Lecture 24 April 5 th, 2005 Magnetic Circuits & Inductance Reading: Boylestad s Circuit Analysis, 3 rd Canadian Edition Chapter 11.1-11.5, Pages 331-338 Chapter 12.1-12.4, Pages 341-349 Chapter 12.7-12.9, Pages 363 367 Note: We do NOT cover RL Transients (12.5-6) Assignment: Assignment #11 Due: N/A
Magnetic Circuits & Inductance Electric Circuit Magnetic Circuit Name Equation Units Units Equation Name Electric Current A Wb Magnetic Flux Electromotive Force (emf) Resistance V I = R I Φ = R = V V At I NI Magneto-motive force (mmf) Ω At/Wb Reluctance R = ρ l A l R = µ A Permittivity ε 0 F/m Wb/Atm µ 0 Permeability Capacitance C Farads Henries L Inductance τ RC seconds seconds R/L τ Electric flux density D Ψ WB/m 2 Φ Magnetic Flux density = Β = A A Electric Field V V/m At/m Intensity E = H = I Magnetic Field Intensity d l Aside: The following 3 constants are exact: Permittivity of free space: ε 0 = 8.85418781762 x 10-12 F/m Permeability of free space: µ 0 = 4π x 10-7 H/m Speed of light: c = 299 792 458 m/s Now ε 0 µ 0 =1/c 2 Coincidence??? April 5 th, 2005 Magnetic Circuits & Inductance 2
Magnetic Circuits & Inductance: In what follows a new circuit element for use in E.P. 155.3 is going to be introduced. It is called an inductor. You will notice during the development that inductance is in some ways similar to previous circuit elements while at the same time having some unusual characteristics! Magnetic circuit analysis is difficult and many assumptions are made. Magnetism: You probably have noticed at some point in your life that opposite poles of magnets attract (and that similar poles of magnets repel). This implies that there exists a magnetic field, analogous to an electric field, responsible for the force. April 5 th, 2005 Magnetic Circuits & Inductance 3
It is noted that magnetic field lines always go from the north pole of a magnet to the south pole of the magnet. Unlike electric field lines, which start on one charge and terminate on another charge, they do not start and end on each pole but are closed loops. Some materials, called ferromagnetic materials, have natural magnetic properties; the classic example is iron. They can affect neighbouring magnetic fields by capturing the magnetic field and causing it to bend. Of course there are other materials, such as plastic, that possess no magnetic properties and have no effect on the magnetic field lines. April 5 th, 2005 Magnetic Circuits & Inductance 4
Electromagnetism: It was noticed in past history that moving charge generates a magnetic field and therefore currents generate a magnetic field. This is the concept that electric motors operate on. The input electrical energy generates a current that creates a magnetic field that interacts with permanent magnets to turn the motor. The Permanent Magnet Moving Coil (PMMC) discussed as part of your E.P. 155.3 meter lectures works on this concept (although on a much smaller scale than an electric motor). The magnetic field strength is given the symbol B and has units of Teslas (T). B is also called the magnetic flux density (recall the previous information on electric flux density). Thus 1 Tesla = (1 Weber/m 2 ) = (1 Wb/m 2 ). Like the electric field E, the B field is a vector field so at all points in space it has a magnitude and direction. Unlike the electric field which is set up due to stationary charge, the magnetic field is set up due to moving charge. April 5 th, 2005 Magnetic Circuits & Inductance 5
Current Carrying Wire: Consider a current carrying wire (current is moving charge). p r I At point p, a perpendicular distance r from a long current carrying wire, the magnetic field strength (magnetic flux density) is µ I B = 2πr where µ is the permeability of the medium between the wire and the point p, and I is the current in the wire. Note that while we would like µ to be constant is it not, especially when we talk about ferromagnetic materials (in part due to an effect called saturation). In the examples and problems that are given as part of E.P. 155.3, the µ values used for the various materials were obtained from the Normal Magnetization Curves (I & II) on Page 37. April 5 th, 2005 Magnetic Circuits & Inductance 6
Example #1: A long straight conductor carries a current of 100A. At what distance from the conductor is the magnetic field caused by the current equal in magnitude to the earth s magnetic field in Saskatoon (about 0.5 x 10-5 T)? April 5 th, 2005 Magnetic Circuits & Inductance 7
Now we have to determine the direction. This is where the concept of a magnetic field starts to diverge from that for an electric field (if it hasn t already). The direction of the magnetic field is given by the Right Hand Rule (RHR). Point your thumb of your right hand in the direction of the current and your fingers will curl in the direction of the magnetic field. In order for us to visualize this we also use arrows to help. If an arrow was coming straight at you would see the tip (designated ) while if it was going away from you would see notch end (designated ). An example diagram indicating the arrow convention is shown below. Note that the above diagram indicates something other than B (i.e., Φ). More on this very soon. April 5 th, 2005 Magnetic Circuits & Inductance 8
Thus for the current carrying wire you would see p r I An end view would show the following Note that the farther we move away from the wire the smaller the magnetic field strength (and magnetic flux density). April 5 th, 2005 Magnetic Circuits & Inductance 9
Current Loop: If we loop the wire with the current in it The direction of the magnetic field (RHR still applies) is as shown. Since B is a magnetic flux density, define Φ as the total flux with units Webers (Wb). A good way to visualize this is that the total flux is the number of magnetic field lines through a surface. B = Φ A. If the current loop is coiled to form more loops (for a total of N loops) as below What happens to the total flux? April 5 th, 2005 Magnetic Circuits & Inductance 10
Note that for an air coil, not all of the flux passes through all of the loops of the coil (since some of it escapes between the loops). If iron is used in the center of the coil (the core) then much more flux will pass through all of the loops since iron tends to capture the flux. The flux that passes through all of the loops is called the flux linkage. It is the flux linkage of a magnetic circuit that gives it its property of inductance. Magnetic Circuits: The flux producing ability of a coil is called its magneto-motive force (mmf). This is simply the current through the wire multiplied by the number of turns in the coil I = NI The units are Ampere-turns (At). The reluctance of a magnetic circuit is its ability to oppose the magnetic flux (this is analogous to resistance which opposes current) l R = µ A where l is the length of the magnetic circuit, A is the cross sectional area of the magnetic circuit and µ is the permeability of the core in the magnetic circuit. The units are Ampere-turns/Wb (At/Wb). Therefore I µani Φ = = R l April 5 th, 2005 Magnetic Circuits & Inductance 11
where Φ is the total flux generated by the magnetic circuit. There is also a Right Hand Rule (RHR) for Φ (in fact we have already seen it). If you place the fingers of your right hand along the current carrying conductors in the direction of the current in a coil, your thumb will point in the direction of the Φ. Note that permeability is analogous to permittivity. Therefore µ = µ r µ 0 where 7 µ 0 = 4π 10 Henries/m is the permeability of a vacuum (free space) and µ r is the relative permeability of the material in the core of the magnetic circuit. The relative permeability for ferromagnetic materials can be quite large (> 1000). Here is a summary of some classes of materials: Class Example Paramagnetic µ r 1 Silver, Copper Diamagnetic µ r 1 Platinum, Aluminium Ferromagnetic µ r >> 1 Iron, Nickel,AlNiCo Ferrite µ >>>> 1 Ceramics r April 5 th, 2005 Magnetic Circuits & Inductance 12
Magnetizing force: While the use of reluctance, R, is comfortable to use, since it looks like resistance, it turns out to be hard to use in reality. The magneto-motive force per unit length is called the magnetic field intensity and is much more commonly used H = I l and since I = NI then NI H = l. The units of H are At/m. Using some of our previous definitions it can now be shown that B = µh. The graphs shown on page 37 are examples of the variation of B with respect to H and show the non-linearity and changing value of µ for some common magnetic circuit materials. April 5 th, 2005 Magnetic Circuits & Inductance 13
Some Magnetic Circuits: There are two coil type magnetic circuits that we are interested in. Solenoid: Note that even with a magnetic core (like iron), the flux has to leave the core and pass through the air to get to the other end of the coil (remember the magnetic field is a continuous loop leaving one end and coming in at the other). This causes a problem. What is it??? Toroid: If we attach the ends of the solenoid to each other there is no (or very little) flux that leaves the core (especially if it is magnetic). Much more useful. April 5 th, 2005 Magnetic Circuits & Inductance 14
Example #2: Consider the copper core toroid with circular cross section shown below. The inside diameter is 8cm and the outside diameter is 12cm. The current through the 2000 turn coil is 1A. What is the total flux in the core? April 5 th, 2005 Magnetic Circuits & Inductance 15
If a conductor is moved through a magnetic field so that it cuts magnetic lines of flux, a voltage will be induced across the conductor, as shown below. The greater the number of flux lines cut per unit time (by increasing the speed with which the conductor passes through the field), or the stronger the magnetic field strength (for the same traversing speed), the greater will be the induced voltage across the conductor. If the conductor is held fixed and the magnetic field is moved so that its flux lines cut the conductor, the same effect will be produced. Faraday s Law: Voltage is induced in a magnetic circuit whenever the flux linkage is changing. The magnitude of the induced voltage is proportional to the rate of change of the flux linkages. Mathematically (note use of lower case) dφ e ind dt April 5 th, 2005 Magnetic Circuits & Inductance 16
where e ind is the time dependent induced voltage. As more turns are used the induced voltage is seen to increase. This leads to dφ e ind = N dt Since Φ is proportional to the current, i, through the magnetic circuit dφ dφ di e = N = N dt di dt The proportionality constant is denoted L, the self inductance of the magnetic circuit 2 di dφ di µ AN di e ind L N dt di dt l = = = dt The units of inductance are Henries. Note that Volt sec ond Volt Henry = = Ampere Ampere. second The denominator is a rate of change of current. April 5 th, 2005 Magnetic Circuits & Inductance 17
Lenz s Law: The polarity of the induced voltage is such that it opposes the cause of the changing flux. Therefore (from Faraday s Law) dφ e ind = N dt. The importance of this is that an inductance opposes changes in its flux. Since the flux is set up by current, the inductor opposes changes in its current (analogous to a capacitor not allowing the voltage across it to change instantaneously). April 5 th, 2005 Magnetic Circuits & Inductance 18
Example #3: Consider the air core solenoid shown below. What is the inductance? April 5 th, 2005 Magnetic Circuits & Inductance 19
Example #4: Consider the iron core solenoid shown below. The relative permeability of iron is µ r =2000µ c. What is the inductance? April 5 th, 2005 Magnetic Circuits & Inductance 20
So far we have looked at simple coil type magnetic circuits. In order to consider more complex circuits we need another law. Ampere s Circuital Law for Magnetic Circuits Just like there is an Ohm s Law for electric circuits there is an equivalent law for magnetic circuits, Ampere s Circuital Law. The nature of the law is essentially the same. Using the following table Electric Circuit Magnetic Circuit V I I Φ R R we can see that V can be replaced by I, I can be replaced by Φ, and R can be replaced by reluctance. We can have rises and drops around a magnetic circuit just like an electric circuit. If the elements are in series I T = I 1 + I 2 + I 3 + I 4 + R T = R 1 + R 2 + R 3 + R 4 + Note that some of the Is will be negative since the sum has to be 0 (like KVL). What about total flux (as shown below)? April 5 th, 2005 Magnetic Circuits & Inductance 21
Example #5: Consider the following magnetic circuit. Ampere s Circuital Law. Use I and H. Apply April 5 th, 2005 Magnetic Circuits & Inductance 22
Example #6: A 1000 turn coil is wound on the center leg of the magnetic circuit shown below. What is the maximum current through it if the total flux in the core must not exceed 2x10-3 Wb? The core is made of laminated sheet steel 4 cm in total thickness. The width of the outer legs is 2.5cm while the inner leg is 5 cm wide. The average path length is 25cm. Use µ sheet steel = 2.283x10-3 H/m. Hint: Symmetry may help you here. April 5 th, 2005 Magnetic Circuits & Inductance 23
Example #7: What is the current, I, needed to get a total flux of 2.4x10-4 Wb? µ sheet steel =1.0x10-3 H/m. April 5 th, 2005 Magnetic Circuits & Inductance 24
Electric Circuits with Inductors: Series inductors: Consider the following diagram. Note the presence of the resistor, R. More on R later. R L1 E L2 The current through both inductors is the same, therefore E = v L + v 1 L 2 di di E = L1 + L2 dt dt Rearranging di E = L + 1 L2. ( ) dt Thus L total = L 1 + L 2 Series inductors are similar to series resistances. April 5 th, 2005 Magnetic Circuits & Inductance 25
Parallel inductors: Consider the following diagram. i a R E i 1 L1 i 2 L2 Note (from KCL) i = i 1 + i 2. and v ab = vl 1 = vl 2. So di1 di2 di L1 = L2 = Ltotal dt dt dt. Noting that di di di2 = 1 + dt dt dt di vl v 1 1 1 L2 = + = v + ab dt L1 L2 L1 L. 2 Thus 1 1 1 L1L 2 = + L total L L or L total = L + L 1 2 b 1 2. Parallel inductances are similar to parallel resistances. April 5 th, 2005 Magnetic Circuits & Inductance 26
Example #8: Find the voltage V 1 and the current through each inductor. April 5 th, 2005 Magnetic Circuits & Inductance 27
Some interesting items: Energy stored in an inductor The energy stored in an inductor is contained within the magnetic field that is created by the current through the inductor W W W W t ( t) = 0 t ( t) = 0 t ( t) = ( t) 0 = L pdt vidt di L dt i ( t ) 0 Li W ( t) = 2 idi ( ) 2 t idt Note that there is no direct relationship to the voltage across the inductor only the current through the inductor. While a capacitor can store energy in its electric field when disconnected from a circuit, an inductor cannot store energy in its magnetic field when disconnected from a circuit (since i = 0). April 5 th, 2005 Magnetic Circuits & Inductance 28
Example #9: Find the energy stored in each inductor in the circuit shown below. April 5 th, 2005 Magnetic Circuits & Inductance 29
Voltage across an inductor: Recall that di v L = L dt. If the current through the inductor is constant di = 0 dt. Therefore is no voltage across the inductor. The significance of this is that for a circuit in which the currents are not changing (i.e., a dc circuit), no voltage exists across an inductor under steady state. As a result of this, under steady state conditions, an inductor is equivalent to a short circuit. Recall in the discussion of series and parallel inductances there was a resistor, R, in the circuit. This is to prevent the inductor(s) from shorting out the battery under steady state conditions. April 5 th, 2005 Magnetic Circuits & Inductance 30
What happens if current is flowing in the inductor and a switch in series with the inductor is opened? di = dt The inductor doesn t like having the current through it change. Therefore it will induce a voltage (whatever it takes) to try and keep the current flowing. This induced voltage can be quite large and will cause a breakdown of the gap between the switch contacts (switch is opening). This can cause a flashover arc. This can be quite spectacular in the dark. April 5 th, 2005 Magnetic Circuits & Inductance 31
Real World Effects: Note: While we discuss the following effects, in E.P. 155.3 we always assume (unless told otherwise) that all inductors are ideal coil inductors. Air Gaps: When an air gap in introduced into a magnetic circuit, some of the flux bulges outside of the area of the core. This is called fringing (just like in a capacitor). See diagram below. The left side shows the air gap with the bulge while the right side shows an ideal air gap. In E.P. 155.3 we ignore the fact that it bulges out. If we didn t, a common adjustment is to increase the length (L) and width (W) of the air gap area by the length of the air gap (l) thus creating a larger area: Area ignoring bulge: A = L x W Area with bulge: A = (L + l) x (W + l) April 5 th, 2005 Magnetic Circuits & Inductance 32
Laminated sheet steel cores: A large number of magnetic circuit cores are made up of laminated sheet steel cores. Laminations Varnish This is because they are cheap and they reduce the power lost through eddy currents. Eddy currents are internal currents within the steel cores due to changes in the flux and as a result of the induced voltage creating a current that circulates around the periphery of a solid core. The use of laminations reduces the power to almost negligible values. The downside to this is that the laminations have to be insulated from one another. This is done with a thin coat of varnish between the laminations. As a result, for any given area of core, the effective area has to be reduced by the total thickness of the varnish (since it is not magnetic). A common adjustment to use is a 10% reduction in area due to the varnish. April 5 th, 2005 Magnetic Circuits & Inductance 33
Actual inductor: Since an inductor is made up of coiled wire, it must have some resistance associated with it (due to the wire in the coil). As well, the closely spaced coils give rise to a stray capacitance between the wires of the coil. Therefore an actual inductor is modeled as below: R C 1 L The value of the resistance for a good inductor is on the order of ohms (Ω) and obviously depends on the wire size used in the coils which in turn depend on the current carrying capability of the inductor. The capacitance is usually ignored. April 5 th, 2005 Magnetic Circuits & Inductance 34
Variation of µ The following graphs are representative of the variation of µ with B and H. They are a) Variation of permeability with respect to H b) Hysteresis and Saturation curve c) Normal magnetization curve (I) d) Normal magnetization curve (II) We will not directly use these graphs in E.P 155.3. a) Variation of permeability with respect to H From graph a) you can see that µ is anything but constant. April 5 th, 2005 Magnetic Circuits & Inductance 35
b) Hysteresis and Saturation curve From graph b) you can see that as H increases the B produced is non-linear and saturates near the ends. As H is then reduced there is a residual B R (top and bottom on B axis) indicating that there is a magnetic field present even if H is 0. This is the case for permanent magnets. The size of the grey area is indicative of the power lost (which causes heating) during cycling around the loop. April 5 th, 2005 Magnetic Circuits & Inductance 36
c) Normal Magnetisation Curve (I) Graph c) represents the initial magnetization (from o- a-b) shown in graph b). d) Normal Magnetization Curve (II) Knowing the value of B you can find the corresponding H using curve c) or d) above. This graph is an expanded version of graph c). April 5 th, 2005 Magnetic Circuits & Inductance 37