2.2 Circuit Equations

Chapter 2.2 Circuit Equations

Physics Lecture Notes – Phys 395 Electronics Book
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Physics Lecture Notes – Phys 395 Electronics Book

  • CHAPTER 2. ALTERNATING CURRENT CIRCUITS23induced EMF. So unlike the capacitor which behaves like an open-circuit in DC circuits, aninductor behaves like a short-circuit in DC circuits.Applications using inductors are less common than those using capacitors, but inductorsare very common in high frequency circuits. We will again skip over the unpleasantness –that non-ideal inductors have some resistance and some capacitance.Inductors are never pure inductances because there is always some resistance inand some capacitance between the coil windings. When choosing an inductor(occasionally called a choke) for a specific application, it is necessary to considerthe value of the inductance, the DC resistance of the coil, the current-carryingcapacity of the coil windings, the breakdown voltage between the coil and theframe, and the frequency range in which the coil is designed to operate. Toobtain a very high inductance it is necessary to have a coil of many turns. Theinductance can be further increased by winding the coil on a closed-loop iron orferrite core. To obtain as pure an inductance as possible, the DC resistance ofthe windings should be reduced to a minimum. This can be done by increasingthe wire size, which of course, increases the size of the choke. The size of the wirealso determines the current-handling capacity of the choke since the work donein forcing a current through a resistance is converted to heat in the resistance.Magnetic losses in an iron core also account for some heating, and this heatingrestricts any choke to a certain safe operating current. The windings of the coilmust be insulated from the frame as well as from each other. Heavier insulation,which necessarily makes the choke more bulky, is used in applications wherethere will be a high voltage between the frame and the winding.The lossessustained in the iron core increases as the frequency increases. Large inductors,rated in henries, are used principally in power applications. The frequency inthese circuits is relatively low, generally 60 Hz or low multiples thereof. In high-frequency circuits, such as those found in FM radios and television sets, verysmall inductors (of the order of microhenries) are frequently used.2.2Circuit EquationsRecall that voltage V is related to current I, via the passive DC circuit element resistanceR, by Ohm’s law V = IR. Analogously, the change in voltage and change in current arerelated to the current and voltage, via the passive AC circuit elements C and L,bydVdt=ICandV = LdIdt.(2.4)Applying the above three equations, along with Kirchoff’s loop rule, to AC circuits resultsin a set of differential equations. These differential equations are linear with constant coeffi-cients and can easily be solved for Q(t), I(t), and V (t). In general the solutions will consistof a transient response and a steady-state response. The transient response describes thereturn to equilibrium after the EMFs change suddenly. The steady-state response describesthe long term behaviour when the circuit is driven by a sinusoidal source.