Inductors and Capacitors Topic 15 Inductors and Capacitors (6.1-6.3)

Magnetic Fields and Current Magnetic fields are created by moving charges So if we vary the current through these coils i.e, current! A loop in the wire concentrates the field …we have our own antenna to generate voltage A bunch of loops concentrates it even more A commercial antenna—its loop generates a voltage across a small gap… when it intercepts a time-varying magnetic field 4/15/2019 Inductors & Capacitors

The Inductor DC Circuits i + v L i + - An inductor is a device that stores electrical energy in a magnetic field caused by an electrical current In the same way that a resistor imposes a constraint between… …the constraint for an inductor is given by: …the voltage across it… …and the current through it… …in the form of Ohm’s Law… DC Circuits So The currents and voltages are unchanging Since v = 0 for any DC current An inductor is a short circuit at DC 4/15/2019 Inductors & Capacitors

The Inductor i v 100 mH + - A 100mH inductor is driven by a time-varying current source 2e-1=2/e Let’s make a quick sketch of the time-varying current i t 10 A/s 10t is a straight line that starts at the origin and rises with a slope of 10 A/s e-5t starts at 1 and decays exponentially i t i t So their product should look like this But we need some scale A 0.736 0.2 sec Which is 0 at .2 secs and infinity 4/15/2019 Inductors & Capacitors

The Inductor The voltage jumps up instantaneously to 1 volt + 100mH + - The voltage jumps up instantaneously to 1 volt What is the voltage? secs volts 1.0 for Graphing the voltage 0.4 0.2 Note that at -0.135 4/15/2019 Inductors & Capacitors

Current for a Specified Voltage + - Suppose we know v(t) but not i. How do we work it out? t0 is often taken to be 0 t t0 Some time when the current is known A later time Some time in the past Now Now Some time in the future 4/15/2019 Inductors & Capacitors

Example 6.2 i v 100 mH + - Find the current and sketch both voltage and current as a function of time 2e-1 v t We already plotted this shape Max at 0.1 We need a scale .736 .1 From the table of integrals in Appendix G Now find the current Replace t with dummy variable τ, which will integrate out 4/15/2019 Inductors & Capacitors

Example 6.2 i + v 100 mH - What about a time scale? secs amps 2 What about a time scale? .528 Max voltage occurs at 0.1 secs .2 .3 .4 .1 4/15/2019 Inductors & Capacitors

Power and Energy in Inductors v L + - i So or So long as di/dt and dw/dt are finite between t0 and t Energy so This condition is satisfied if i(t) is continuous over that interval 4/15/2019 Inductors & Capacitors

Power and Energy in Inductors v L + - i For well behaved currents The energy at time t only depends on the current at time t …plus the initial conditions If we can find a time, t0, where both w(t0) and i(t0) are 0, then Can we guarantee to find such a time? How about when it was built? So we write more simply (for i continuous with no initial energy or current) 4/15/2019 Inductors & Capacitors

Inductors & Capacitors Assessment 6.1 ig v 4mH + - a) Find v(0) So b) The time >0 when v(t) passes through 0 c) The expression for the power delivered to the inductor 4/15/2019 Inductors & Capacitors

Inductors & Capacitors Assessment 6.1 ig v 4mH + - c) The expression for the power delivered to the inductor 4/15/2019 Inductors & Capacitors

Inductors & Capacitors d) The time when the power delivered to the inductor is maximum Let (d) Plugging .411 mS into power equation yields 32.7 watts 4/15/2019 Inductors & Capacitors

e) The maximum energy stored Hey! We did that already! e) The maximum energy stored We found where v went to 0 Max energy occurs for max current So take derivative of i wrt t and set to 0 From part (b), v (t) went to 0 at 1.54 mS so di/dt must also go to 0 at the same time. Which is the answer to part f 4/15/2019 Inductors & Capacitors

The Capacitor The capacitor is a device for holding charge q=Cv The proportionality constant is known as the dielectric constant A vacuum and many materials have a basic dielectric constant ε0… C is a measure of how much charge a capacitor can hold for a given voltage—its capacity But some special materials—known as dielectrics—have a much bigger constant The bigger the area of the plates the more charge can be held The closer they are, the more electrical attraction & the more charge can be held 4/15/2019 Inductors & Capacitors

Cutaway of an electrolytic capacitor (big & cluncky, polarized, poor at high frequencies) An old-fashioned radio air-gap tuning capacitor A modern tantalum capacitor 4/15/2019 Inductors & Capacitors

The Capacitor DC Circuits i + v C - The constraint on v and i for a capacitor is given by: DC Circuits (unchanging currents and voltages) Since i = 0 for any DC voltage So i A capacitor is an open circuit at DC This is hardly surprising…. ….where’s a current going to go? 4/15/2019 Inductors & Capacitors

Voltage for a Specified Current + - Suppose we know i(t) but not v. How do we work it out? Again, t0 is often taken to be 0 t t0 Some time when the voltage is known A later time Some time in the past Now Now Some time in the future 4/15/2019 Inductors & Capacitors

Power and Energy in Capacitors v C i + - So or So long as dv/dt and dw/dt are finite between t0 and t—i.e v(t) must be continuous Energy so For v continuous and w(t0) & v(t0) zero or 4/15/2019 Inductors & Capacitors

Inductors & Capacitors Assessment 6.2 v 0.6µF i + - Find (a) i(0), (b) the power delivered to the capacitor at t = π/80 mS and (c) the energy stored in the cap at that same time (a) The text has this wrong (b) (c) 4/15/2019 Inductors & Capacitors

Combining Inductors v1 v2 v3 i + - + - + - + L1 L3 L2 All the inductors share the same current i.e., they are in series Just like resistors! So inductors in series add 4/15/2019 Inductors & Capacitors

Combining Inductors v + - i L1 L2 L3 i1 i2 i3 All the inductors share the same voltage i.e., they are in parallel where or Just like resistors again! 4/15/2019 Inductors & Capacitors

Combining Caps v + - i C1 i1 i2 i3 C2 C3 All the capacitors share the same voltage i.e., they are in parallel Which is clearly equivalent to where This is the opposite of resistors! So capacitors in parallel add 4/15/2019 Inductors & Capacitors

Combining Caps v + - i C1 v1 v2 v3 C2 C3 All the capacitors share the same current i.e., they are in series Again, opposite resistors! where or 4/15/2019 Inductors & Capacitors

Combining Caps v + - i C1 v1 v2 v3 C2 C3 So this v Ceq i + - Can be replaced by this where But it is understood that That is, C1||C2 stands for a computation, not a circuit configuration 4/15/2019 Inductors & Capacitors

Combining Caps v + - i C1 i1 i2 i3 C2 C3 This makes good physical sense Total capacitance increases Combining caps in parallel effectively increases their area, increasing their capacity to hold charge Combining them in series effectively increases their gap, decreasing their capacity to hold charge v + - i C1 v1 v2 v3 C2 C3 Total capacitance decreases 4/15/2019 Inductors & Capacitors