EE 42/100 Lecture 16: Inductance. Rev B 3/15/2010 (8:55 PM) Prof. Ali M. Niknejad

Transcription

1 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 1/23 EE 42/100 Lecture 16: Inductance ELECTRONICS Rev B 3/15/2010 (8:55 PM) Prof. Ali M. Niknejad University of California, Berkeley Copyright c 2010 by Ali M. Niknejad

2 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 2/23 Inductance Inductance is a fundamental property of circuits, much like capacitance. While capacitors relate charges and potential, inductance relates currents and magnetic flux. In fact there is an important duality between inductance and capacitance that you ll quickly recognize. You can simply interchange the role of current/voltage in the circuit equations and you ll go from capacitance to inductance! This duality is only broken due to the absence of magnetic charge.

3 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 3/23 Inductors While every circuit has inductance like it or not while an inductor is an element that is purposefully designed to maximize the magnetic energy storage, which is related to the inductance. Coils and spiral inductors (planar geometry) are common inductors. At lower frequencies (below 100 MHz), magnetic cores can be used to boost the inductance value. The circuit schematic symbol for an inductor is made to resemble a coil, it s most common incarnation.

4 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 4/23 Short Circuit Experiment Suppose we short circuit a voltage source with real wires. We would expect the current flow through the wires to be infinitely large, or at least only limited by the battery (or source) internal resistance. In practice, we observe a ramping current [recall the ramping voltage of a capacitor]. Also, when the source voltage drops to zero, the current does not go away. In absence of loss (resistance) it circulates indefinitely! Since there is no

5 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 5/23 Short Circuit Experiment Also, when the source voltage drops to zero, the current does not go away. In absence of loss (resistance) it circulates indefinitely! When the inductor current is ramping, energy is delivered by the source. Where does it go? When the current circulates, there is zero power since the voltage is zero. If we try different wires (of different length or loops of different shapes), we find that the current still ramps, but the ramp rate is different. Longer wires with larger cross-sectional areas have slower ramp rates.

6 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 6/23 Basic Physics of Inductance Inductance is related to the magnetic field and flux generated around any current carrying conductor, including wires. The energy of the voltage source goes into building a magnetic field, and the energy stays in the field. The magnetic flux is proportional to the current (for linear media) and it is defined by:

7 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 7/23 Inductance Units If we assume that the media is linear, so the flux is proportional to the current, Φ = LI, we have the units L = Φ I = Tesla meters2 Amperes or more simply, L has units of Henrys (H). Recall that we defined capacitance in an analogous way C = Q V

8 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 8/23 Faraday s Law Faraday discovered that when the flux is changed, a voltage is induced around the loop of the current. In fact, flux is generated around any loop (even space) if the flux is changing. This is called magnetic induction (hence the name inductor). This may take some adjustment. Before we dealt with the fact that if you moved in any closed path, the energy gained or loss was zero. Now we see that this is only true in places where the magnetic field is zero or constant.

9 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 9/23 Generators Magnetic induction is the basic mechanism by which electrical generators work. A rotating loop in a static magnetic field will see a flux change, which induces a voltage across the loop, which can do work to drive an external circuit. Mechanical work is converted to electricity.

10 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 10/23 Voltage Pulse Experiment Based on what we have learned, we can now decipher the behavior of the voltage pulse experiment. We see that the behavior is in fact consistent with Faraday s law. The voltage generates a current which in turn generates magnetic flux. As the flux builds up at each step, there is a voltage induced in the loop which opposes the voltage applied to the circuit. This is why the current build up is gradual and not instantaneous.

11 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 11/23 Poor Man s Current Source Just like a capacitor is a poor substitute for a voltage source (because it s voltage droops as you draw current), an inductor is also a poor substitute for a current source. If we apply zero volts across it, the current circulates in a loop, just like a current source. But if we try to drive a load, the current will droop as well. If we connect a load across the inductor terminals, then v = ir = L di dt The solution is a decaying exponential function, much like a discharging capacitors.

12 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 12/23 Magnetic Energy The energy delivered to an inductor is stored in the magnetic fields, much like a capacitor that stored energy in the electric fields. In our experiment, as we connect the constant voltage across the inductor, it has to do more and more work because it is creating more magnetic flux with time. In a capacitor this work was related to the basic electrical repulsion of like charges balanced by the electrical attraction to opposite charges on the second plate (or ground). Here there are no magnetic charges, but the same role is played by magnetic flux.

13 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 13/23 Current Source Experiment Suppose we connect a current source to an inductor to control the flux. When the current is steady (after we wait long enough), the current source does not do any additional work to maintain the current since the voltage drops to zero. Then suppose we ramp the current to a new value. Since the current is increasing, a voltage is induced across the source which means the current source is doing work. The current is changing the flux, which requires work. Eventually when the new flux settles, the current source ceases to do any additional work. The work only was required to change the flux.

14 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 14/23 Inductance Analogy Using the fluid flow analogy, inductance represents the momentum of the fluid. Changing the rate of flow requires work because it changes the momentum of the fluid. If we suddenly interrupt the fluid flow, we find that there is a great force created by the fluid as it stops and loses it s kinetic energy. Suppose that we take an emptying water tank (capacitor) and connect it to a long frictionless pipe which drains the fluid. If we close the valve suddenly, what happens to the fluid in the capacitor? The water direction reverses and flows back into the capacitor! This process repeats and the water level oscillates.

15 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 15/23 Series/Parallel Connection Using KVL and KCL, it s easy to deduce how to treat parallel and series inductors.

16 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 16/23 KCL with Inductors Writing KVL with inductors is also easy, because the voltage across an inductor depends on the rate of change of current.

17 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 17/23 KVL with Inductors KCL with an inductor requires an integration step (assume magnetic linear media) i L (t) = i L (0) + 1 L Z t 0 v L (τ)dτ Notice that the inductance has memory. The current at a given time depends on the initial current and the net voltage applied across the inductor.

18 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 18/23 R-L Circuits A circuit with a single inductor and resistors is a simple first-order system that is very similar to an RC circuit. Note that there is a characteristic time associated with the circuit τ = GL = L/R As loss goes to zero, the time to disflux an inductor goes to infinity.

19 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 19/23 Example I: Inductor Disflux In this circuit we initially run a current I through the inductor. Then we disconnect the current source but instead connect a resistor. Energy is now lost to the resistor (heat) and so the magnetic flux decays. This means the current decays exponentially.

20 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 20/23 Flash Bulb Voltage Generator In the above circuit, the battery voltage is too small to deliver sufficient current to the bulb to generate a flash. Instead, the capacitor is charged by interrupting the current flow in circuit periodically. This creates a large voltage kick-back from the inductor, which charges the capacitor to a high voltage. The diode is used to avoid discharging the capacitor.

21 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 21/23 Example II: Voltage Boost In this circuit we periodically short connect a voltage across the inductor and then re-route the current through a load. Since the inductor wants to keep the current constant (inertia), it kicks up a voltage across the diode, which can be a voltage much larger than the source voltage. The inductor can boost the output voltage to any desired value, by controlling the duty cycle of the switch.

22 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 22/23 Example II (cont) During the on period, the inductor current increases linearly I L t = V s L During the end of the period, the current has increased by I L,on = 1 L d T L V s where T is the period of the waveform controlling the switch and d is the duty cycle, a number between 0 and 1 which we shall see plays an important role. If d is equal to 0.5, the waveform has 50% duty cycle. During the off period, the voltage across the inductor is given by V s V o = L di L dt At the end of the off period, the current is therefore given by I off = (V s V o )(1 d)t L

23 A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 16 p. 23/23 Example II (3) Since the inductor is driven by a periodic waveform and we assume steady state conditions, then the energy stored in the inductor is the same in each cycle (E = 1 2 LI2 L ). That means the net current through the inductor is zero I L,on + I off = 0 I L,on + I off = V sdt L + (V s V o )(1 d)t L We have an extremely nice and simple result V o V s = 1 1 d The duty cycle controls the voltage step up!