1. Inductors and Transformers
AC Circuits I
AC Circuits I

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Example If
AC Circuits I

11. 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

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

13. 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

14. Summary
AC Circuits I

15. 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

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

17. 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

18. 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

19. 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

20. 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 ° angle to the turns of the coil
AC Circuits I

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

22. 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!!!

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

24. 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

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

26. 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

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

28. 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

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

30. Transformer Classifications
AC Circuits I

31. Dot Notation
AC Circuits I

32. 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

33. Turns Ratio
AC Circuits I

34. 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

35. 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

36. 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

37. 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

38. Transformers
AC Circuits I

39. 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)

40. 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

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

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

43. 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

44. 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

45. Inductor Quality (Continued)

46. 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

47. Types of Inductors
Toroids

48. Types of Inductors
Chokes