This chapter has extended the material presented in Chapter 9 on inductance into the realm of magnetic circuits. We began with the concept of electromagnetic induction, which in simple terms states that if a conductor experiences changing magnetic lines of force, then a voltage will be induced in that conductor. This concept is exploited by an array of different devices including electric motors, generators, relays, loudspeakers, microphones, and other transducers and sensors. For example, dynamic loudspeakers and microphones both make use of a coil of wire suspended in a fixed magnetic field. In the case of the loudspeaker, the signal fed into the coil creates a changing magnetic field that interacts with the permanent magnet's field and causes an attached diaphragm to vibrate against the air, which in turn creates sound. A dynamic microphone works in an opposite manner. Sound waves cause a diaphragm to vibrate back and forth. This diaphragm is attached to a coil, and the motion of the coil within the permanent magnet's field causes a voltage to be induced in the coil. This voltage can then be amplified by electronic circuitry.

Magnetic circuits are constructed by creating one or more coils of wire wrapped around a core that is made of a ferromagnetic material such as steel. A key part of analyzing magnetic circuits is Hopkinson's law (also called Rowland's law), which is the magnetic circuit version of Ohm's law. In this analogy, magnetic flux is likened to current flow, magnetic reluctance stands in for resistance, and magnetomotive force is analogous to electromotive force (i.e., voltage). Further, the magnetomotive force is the product of the number of turns in a coil and the current through said coil, or \(NI\). When current is passed through the coil, a magnetizing force, \(H\), and an associated magnetic flux are created. Given this, the core material will have a corresponding flux density, \(B\). The linkage between \(B\) and \(H\) is found through a \(BH\) curve for that particular material. \(BH\) curves are non-linear for ferromagnetic materials. They also exhibit hysteresis, meaning that the recent operational history of the material plays a role in its current state. Every element of the core has a certain reluctance and thus exhibits an analogous “voltage drop”. These “drops” must add up to the \(NI\) “rises” of the coil(s), in a KVL analogy.

Transformers use one or more primary coils to transform an input voltage into one more output voltages appearing on secondary coils. The ratio between the number of turns of the primary coil to the number of turns of the secondary coil is called the turns ratio, \(N\). The secondary voltage is equal to the primary voltage divided by \(N\) while the secondary current is equal to the primary current times \(N\). If \(N\) is greater than one, we have a step-down transformer (reducing the voltage), and if \(N\) is less than one, we have a step-up transformer (increasing the voltage). Ideally, transformers do not dissipate power. Thus, they do not have a power rating, but instead have a VA (volt-amps) rating to indicate their capacity. Transformers can also be used for impedance matching to increase the efficiency of a system. The impedance seen by the source on the primary is equal to the secondary impedance times the square of the turns ratio. This is called the reflected impedance.

## Review Questions

1. Describe the concept of electromagnetic induction.

2. Describe Ohm's law for magnetic circuits (Hopkinson's or Rowland's law).

3. Describe hysteresis.

4. Define residual magnetism and coercivity.

5. Outline the operation of a relay.

6. What are some of the uses of transformers?

7. What is turns ratio? How does it affect voltage and current in the primary and secondary?

8. What is the difference between a step-up and a step-down transformer?

9. What is reflected impedance?