Inductors and Capacitors PDF

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Introduction to what inductors and capacitors are and how to solve currents and voltage through them....

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Cha Chap pte terr 6: Ind Induc uc ucto to torrs and Cap Capac ac acit it itor or orss Tuesday, September 22, 2015

5:50 PM

Table of contents. 1. Introduction 2. Inductors 3. Capacitors 4. Series and parallel combinations 5. Mutual inductance 6. Complex numbers I. Introduction 1. Up to now… ○ DC circuits -- constant with time 2. Next: ac circuit -- sinusoidal

Which waveform is dc, sinusoidal and complex? 3. Why we care about sinusoidal signals

4. Complex waveforms -- Fourier series ○ General equation to represent ANY waveform:

e.g. Pulse waveform

As n increases, amplitude goes down and frequency goes up.

Add the 5 waveforms together

Similarly, we can get other waveforms

*Right now, we consider only one cosine --> "ac analysis" II. New circuit elements 1. Capacitor and Inductor: very different behavior at dc and ac Depends on magnetic and electric fields: time-dependent

2. Inductor: a coil of metal wire a. Magnetic field generated when current I flows b. Coil of wire will concentrate the magnetic field c. The magnetic field strength is a function of dI/dt Ampere's law:

Faraday's law:

d. Symbol, inductance, terminal voltage, and unit Inductance: reflect the ability of a metal coil to resist the change of current.

L = inductance

units = Henry, H

Q1: Match the current and voltage waveforms

Q2: L = 5 mH. What is v(t)?

e. Inductance value

f. Inductor energy 1) Passive device, can't deliver energy 2) Do not consume energy either 3) It stores energy Stored power:

Energy stored

Q3: How much energy is stored in an inductor of 20mH with a current of 80mA?

Q4: 1. What is the voltage on an inductor if di/dt=0 A/s? Is it a short or an open situation? 2. Can the current through an inductor change instantaneously? 3. Will an inductor dissipate energy?

3. Capacitor: two metal plate with insulator in the middle a. Separated charge generate electric field b. Parallel plates will concentrate the electric field c. The current flow through a capacitor is a function of dv/dt

d. Symbol, capacitance, ac current and unit Capacitance: the ability of the capacitor to hold charges

C = capacitance units = Farad, F = C/V Q5: Match the voltage and current waveforms

Q6: C=5mF. What is i(t)?

4. Capacitance value

Q7: How do you get the most capacitance for a given physical volume?

5. Capacitor energy a. Passive device, can't deliver energy b. Do not consume energy either c. It stores energy Power stored:

Energy stored:

Q8: How much energy is stored in a capacitor of 180mF with a voltage of 3.5 V?

Q9: 1. What is the current on a capacitor if dv/dt=0 V/s? Is it a short or an open situation? 2. Can the voltage on a capacitor change instantaneously? 3. Will a capacitor dissipate energy?

III. Series and parallel combinations 1. Combining inductor e.g. if an inductor has inductance L=100nH

If two identical inductors are connected together, what is the total inductance

If n inductors are in series:

If n inductors are in parallel:

Q10: Find

for each case.

Q11: Inductor at dc A. Draw the equivalent circuit under dc conditions. B. Determine the current through each inductor C. Determine the energy stored in each inductor

2. Combining capacitors e.g. if a capacitor has capacitance of 60pF

What is the total capacitance?

If n capacitors are in parallel:

If n capacitors are in series:

Q12: Find

for each case.

Q13: Capacitors at dc A. Draw the equivalent circuit under dc conditions B. Determine the voltage across each capacitor C. Determine the charge on each capacitor D. Determine the energy stored in each capacitor

IV. Mutual inductance 1. Key component in a power distribution system - transformers

2. Transformer structure Use mutual inductance: two inductors, shared core Primary: apply a voltage, VP, creates magnetic field in core, Ampere's law Secondary: magnetic field induces a voltage, VS, Faraday's law of induction& Len's law Right-hand rule:

Step up transformer: VS > VP Step down transformer: VS < VP

3. Mutual inductance A. Polarity of winding

*Need to know the wrapping directions of the windings

1) Arbitrarily select one terminal—e.g. D terminal—of one coil and mark it with a dot 2) Assign a current into the dotted terminal and label it 3) Use the right-hand rule to determine the direction of the magnetic field and label this field 4) Arbitrarily pick one terminal of the second coil—e.g. terminal A—and assign a current int this terminal, name it 5) Use the right-hand rule to determine the direction of the flux established and label this flux 6) Compare the directions of the two fluxes a. If the fluxes have the same reference direction, place a dot on the terminal of the second coil where the test current enters. b. If the fluxes have different reference directions, place a dot on the terminal of the second coil where the test current leaves. B. Determine the polarity of winding experimentally.  DC source, resistor, switch and voltmeter  The coil terminal connected to the positive terminal of the source receives a polarity mar  Observe the voltmeter while closing the switch: □ If the voltmeter deflect upscale momentarily, the coil terminal connected to the positive terminal of the voltmeter receives the polarity mark □ If the voltmeter defect downscale momentarily, the coil terminal connected to the negative terminal of the voltmeter receives the polarity mark

C. Dot convention method -- polarity of induced voltage -- Self-inductance L and mutual inductance M both contribute to the voltage across the inductor.

a. When the reference direction for a current enters the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is positive at its dotted terminal. b. When the reference direction for a current leaves the dotted terminal of a coil, the reference polarity of the voltage that it induces in the other coil is negative at its dotted terminal.

Q14: Write the KVL equation for the primary and secondary loops, separately.

V. Complex numbers 1. Invented to extract the square roots of negative numbers e.g. i or j is used to represent the solution of 2. Useful to solve problems that have no solution in a real number system 3. Notation of complex numbers

a. Rectangular form: e.g. 2+j3; 5-j5; -3+j2 b. Polar form: where e.g. ; ; ; Note: 4. Conversion between rectangular and polar form Q15: Convert to rectangular representation and plot the point (pick 2): A. 8.5 45° B. 10 -53° C. 10 130° D. 6 -150° E. 4.0 90° 5. Graphical view of complex numbers ○ Complex plane a. real axis = horizontal = x-axis b. imaginary axis = vertical = y-axis

○ Magnitude: always positive ○ : positive real axis ○ Phase: ○ Four quadrants Q16: What is the range of phase for each quadrant Quadrant Range of angle

Sign of (b/a)

Range of

I II III IV

6. Rectangular to polar conversion ○ Pythagorean Theorem - works for all quadrants

Correction to

ranges between -90° and +90°

Problems:

But the angle needed is from -180° to +180°

Correction:

Q17: Determine the magnitude and angle of the polar representation for the following numbers. A. 2 + j 7 B. -4 + j 5 C. -3 - j 2 D. 6 - j 3 7. Complex number arithmetics ○ Addition: (a + j b) + (c + j d) = (a + c) + j (b + d) ○ Multiplication: (c / q ) * (d / f ) = c * d / q + f ○ Division: (c / q ) / (d / f ) = c / d / q - f ○ Complex conjugate:

Q18: Find and plot: A. (2 + j 7) + (-4 + j 5) B. (-3 - j 2) + 10 130° C. (7 70° ) * (3 -50° ) D. (7 70° ) * (6 - j 3)

E. 10

130° / 6

-150°...