no tagsLecture 030: Using InductorsSteveSekula, 10 November 2010 (created 8 November 2010)Goals of this lecture:Learn how inductors work in circuits to resist changeDescribe this behavior mathematicallySelf-InductanceLast time, we discussed the idea that a device that stores a magnetic field,like a solenoid, generates a field and thus establishes a flux through itsown area. This device is called an inductor. Any change in that flux - forinstance, the decrease or increase of current through the solenoid - will beresisted. This is called self-inductance.We defined the inductance, , using what should be a linear relationshipbetween the current in the inductor and the flux in the inductor:LWhat happens if the flux changes in response to a change in currentthrough the inductor?The behavior of an inductorLet's imaging an inductor with a steady current passing through it. Nowchange the current suddenly (increase or decrease - it doesn't matter).How does the change in flux relate to the inductance of the device?General Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) LectureNotesL =IÈBGeneral Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...1 of 7 11/10/2010 01:53 PMIf the current changes, the flux changes in our inductor:We know from Faraday's Law that if the flux changes, an EMF will beestablished that OPPOSES the change:And inserting Faraday's Law we find the EMF induced in the inductor:This is the form of Faraday's Law for an inductor with self-inductance .Reminder: the inductance has units of , which is given thespecial name "Henrys" in honor of the American Joseph Henry whoco-discovered magnetic inductance. Typical inductors in electronicapplications have values in the range of micro-Henrys up to severalHenrys.The EMF induced in an inductor when the current changes through theinductor is called back emf, and again it's an EMF that is established toresist the change in current flow. Consider the following situation:Draw a vertical inductor symbol and an arrow pointing downrepresenting currentCurrent is flowing down through an inductor in a steady state.Suddenly the current begin to decrease. What direction is theback-EMF in the inductor (indicate by labeling the two sides of theinductor with a "plus" or "minus", like a battery, to indicate thedirection of the back EMF.Current is flowing down through an inductor in a steady state.Suddenly the current begins to increase. What direction is the backdtdÈB= LdtdIE = ÀdtdÈBE L L= ÀdtdÈB= ÀdtdIL L T =A Á m22General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...2 of 7 11/10/2010 01:53 PMEMF now?This isn't just math. Rapid changes in current through an inductor cancause currents to flow through a circuit that burn out electronic devices inthe circuit. For instance, our TA (Dennis) experienced this on HomecomingWeekend when some buddies of his starting flipping on and off the solenoidvalve to his confetti cannon. This fried the electronics in the device. Why?The solenoid valve, which opens and closes with the flip of a switch andallows compressed air to leave the cannon cylinder, stores energy in amagnetic field which is released when the switch is opened and closed.Done too rapidly, this will fry electronics in the device (which it did).Inductors in CircuitsLet's write down a simple circuit containing a battery, a switch, a resistor,and an inductor. Let's consider the circuit with the switch originally openand no current anywhere in the system.QUESTION: If the switch is closed, what will be the current flow in thecircuit just afterward?ANSWER: inductors resist changes in current - that includes whenthe current is originally zero and then a battery tries to begindriving current through the system. Thus at very, very early timesafter the switch is closed we expect the current to be zero in thecircuit.QUESTION: Long after the switch is closed, what will be the currentflow in the circuit?ANSWER: while the inductor will at first resist the change incurrent, the current should increase over time until it achieves asteady state. At this point, the current flow is no longer changingand the inductor is no longer resisting. Thus for very long times,we expect the inductor to behave like a perfect conductor, offeringno resistance to current flow.At any moment in time, we can analyze the behavior of the circuit usingKirchoff's Loop Law. Starting from just before the battery, the change inGeneral Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...3 of 7 11/10/2010 01:53 PMvoltage through the circuit will be:It might at first seem strange that there is not a minus sign on the lastterm (the inductor EMF) - but, keep in mind that the definition of the EMFcontains a minus sign,We'll let that equation give us the minus sign, rather than adding it in byhand.We can then ask, "How do quantities in such a circuit change with time?"We can answer this question by taking the time derivative of the equation:The battery voltage is constant, so we just have to deal with the other twoterms:We can then substitute for the first-derivative of the current using theinductor form of Faraday's Law:Our final equation for the behavior of the circuit is then:E R 0À I + EL= 0E L L= ÀdtdI ddt(E R )0À I + EL= 0R dtdI=dtdELE L ! L= ÀdtdIÀdtdI= ÀLELGeneral Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Documents/Notebooks/PHY1308...4 of 7 11/10/2010 01:53 PMWe've encountered this equation before, when we discussed RC circuits inclass. It's a first-order differential equation, because it involves both andthe first-derivative of . I'll again demonstrate how one solves a first-orderdifferential equation of this sort (which appear in all kinds of places - cellculture population, etc.).1. Try to collect terms involving a quantity and its differential all onone side.2. Integrate both sides to try to isolate the variable.Integrate both sides (for the range t=0,t and ):3. This involved a natural log. We can "undo" that using the EulerNumber.Raise both sides of the equation into the exponent of the Euler Number.Thus "undoes" the natural logarithm on the right side.À E LRL=dtdELE LE LÀ dt LR=ELdELE E ; L= À0ELÀ t LRZ0td =ZELÀE0ELdELÀ (t ) n(E ) n(ÀE ) LRÀ 0 = lLÀ l0À t n LRÀ lÒELÀE0ÓGeneral Physics - E&M (PHY 1308) - Lecture Notes
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