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EXAMPLE 168 Impedance Transformer
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Find the equivalent load impedance seen by the voltage source (ie, re ected from secondary to primary) for the transformer of Figure 1635
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V1 ~ N1 N2
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Known Quantities: Transformer turns ratio, Find: Re ected impedance, Z2
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Figure 1635 Ideal transformer
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Assumptions: Assume an ideal transformer Analysis: By de nition, the load impedance is equal to the ratio of secondary phasor
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voltage and current: Z2 = V2 I2
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To nd the re ected impedance we can express the above ratio in terms of primary voltage
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Part III
Electromechanics
and current: Z2 = V1 V2 1 V1 = = 2 I2 I1 I1
where the ratio V1 /I1 is the impedance seen by the source at the primary coil, that is, the re ected load impedance seen by the primary (source) side of the circuit Thus, we can write the load impedance, Z2 , in terms of the primary circuit voltage and current; we call this the re ected impedance, Z2 : Z2 = 1 V1 1 1 = 2 Z1 = 2 Z 2 2 I 1
V1 ~ 2Z2 N1 N2 V2
Thus, Z2 = 2 Z 2 Figure 1636 depicts the equivalent circuit with the load impedance re ected back to the primary
Comments: The equivalent re ected circuit calculations are convenient because all
circuit elements can be referred to a single set of variable (ie, only primary or secondary voltages and currents)
Check Your Understanding
169 The high-voltage side of a transformer has 500 turns, and the low-voltage side has 100 turns When the transformer is connected as a step-down transformer, the load current is 12 A Calculate: (a) the turns ratio ; (b) the primary current 1610 Calculate the turns ratio if the transformer in Check Your Understanding 169 is used as a step-up transformer 1611 The output of a transformer under certain conditions is 12 kW The copper losses are 189 W and the core losses are 52 W Calculate the ef ciency of this transformer 1612 The output impedance of a servo ampli er is 250 The servomotor that the ampli er must drive has an impedance of 25 Calculate the turns ratio of the transformer required to match these impedances
ELECTROMECHANICAL ENERGY CONVERSION
From the material developed thus far, it should be apparent that electromagnetomechanical devices are capable of converting mechanical forces and displacements to electromagnetic energy, and that the converse is also possible The objective of this section is to formalize the basic principles of energy conversion in electromagnetomechanical systems, and to illustrate its usefulness and potential for application by presenting several examples of energy transducers A transducer is a device that can convert electrical to mechanical energy (in this case, it is often called an actuator), or vice versa (in which case it is called a sensor) Several physical mechanisms permit conversion of electrical to mechanical energy and back, the principal phenomena being the piezoelectric effect,3 consisting of the generation of a change in electric eld in the presence of strain in
3 See
Focus on Measurements: Charge Ampli ers in 12
16
Principles of Electromechanics
certain crystals (eg, quartz), and electrostriction and magnetostriction, in which changes in the dimension of certain materials lead to a change in their electrical (or magnetic) properties Although these effects lead to many interesting applications, this chapter is concerned only with transducers in which electrical energy is converted to mechanical energy through the coupling of a magnetic eld It is important to note that all rotating machines (motors and generators) t the basic de nition of electromechanical transducers we have just given Forces in Magnetic Structures Mechanical forces can be converted to electrical signals, and vice versa, by means of the coupling provided by energy stored in the magnetic eld In this subsection, we discuss the computation of mechanical forces and of the corresponding electromagnetic quantities of interest; these calculations are of great practical importance in the design and application of electromechanical actuators For example, a problem of interest is the computation of the current required to generate a given force in an electromechanical structure This is the kind of application that is likely to be encountered by the engineer in the selection of an electromechanical device for a given task As already seen in this chapter, an electromechanical system includes an electrical system and a mechanical system, in addition to means through which the two can interact The principal focus of this chapter has been the coupling that occurs through an electromagnetic eld common to both the electrical and the mechanical system; to understand electromechanical energy conversion, it will be important to understand the various energy storage and loss mechanisms in the electromagnetic eld Figure 1637 illustrates the coupling between the electrical and mechanical systems In the mechanical system, energy loss can occur because of the heat developed as a consequence of friction, while in the electrical system, analogous losses are incurred because of resistance Loss mechanisms are also present in the magnetic coupling medium, since eddy current losses and hysteresis losses are unavoidable in ferromagnetic materials Either system can supply energy, and either system can store energy Thus, the gure depicts the ow of energy from the electrical to the mechanical system, accounting for these various losses The same ow could be reversed if mechanical energy were converted to electrical form
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