Inductors

1. Inductors A coil of wire can create a magnetic field if a current is run through it. If that current changes (as in the AC case), the magnetic field created by the coil will change. Will this changing magnetic field through the coil cause a voltage to be created across the coil? YES! This is called self-inductance and is the basis behind the circuit element called the inductor.

2. Inductors Since the voltage created depends in this case on the changing magnetic field, and the field depends on the changing current, we have: Vinductor = -L dI /dt where the L (called the inductance) depends on the shape and material (just like capacitance and resistance).

3. Inductors Vinductor = -L dI /dt Here the minus sign means that when the current is increasing, the voltage across the inductor will tend to oppose the increase, and it also means when the current is decreasing, the voltage across the inductor will tend to oppose the decrease.

4. Units: Henry From Vinductor = -L dI /dt L has units of Volt / [Amp/sec] which is called a Henry: 1 Henry = Volt-sec / Amp . This is the same unit we had for mutual inductance before (recall the transformer).

5. Solenoid type inductor For a capacitor, we started with a parallel plate configuration since the parallel plates provided a uniform electric field between the plates. In the same way, we will start with a device that provides a uniform magnetic field inside the device: a solenoid.

6. Solenoid inductor Bsolenoid = mnI where n = N/Length . Recall Faraday’s Law: V = d/dt [ BdA ] . Since the I in the B is a constant with respect to the variable of integration (dA), we have: V = L dI/dt where L =  (mn) dA , and since B is uniform over the area for the solenoid (and in the same direction as area): L = (mn)A. But the area is for each of N loops, so: L = m N2A/Length where Aeach loop = pR2. L = mpN2R2 / Length.

7. Size of a Henry From the inductance of a solenoid: L = mpN2R2 / Length we see that with vacuum inside the solenoid, m becomes mo which has a value of 4p x 10-7 T-m/A, a rather small number. R will normally be less than a meter, and so R2 will also make L small. However, N can be large, and m can be a lot larger than mo if we use a magnetic material. Altogether, a henry is a rather large inductance.

8. Inductors Lsolenoid = mpN2R2 / Length. As we indicated before, the value of the inductance depends only on the shape (R, Length, N) and materials (m). The inductance relates the voltage across it to the changing current through it: V = - L dI/dt . We again use Lenz’s Law to give us the direction (sign) of the voltage.

9. Energy Stored in an Inductor We start from the definition of voltage: V = PE/q (or PE = qV). But since the voltage across an inductor is related to the current change, we might express q in terms of I: I = dq/dt, or dq = I dt. Therefore, we have: Estored = S qi Vi =  V dq =  V I dt and now we use VL = L dI/dt to get: Estored=  (L dI/dt) I dt =  L I dI = (1/2)LI2.

10. Review of Energy in Circuits There is energy stored in a capacitor (that has Electric Field): Estored = (1/2)CV2 . Recall the V is related to E. There is energy stored in an inductor (that has Magnetic Field): Estored = (1/2)LI2 . Recall the I is related to B. There is power dissipated (as heat) in a resistor: Elost = RI2 .

11. Review of Circuit Elements Resistor: VR = R I where I = Dq/Dt Capacitor: VC = (1/C)q (from C = q/V) Inductor: VL = -L DI/Dt We can make an analogy with mechanics: q is like x; t is like t, I = Dq/Dt is like v = Dx/Dt; DI/Dt is like a = Dv/Dt V is like F; C is like 1/k (spring); R is like air resistance, L is like m.

12. RL Circuit What happens when we have a resistor in series with an inductor in a circuit? From the mechanical analogy, this should be like having a mass with air resistance. If we have a constant force (like gravity), the object will accelerate up to a terminal speed (due to force of air resistance increasing up to the point where it balances the gravity). SF=ma -bv - mg = ma, or m dv/dt + bv = -mg

13. Mechanical Analogy: mass falling with air resistance

14. RL Circuit (cont.) If we connect the resistor and the inductor to a battery and then turn the switch on, from the mechanical analogy we would expect the current (which is like velocity) to begin to increase until it reaches a constant amount.

15. LR Circuit - qualitative look From the circuit point of view, initially we have zero current so there is no VR(voltage drop across the resistor). Thus the full voltage of the battery is trying to change the current, hence VL = Vbattery, and so dI/dt = Vbattery /L. However, as the current increases, there is more voltage drop across the resistor, VR, which reduces the voltage across the inductor (VL = Vbattery - VR), and hence reduces the rate of change of the current!

16. RL Circuit:Differential Equation To see this behavior quantitatively, we need to get an equation. We can get a differential equation by using the Conservation of Energy (SVi = 0): Vbattery - Vresistor - Vinductor = 0. [This looks like an ordinary algebraic equation. But all the V’s are not constants: we have relations for Vresistor and Vinductor.]

17. RL Circuit:Differential Equation Vbattery - Vresistor - Vinductor = 0. With Vbattery = constant, Vresistor = IR, and Vinductor = L dI/dt, we have the differential equation (for I(t)): Vbattery - IR - L dI/dt = 0. This can be rewritten as: IR + L dI/dt = Vbattery which is an inhomogeneous first order differential equation, just like we had for the mechanical analogy: m dv/dt + bv = -mg .

18. RL Circuit (cont.) IR + L dI/dt = Vbattery The homogeneous equation is:LdI/dt + RI = 0. This has a dying exponential solution: IH(t) = Io e-(R/L) t . The inhomogeneous equation is: L dI/dt + RI = Vbattery . This has the simple solution:II(t) = Vbattery / R.

19. RL Circuit (cont.) I(t) = IH(t) + II(t) = Io e-(R/L) t + Vbattery / R . To find the complete solution (that is, find Io , we apply the initial conditions: I(t=0) = 0 = Io + Vbattery/R so Io = -Vbattery/R . Therefore, we have: I(t) = [Vbattery/R]*[1 - e-(R/L) t ] .

20. RL Circuit I(t) = [Vbattery/R]*[1 - e-(R/L) t ] . As time goes on, the current does increase and finally reaches the value Vbattery /R which is what it would be without the inductor present. The graph on the next slide shows this function when Vbattery = 24 volts; R = 24 W, and L = 1 H. Note that the max current is 1 Amp, and the time scale is in milliseconds.

21. RL Circuit: I(t) versus t

22. Mechanical Analogy: mass falling with air resistance

23. RL Circuit Note that this graph of I(t) versus t looks qualitatively just like v(t) versus t for the mass falling under constant gravity with air resistance. According to the analogy, the inductance acts like the mass, and the resistance acts like the coefficient for air resistance.

24. R, L and C in a circuit We have already considered an RC circuit (in Part 2) and an RL circuit (just now). We looked at these cases for a circuit in which we had a battery and then threw the switch. However, the main reason these circuit elements are important is in AC circuits. We look next at the case of the three elements in a series circuit with an AC voltage applied.

25. LRC Circuit: Oscillations Newton’s Second Law: S F = ma can be written as: S F - ma = 0 . With a spring and resistance, this becomes: -kx -bv -ma = 0 This is like the equation we get from Conservation of Energy when we have a capacitor, resistor and inductor: S V = 0 , or (1/C)Q + RI + L dI/dt = 0 , where VC = (1/C)Q, VR = RI, and VL = L dI/dt.

26. Resonance If we put an inductor and a capacitor with an AC voltage, we have the analogy with a mass connected to a spring that has an oscillating applied force. In both of these cases (mechanical and electrical), we get resonance. We’ll demonstrate this in class with a mass and spring. This, it turns out, is the basis of tuning a radio!

27. LRC Circuit: Differential Equation Starting with Conservation of Energy (SVi = 0) and using VR = IR, VC = Q/C, VL = LdI/dt , and applying a sine wave voltage (AC voltage), we get: (1/C)Q + RI + LdI/dt = Vosin(wt) Putting this in terms of Q (I = dQ/dt): (1/C)Q + RdQ/dt + Ld2Q/dt2 = Vosin(wt) we have a second order linear inhomo-geneous differential equation for Q(t).

28. LRC Circuit: Differential Equation (1/C)Q + RdQ/dt + Ld2Q/dt2 = Vosin(wt) We again look at the homogeneous solution (that is, without the applied voltage), and then try to find an inhomogeneous solution. If the resistance is small, the homogeneous equation becomes fairly simple (1/C)Q + Ld2Q/dt2 = 0and has the solution: Q(t) = Qosin(wot) .

29. LC Circuit When we substitute this expression Q(t) = Qosin(wot) into the diff. Eq., (1/C)Q + Ld2Q/dt2 = 0 we find the “natural frequency”: wo = [1/LC] And just like the case of the mass on the spring (mechanical analogy), when the applied frequency approaches the “natural frequency”, the amplitude of the resulting oscillation gets very large (it resonates).

30. LRC Circuit By placing a resistor in the circuit, the differential equation becomes a little harder. We need to consider either both sines and cosines, or we need to consider exponentials with imaginary numbers in the exponent. This can be done, and reasonable solutions can be found, but we will not pursue that here. We will pursue an alternative way.

31. LRC Circuit:Impedance An alternative way of considering the LRC Circuit is to use the concept of impedance. The idea of impedance is that all three of the major circuit elements impede the flow of current. A resistor obviously limits the current in a circuit. But a capacitor and an inductor also limit the current in an AC circuit.

32. LRC Circuit The basic idea we will pursue is that in a series circuit, • the same current flows through all of the elements: I = Io sin(t) ; and • the voltages at any instant add up to zero around the circuit: VR(t) + VC(t) + VL(t) = VAC(t) .

33. Resistance Because of Ohm’s law (VR = IR), we see that the current and the voltage due to the current are in phase (that is, when the oscillating voltage is at a maximum, the oscillating current is also at a maximum. VR = Vro sin(t) I = Io sin(t) Vro = IoR

34. Capacitive Reactance For a capacitor: VC = (1/C)Q, and with I = dQ/dt, or Q = I dt, if I = Iosin(wt), then Q = (-Io/w) cos(wt), so VC = -(1/wC)Iocos(wt). Note that cosine is 90o different than sine - we say it is 90o out of phase. This means that VC is 90o out of phase with the current! Note that the constant (1/wC) acts just like R. We call this the capacitive reactance, XC = (1/wC) .

35. Voltage across the capacitor VC = -VCo cos(t) I = Io sin(t) VCo = IoXCwhere XC = 1/C

36. Inductive Reactance For an inductor, VL = L dI/dt, if I = Iosin(wt), then dI/dt = Iow cos(wt). Note that cosine is 90o different than sine - we say it is 90o out of phase. This means that VL is 90o out of phase with the current. Since VL = L dI/dt, VL = wLIo cos(wt) , and we see that the constant wL acts just like R. We call this the inductive reactance, XL = wL .

37. Voltage across the inductor VL = VLo cos(t) I = Io sin(t) VLo = IoXLwhere XL = L

38. LRC in series Note that in a series combination, the current must be the same in all the elements, while the voltage adds. However, the voltage must add to zero across a complete circuit at every instant of time. But since the voltages are out of phase with each other, the amplitudes of the voltages (and hence the rms voltages) will not add up to zero!

39. LRC in series For: I = Io sin(t) VR = IR = RIo sin(t) VC = (1/C)*Q = -(1/C) Io cos(t) VL = L*dI/dt = (L) Io cos(t) VR + VC + VL = VAC = Io [R sin(t) – (1/C) cos(t) + L cos(t)] = Io Z sin(t+θ) = Vo sin(t+θ)

40. Impedance In 2-D space, x and y are 90o apart. We combine the total space separation by the Pythagorean Theorem: r = [x2+y2]1/2 . If we add up the V’s, this is equivalent to adding up the reactances. But we must take the phases into account. For the total impedance, Z, we get : Vrms = Irms Z where Z = [R2 + (wL - 1/wC)2]1/2 . (Note that we had to subtract XL from XC because of the signs involved.)

41. Resonance Vrms = IrmsZ where Z = [R2 + (wL-1/wC)2]1/2 Note that when (wL - 1/wC) = 0, Z is smallest and so I is biggest! This is the condition for resonance. Thus when w = [1/LC]1/2, we have resonance. This is the same result we would get using the differential equation route. Note that this is equivalent to the resonance of a spring when w = [k/m]1/2.

42. V(t) versus Vrms From Conservation of Energy, SVi = 0. This holds true at every time. Thus: VC(t) + VR(t) + VL(t) = Vosin(wt+qo) . But due to the phase differences, VC will be maximum at a different time than VR, etc. However, when we deal with rms voltages, we take into account the phase differences by using the Pythagorean Theorem. Thus, VC-rms + VR-rms + VL-rms > VAC-rms .

43. Net Power Delivered What about energy delivered? For a resistor, electrical energy is changed into heat, so the resistor will remove power from the circuit. For an inductor and a capacitor, the energy is merely stored for use later. Hence, the average power used is zero for both of these. Note that Power = I*V, but I and V are out of phase by 90o for both the inductor and capacitor, so on average there is zero power delivered.

44. Net Power Delivered Thus, even though we have V=IZ as a generalized Ohm’s Law, power is still: Pavg = I2R (not P=I2Z).

45. Computer Homework There is a computer homework program on Inductance on Vol. 4, #4, that gives you practice on DC and AC behaviors of inductors.

46. Magnetism in Matter Just as materials affect the electric fields in space, so do materials affect the magnetic fields in space. Recall that we described the effect of materials on the electric fields with the dielectric constant, K. This measured the “stretchability” of the electric charges in the materials. This stretching due to applied electric fields caused electric fields itself.

47. Two main effects Atoms have electrons that “orbit” the positive nuclei. These “orbiting” electrons act like little current loops and can create small magnets. Effect 1: A diamagnetic effect similar to the dielectric effect. Effect 2: An aligning effect.

48. Diamagnetic Effect Effect 1: By Lenz’s law, when the applied magnetic field changes, there is a tendency in the circuit to resist the change. This effect tends to create a magnetic field opposing the change. In this effect, the material acts to reduce any applied external magnetic field. This is similar to the dielectric effect that leads to the dielectric constant for electric fields.

49. Aligning Effect Effect 2: Since the normal “currents” due to the “orbiting” electrons act like tiny magnets, these magnets will tend to align with an external magnetic field. This tendency to align will tend to add to any external applied magnetic field. Heat tends to destroy this ordering tendency.

50. Net Result When both of these effects (Lenz’s law and aligning) are combined, we find three different types of results: 1. Diamagnetic: Magnetic field is slightly reduced in some materials 2. Paramagnetic: Magnetic field is slightly increased in some materials 3. Ferromagnetic: Magnetic field is greatly increased in a few materials