capacitive and Inductive Effects A practical case of problems caused by stray L and C is a long cable used for rf signals.If the cable is rolled in a coil to save space, a serious change in the electrical characteristic of the line will take place.For twin-lead or coaxial cable feeding the antenna input to a television receiver, the line should not be coiled because the added L or C can affect the signal.
6719-15: Measuring and Testing Inductors The capacitor-inductor analyzer can also test the quality (Q) of the inductor by using a ringing test.Another test instrument that is capable of measuring inductance L, capacitance C, and resistance R, is an LCR meter.A typical LCR meter is shown in Fig (previous slide).This is a handy piece of test equipment, however, most LCR meters only measure the value of a component.
6919-15: Measuring and Testing Inductors Fig : The internal dc resistance ri of a coil is in series with its inductance L.A coil has dc resistance equal to the resistance of the wire used in the winding.As shown in Fig , the dc resistance and inductance of a coil are in series.Although resistance has no function in producing induced voltage, it is useful to know the dc coil resistance because if it is normal, usually the inductance can also be assumed to have its normal value.