Inductors and Transformers AC Circuits I AC Circuits I

Inductance The ability of a component with a changing current to induce a voltage across itself or a nearby circuit by generating a changing magnetic field Inductor – a component designed to provide a specific measure of inductance (coil) AC Circuits I

Inductance The Effect of Varying Current on a Magnetic Field When current passes through an inductor (coil), magnetic flux is generated Flux density is found as AC Circuits I I B where B = the flux density, in webers per square meter (Wb/m2) m = the permeability of the core material NI = the ampere-turns product ℓ = the length of the coil, in meters

Inductance When an AC current is applied through the coil, The magnetic field expands as current increases The magnetic field contracts as current decreases AC Circuits I

Faraday’s Laws of Induction Law 1: To induce a voltage across a wire, there must be a relative motion between the wire and the magnetic field. Law 2: The voltage induced is proportional to the rate of change in magnetic flux encountered by the wire. Law 3: When a wire is cut by 108 perpendicular lines of force (1Wb) per second, 1 V is induced across that wire. AC Circuits I Self-inductance

Lenz’s Law 1834, Heinrich Lenz – derived the relationship between a magnetic field and the voltage it induces Lenz’s Law – an induced voltage always opposes its source AC Circuits I

Lenz’s Law An increase in the inductor current causes the magnetic field to expand. As the magnetic field expands, it cuts through the coil, inducing a voltage. The polarity of the voltage (CEMF) opposes the increase in current. AC Circuits I

Lenz’s Law An decrease in the inductor current causes the magnetic field to collapse. As the magnetic field collapses, it cuts through the coil, inducing a voltage across the component. The polarity of the voltage opposes the decrease in current. AC Circuits I

Induced Voltage can be found as where vL = the instantaneous value of induced voltage L = the inductance of the coil, measured in henries (H) = the instantaneous rate of change in inductor current (in amperes per second) AC Circuits I

Example If AC Circuits I

Unit of Measure – henry (H) Inductance is measured in volts per rate of change in current When a change of 1A/s induces 1V across an inductor, the amount of inductance is said to be 1 H AC Circuits I

Example If current is changing at a rate of 45mA/s through a 10mH inductor, Calculate the induced voltage? AC Circuits I

Make your own inductor The inductance of a coil can be characterized by the following equation: where L =inductance m = the permeability of the core material N2 = the square of the number of turns A = cross-sectional area of the inductor core in cm2 ℓ = the length of the coil, in meters AC Circuits I

Summary AC Circuits I

Overview What is an inductor? What is the relationship between flux density and current? How many Laws did Faraday postulate and what are they? What is Lenz’s Law? What is the unit measure of inductance? Two inductors have identical physical characteristics except that one of them has an air core and the other has an iron core. Which one will have the higher value of inductance? AC Circuits I

The Phase Relationship Between Inductor Current and Voltage Sine-Wave Values of reaches its maximum value when i = 0 AC Circuits I

The Phase Relationship Between Inductor Current and Voltage The Phase Relationship Between Inductor Voltage and Current Voltage leads current by 90° Current lags voltage by 90° Ideal Inductors do not dissipate power!!!! They store energy in a magnetic field AC Circuits I

Connecting Inductors in Series and Parallel Series-Connected Inductors where Ln = the highest-numbered inductor in the circuit Parallel-Connected Inductors AC Circuits I where Ln = the highest-numbered inductor in the circuit

Mutual Inductance When one inductor is placed in close proximity to another, the flux produced by each coil can induce a voltage across the other Energy is transferred from one coil to another i.e. coupled AC Circuits I

Mutual Inductance (Continued) Amount of mutual inductance: Coefficient of Coupling (k) – a measure of the degree of coupling that takes place between two or more coils where 1 = the amount of flux generated by L1 2 = the amount of 1 that passes through L2 at a 90° angle to the turns of the coil AC Circuits I

Mutual Inductance (Continued) The Effects of Mutual Inductance on LT AC Circuits I

Inductive Reactance (XL) Inductive Reactance (XL) – the opposition (in ohms) that an inductor presents to a changing current Calculating the Value of XL AC Circuits I Inductors oppose Current. For the circuit shown, But measured resistance is 0.21Ω!!!! How come!!!

XL and Ohm’s Law Example: Calculate the total current below AC Circuits I

Resistance, Reactance and Impedance Resistance is a static value. real resistance Reactance is a dynamic value Reactance is a function of frequency Reactance is an imaginary resistance Resistance and Reactance are similar…both measured in ohms They however cannot be added together directly Must combine them into an impedance (Z) Impedance is the total opposition to current in an ac circuit, consisting of resistance and/or reactances. More on this in the next chapter AC Circuits I

Inductive Reactance (XL) Series and Parallel Reactances AC Circuits I

Review What is mutual inductance? What do we mean when we say two components are coupled? What is the Coefficient of Coupling (k) ? What is the opposition of current provided by the inductor? and how would you measure it ? Why is the resistance of an inductor typically low? What is impedance ? Show how the units of measure in 2fL resolve themselves to yield a result measured in ohms. AC Circuits I

Transformers Nope, not this kind of transformer!! AC Circuits I

Transformer A two-coil component that uses electromagnetic induction to pass an ac signal from its input to its output while providing dc isolation between the two AC Circuits I

Transformer Construction and Symbols Construction - Two coils Primary – input Secondary – output Schematic Symbols AC Circuits I

Transformer Classifications AC Circuits I

Dot Notation AC Circuits I

Transformer Operation Changing magnetic field in the primary windings induces a voltage in the secondary windings Primary and secondary windings are not physically connected AC Circuits I

Turns Ratio AC Circuits I

Voltage and Current Transformer Relations Voltage in the secondary winding is determined by the primary voltage and turns ratio For an ideal transformer power in the primary winding is completely transfered to the secondary winding : Therefore current varies inversely with turns ratio AC Circuits I

Voltage and Current Transformer Relations If a transformer as a 10:1 winding ratio (stepdown) Voltage in secondary winding is 10x smaller than voltage in primary winding Current in secondary winding is 10x larger than current in primary winding If a transformer as a 1:10 winding ratio (stepdown) Voltage in secondary winding is 10x larger than voltage in primary winding Current in secondary winding is 10x smaller than current in primary winding AC Circuits I

Examples Determine the VS (Secondary winding voltage) for a transformer whose NP/NS ratio is 7:1. Assume that VP (Primary winding voltage is 120Vac(VRMS) Determine the secondary current (IS) of a transformer whose IP is 100mA ,VP is 120VRMS and whose whose NP/NS ratio is 15:1. AC Circuits I

Power Transfer Ideal Conditions: PP = PS or IPVP = ISVS In Practice: Secondary power is always slightly lower because of a number of power losses Losses Copper Loss (I2R loss) Resistance of the copper wire used in windings Loss Due to Eddy Currents Magnetic flux induces a current in the iron core called an eddy current. This current travels in a circular through the core. Resistance of the core causes further power loss Hysteresis Loss Energy expended to overcome Hysteresis loss AC Circuits I

Transformers AC Circuits I

Primary Impedance (ZP) Proportional to the square of the turns ratio Can be derived from current and voltage relations AC Circuits I where ZS = the total opposition to current in the secondary (generally assumed to equal the opposition provided by the load)

Transformer As An Impedance Matching Circuit Need to ensure that maximum power is delivered to the load According to Maximum Power Transfer (MPT) this only happens when RS = RL How can we achieve Maximum power transfer when RS ≠ RL ? Use a Transformer as an impedance matching circuit (buffer)! Set ZP to RS and set ZS to RL. The Transformer that will ensure MPT is one whose primary to secondary ratios satisfy: AC Circuits I

Transformer As An Impedance Matching Circuit - Example Z P = Zin = RS = 100 ZS = Zout = RL = 4 NP:NS = 5:1 AC Circuits I

Center-Tapped Transformer Voltage from S1 to center or S2 to center is half VS (secondary winding voltage). AC Circuits I

Inductors - Apparent Power (PAPP) Energy in an inductor is actually stored in the electromagnetic field generated by the inductor Energy not dissipated Called reactive power – units of measure: volt-ampere-reactive (VAR) Power is dissipated only through resistance – winding resistance, RW Energy dissipated as heat Called true power – units of measure: watts (W) Apparent Power – the combination of resistive (true) and reactive (imaginary) power – units of measure: volt-amperes (VA) Example on board

Inductor Quality (Q) Is a figure of merit that indicates how close the inductor comes to the power characteristics of an ideal component where PX = the reactive power of the component, measured in VARs PRw = the true power dissipation of the component, measured in watts

Inductor Quality (Continued)

Types of Inductors Air-Core: low Q Iron-core: higher Q, limited to low frequencies due to power losses Ferrite Core: highest Q, used in higher frequencies, low power loss

Types of Inductors Toroids

Types of Inductors Chokes