Physics 331 Review Sheet

Capacitors and Capacitance

A Capacitor is a device which stores charge. The capacitance of a capacitor depends on the charge on the capacitor (Q) and the voltage across the capacitor (V). Q C = [F] (Farads) V A parallel plate capacitor has capacitance A C = where A=Area of the plates, d = distance between the plates,  = permitivity d

Capacitors in Series: The charge QT= Qn on each capacitor remains the same but the voltage across each capacitor is different. This is similar to resistors in series, each resistor has a different voltage drop but the current though each is the same. 1 1 1 1     ..... CT C1 C2 C3 Capacitors in Parallel: The voltage across each capacitor remains the same but the charge on each is different.

CT  C1  C 2  C3  ..... Transients in Series RC Networks: A transient is quantity that is temporary, or short- lived. When a voltage source is connected across a capacitor the current that flows is short lived or transient.

Charging and Discharging a Capacitor:

R1 1 2 E C R2

At time t = 0s the switch is moved to position 1 and the capacitor begins to charge.

t - t t E RC - - i(t)  e v (t)  E e RC v (t)  E(1 - e RC )  = RC R R C At some later time the switch is moved to position 2. (We start again at time t = 0 s for our equations)

t - t t Vo   - - i(t)   e v (t)   V e   v (t)  V e   ' = RTC R R o C o

Where Vo is the voltage across the capacitor just before the switch was moved to position 2.

Series-Parallel RC Networks: If you have components that are not in series or parallel you must find the Thevenin Equivalent Circuit first and then apply the above equations. If this method needs to be used it will be stated in the problem.

Initial and Steady-State Values: Initially the Capacitor behaves like a short circuit. 1) replace all capacitors by shorts 2) find the initial values for current and voltage Under steady-state the current is no longer changing and the capacitor behave like and open circuit. 1) replace all capacitors by open circuits 2) find the steady-state values for current and voltage

Energy Stored in a Capacitor Q 2 1 1 W =  C V 2  Q V 2 C 2 2

Inductors and Inductance

An Inductor is a device which resists a change in the current. The inductance of an inductor is dependent on the voltage across the inductor and the change in current through the inductor. d i VL = L (Inductance is measured in Henries [H]) d t The inductance of a coil can be found by knowing the number of turns [N], the Area [A], and the length of the coil [l].  N 2 A L = [H] l Inductors in Series: The current through each inductor is the same just as it is for resistors. The voltage across each inductor is different.

LT = L1 + L2 + L3 +. . . . Inductor in Parallel: The voltage across each inductor is the same just as it is for resistors. The current through each is different. 1 1 1 1     ..... LT L1 L 2 L3 Transients in Series RL Networks:

R1 1 2

E R2 L

At time t = 0 s the switch is moved to position 1. t t t - -  E L v (t)  E e  v (t)  E(1 - e  ) i(t)  (1  e  )  = L R R R At some later time the switch is moved to position 2. The inductor wants to keep the current the same as it was at position 1. (We start again at time t = 0 s in our equations) t - L i(t)  I e   vR(t) = R i(t) vL(t) =  RT i(t) ' = o R T

Io is the current through the inductor just before the switch was moved to the new position.

Series-Parallel RC Networks: If you have components that are not in series or parallel you must find the Thevenin Equivalent Circuit first and then apply the above equations. If this method needs to be used it will be stated in the problem.

Initial and Steady-State Values: Initially the Inductor behaves like an open circuit. 1) replace all capacitors by open circuits 2) find the initial values for current and voltage Under steady-state the current is no longer changing and the inductor behaves like and short circuit. 1) replace all inductor by short circuits 2) find the steady-state values for current and voltage

Energy Stored in an Inductor

1 2 W = LI 2 Chapter 15: AC Fundamentals

1) Review and know how to use the trigonometry functions 2) Know the relationship between the waveform parameters a) period T b) frequency f = 1/T c) angular frequency or angular velocity  = 2  f d) amplitude (peak value) and peak to peak value e) phase angle 3) Be able to analyze a wave if given a picture. (Be able to identify the above values for a sine wave) 4) Know what a phase shift is and how to determine the phase shift between two waves

PROBLEMS You should review the examples that were covered in class as well as your homework. This is only an example of a few types of problems. This is by no means a complete review of all of the types of problems you were asked to do for homework.

1. Answer the following questions about the circuit shown below.

5 k a

30 F

15 V 10 F 20 F

60 F

b (a) What is the total equivalent capacitance across the terminals a and b? Redraw the circuit with the equivalent capacitance.

(b) If the switch is closed at time t = 0 s what is the time constant?

(c) Write the expression for the current through the resistor as a function of time.

(d) What is the current after 1.5 time constants have elapsed? (t = 1.5) 2. Find the current through each resistor under initial and steady state conditions. Also determine the energy stored in the inductor and capacitor under initial and steady state conditions. Show work when ever possible.

L = 0.5 H

C = 2mF R3 = 40  24 V R2 = 20 

R1 = 15 

(b) At what time is the voltage across the inductor equal to 6 V? (vL(t) = 6V)

3. (a) At time t = 0 s the switch is closed. Determine the voltage across the inductor as a function of time. You must find the Thevenin equivalent circuit first.

30 V 10  1 mH 24 15  6 

(b) At what time is the voltage across the inductor equal to 6 V? (vL(t) = 6V) 4.

R1= 5 k 1

50 V R2= 12 k C = 25F

(a) The switch is moved to position 1 at time t = 0s. Write down the expression for the current through the resistor and the voltage across the capacitor as functions of time.

(b) What is the voltage across the capacitor at t = 250 ms?

(c) Suppose the switch is moved to position 2 after the switch had been in position 1 for a time of t = 250 ms. (The capacitor is not fully charged when the switch is moved). Write down an expression for the current and voltage across the capacitor as a function of time assuming we start at time t = 0 s again for switch 2.

5. The following wave describes the current produced by an AC source. Answer the following questions.

18ms 26ms 135 10ms

2ms 6ms 14ms 22ms

50 mA

a) What is the frequency of oscillation of the current? b) Write down the current of the AC source in the time domain, i(t). c) What is the current of the source at a time of 12.5 ms? d) If the current given above is the current through a 1.5 k resistor what must be the voltage across the resistor, v(t)?