Chapter 6 Inductance and Capacitance

Chapter 6: Inductance and Capacitance We introduce here the two remaining basic circuit elements: the inductor and the capacitor. The behavior of the inductor is based on the properties of the magnetic field generated in a coil of wire. In fact, the inductor is basically a coil of wire. Ampere’s Law: current in a coil  magnetic field Faraday’s Law: Time-varying magnetic field  induced voltage (emf) In circuits that we will study, the time-varying magnetic field is produced by a changing current. The behavior of the capacitor is based on the properties of the electric field created in a dielectric (non-conductor) placed between two conductors. The capacitor is basically a nonconductor sandwiched between two conductors. Energy can be stored in, but not generated by, an inductor or a capacitor, so these are passive devices. The inductor stores energy in its magnetic field; the capacitor stores energy in its electric field. 6.1 The Inductor Circuit symbol L iL + vL - There is a relationship between current and voltage for an inductor, just as there is for a resistor. However, for the inductor, the voltage is related to the change in the current, as follows. vL  L L iL + vL diL dt This relationship holds when the voltage and current are drawn in the passive sign convention. When they are in the active sign convention, we need a ‘-‘ sign: - vL   L diL dt The relationship between current and voltage involves the time derivative of the current. This is because a changing current produces a changing magnetic field, which induces a voltage. Units: voltage  L  current time  volt  sec  Henry  H  Amp Very Important Points: 1. vL  L diL  if current iL is constant (but NOT necessarily 0), there is no voltage. In other dt words, under constant current conditions, the inductor is a short. 2. An instantaneous change in current would generate an infinite voltage! Therefore we assume (and in reality this is always the case) that in an inductor, there cannot be an instantaneous change in current. Here is what we mean by “instantaneous change in current”: iL  t  di L dt  Here, the current changes from one value to another over a time span of 0 [s] at t1, i.e., instantaneously. This produces a derivate, and hence a voltage, that is infinity large. We can’t have this!! t1 t t1 Current in terms of voltage The current-voltage relationship we discussed above gives the inductor voltage if we know the inductor current. But sometimes we have the inductor voltage and need to find the current So we need to integrate… vL  t   L diL dt vL  t dt   iL  t  Integrate both sides:  t t0 iL  t   iL  t0   iL  t0  LdiL 1 t vL  t dt L t0 Do a little algebra, assume that t0 = 0, and voila! iL  t   1 t vL  t  dt  iL  0  L 0 We have integrated the voltage from an “initial” time t0 to the “final” time t (which is arbitrary). If we know the value of the current at the initial time t0, we can find the current as a function of time. In the last equation above, we assume for simplicity that the initial time is 0. The current-voltage relationship is a first-order differential equation. To solve it (meaning that we find a complete numerical expression for the current as a function of time) we need to know the initial condition iL(t0). This will either be given, or we have to have a way to find it. Power and Energy vL + iL - Not surprisingly, we will want to know about energy stored in the inductor and about power delivered to/from the inductor. In what follows we assume passive sign convention. pabs  vL  iL  iL  L  LiL diL dt diL dw  dt dt Now we can integrate…  w 0 iL dw   LiL diL 0 So the energy stored in an inductor that is carrying a current iL is.. 1 W  LiL2 2 Units units of w   L  iL  2  H  Amp 2  Volt.sec. Amp  Volt  Coul But we know that 1Volt  1 Joul Coul units of w   Joul  So w is an energy, as we expected. Construction: We can make an inductor by wrapping a coil of wire around a core of magnetic material. Modeling: Any physical device that involves a coil of wire can be modeled using inductance. An obvious example is a motor, whose windings have an inductance. More generally, a device with a current-induced magnetic field that interacts like an inductor will have inductance. 6.2 The Capacitor Circuit symbol C There is a relationship between current and voltage for a capacitor, just as there is for a resistor. However, for the capacitor, the current is related to the change in the voltage, as follows. iC  C iC vC + - dvC dt C This relationship holds when the voltage and current are drawn in the passive sign convention. When they are in the active sign convention, we need a ‘-‘ sign: iC  C + vC iC - dvC dt The relationship between current and voltage involves the time derivative of the voltage. This is because a changing voltage produces a changing electric field, which induces a current. Very Important Points: 1. iC  C dvC  if voltage vC is constant (but NOT necessarily 0), there is no current. In other dt words, under constant voltage conditions, the capacitor is an open circuit. 2. An instantaneous change in voltage would generate an infinite current! Therefore we assume (and in reality this is always the case) that in a capacitor, there cannot be an instantaneous change in voltage. What is meant by “instantaneous change in voltage” can be seen by looking at the graph above for instantaneous change in inductor current – just substitute vC for iL. Units units of C = Amp  sec Coul = = Farad [F] Volt Volt Voltage in terms of Current The current-voltage relationship we discussed above gives the capacitor current if we know the capacitor voltage. But sometimes we have the capacitor current and need to find the voltage. So we need to integrate… iC  C dvC dt t vC  t  t0 vC  t0    iC dt    vC  t   vC  t0   vC  t   CdvC 1 t iC dt C t0 1 t iC dt  vC  0  C 0 Power and Energy We will want to know about energy stored in the capacitor and about power delivered to/from the capacitor. In what follows we assume passive sign convention. C iC + - vC pabs  vC  iC pabs  vC  C pabs  CvC Now we integrate… dvC dt dvC dw  dt dt  w 0 t dw   CvC dvC 0 So the energy stored in a capacitor that has a voltage vC across it is 1 w  CvC2 2 Units units of w  C   vC  2  F  Volt 2  Coul  Volt   Joul  Construction: We can make a capacitor by sandwiching an insulator between two conductors. Modeling: Any physical device that involves conducting plates or wires with insulation between them can be modeled using capacitance. Two wires stranded together that connect two devices will have capacitance. Semiconductor devices are often made from some combination of metal and semiconductor layers that have capacitance. More generally, a device with an electric field that interacts like a capacitor will have capacitance. 6.3 Series – Parallel Combinations of Inductance and Capacitance Inductors in Series + vL L1 v1 + i1 - - L2 v2 + L3 - i2 v3 + i3 Since i1  i2  i3 , we have vL  v1  v2  v3  ( L1  L2  L3 ) Leq  L1  L2  L3 diL dt - In general Leq   Ln n Inductors in Parallel iL vL i2 i1 + L1 i3 L2 L3 - Since iL  1 t vL  t  dt  iL  t0  L t0 we have 1 1 1 t iL  i1  i2  i3       vL  t  dt  i1  t0   i2  t0   i3  t0  t  L1 L2 L3  0 iL  1 Leq  v  t  dt  i t  t t0 L L 0 1 1 1 Leq       L1 L2 L3  iL  t0   i1  t0   i2  t0   i3 t0  In general 1 1  Leq n Ln Capacitors in Series + vC C2 C1 + v1 - - C3 + v2 vC  + v3 - - 1 t iC dt  vC  t0  C t0  1 1 1  t vC       iC dt  v1  t0   v2  t0   v3  t0  t  C1 C2 C3  0 1 1 1 1    Ceq C1 C2 C3 In general 1 1  Ceq n Cn Capacitors in Parallel iC + vC i1 C1 i2 C2 i3 C3 - dvC dt iC  C iC  i1  i2  i3   C1  C2  C3  iC  Ceq dvC dt In general Ceq   Cn n dvC dt