1

Chapter 4

“Electrical Principles” Bill Ryan, KJ6IGX and Glen

Rikerd, NO6W Discussion Leader

Chapter 4

Radio Mathematics

2

A General Principle

•Each question should be approximately as difficult as the other questions.

General Principles

•Question statements that involve more difficult subjects generally can be expected to have easier answers.

Look for Simplicity

•Calculations are set up to use simple numbers or combinations of values

•Estimate the answers

Central Ideas

•Resonance•Triangles•Ratio•Handling Exponents

Resonance

•Capacitive Reactance and Inductive Reactance cancel each other

Triangles

•The test uses simple triangles that can be solved by sight

Ratio

The ratio of a triangles sides often tells you how to find the answer

Handling Exponents

• When multiplying, add the exponents

• When dividing, change the sign and then add the exponents

• For square roots, cut the exponent in half

Question E5C11

• Impedance has a real part (resistance) and an imaginary part (reactance)

• The horizontal axis is the real part

• The vertical axis is the imaginary part

Polar Coordinates

•Polar Coordinate System has same information as the rectangular axis

•Determined by a distance from a fixed point and an angle with a fixed direction

Phase Angle

Question E5B06: Time Constant

• Time it takes for a capacitor to charge or discharge by 63.2% is called the time constant

Question E5B06: Time Constant

• Tau = Resistance x Capacitance

Handling Exponents

•1 megohm x 1 microfarad =

•1 x 106 x 1 x 10-6 =

•106 + (-6) = 100 = 1 second

E5B06: Time Constant Calculation

• Given 800 V DC charge decreases to 294 V DC. (Looks like – 63.2%)

• R x C = 1 megohm x 450 microfarads

• 1 x 106 x 450 x 10-6 = 450 seconds

Phase

• Relationship of current and voltage is the phase of the two waves

• Phase means time

• The effect that occurs first leads the second, the trailing effect lags the first

Voltage jumps from coil to coil

Since voltage does not travel along the whole wire, it moves faster than the current in an inductor

Voltage leads current in an inductor

• Voltage must build up in a capacitor. This causes a delay. Current seems to go right through the capicator

Current leads voltage in a capacitor

• When reactance is inductive, voltage leads current

• When reactance is capacitive, current leads voltage

Question E5B10: Phase Relationship

What is the Relationship between current through and voltage across an inductor

• Voltage moves from loop to loop, current must move through the wire

• In this question, voltage leads current by 90 degrees

Understanding Right Triangles:The central angle is the phase angle

• Two sides are equal: opposite angles = 45 degrees

• Opposite side longer than adjacent: central angle is more than 45 degrees

• Opposite side shorter: central angle is less than 45 degrees

Equal sides of triangle, phase angle = 45

• When reactance and resistance are the same number of ohms, then the triangle has equal sides

• 100 ohms• 45• 100 ohms

Reactance smaller, phase angle = <45

• When reactance is smaller than the resistance, the phase is less than 45

• 250 ohms• • < -45 100 ohms

• Net Capacitative Reactance is a negative phase angle

Reactance larger than resistance

• When reactance is larger than the resistance, the phase is more than 45

• 100 ohms - > -45

• 250 ohms XC

• •

Question E5B13

• Resistance = 1000 ohm• Reactance is inductive 250 ohms• Phase angle is less than 45 degrees

and voltage leads the current

• Inductive Reactance• 250 ohms• <45• Resistance 1000 ohms

Question E5B07

Resistance = 1000 ohm• Reactance is capacitive 250 ohms• Phase angle is less than 45 degrees

and current leads the voltage 1000• <-45 250

• Capacitive reactance is always drawn downwards to show a negative angle

Question E5B13 Complex Impedance

• Phase angle between voltage across and current through series RLC circuit

• XL =500 ohms, XC = 250 ohms R = 1000 ohms

• Net reactance: XL – XC = 250 ohms XL

• Phase angle is less than 45 degrees and voltage leads current

Tank Circuits

• Parallel RLC circuits at resonance have a high impedance and appear as an open circuit.

• It also has a circulating current and builds a maximum voltage at resonance.

3 4 5 Right Triangles

• Pythagorean Theorem for series RLC

• If shortest sides are 3 and 4 units long, then the longest side is 5 units

• Resistance = 400 ohms and • Reactance = 300 ohms

• Impedance = 500 ohms

Pythagorean Triangles

• Any combination of 300 and 400 for the sides will produce an impedance of 500

• 500• 300•

• 400•

30 60 90 Triangles

• When the phase angle is 30 degrees, the reactance is always half of the impedance

• 200 • 30 100• 186

Equilateral Triangles

When the phase angle is 45 degrees, the impedance is 1.41 times either the reactance or resistance

141 100 45 100

Question: E5D19 Power Factor

• Real Power = PF x Apparent Power

• Power Factor (PF) = 0.71

• Apparent Power = 500 watts

• Watts consumed = 0.71 x 500 watts = 355 watts

Resonant Frequency

• Same for Series and Parallel Circuits•

Fr = 1

2 pi L x C

Resonant Frequency Strategy

•Calculate L x C alone first•Take square root•Plug values into formula•Estimate the result

Question E5A14 Resonant Freq

• L x C = • 40 microhenrys x 200 picofarads =

• 40 x 10-6 x 200 x 10-12 = 8,000 x 10-12

• Take square root of 8,000 x 10-18 =

• Square root of 80 x 10-16 = 9 x 10-8

Plug in values and estimate for Fr

Fr = 1 = 10+8

6 x 9 x 10-8 50

Fr = .02 x 10+8 = 2 x 10+6 = 2 MHz estimate

Answer to Question E5A14

•Estimate of Fr = 2 MHZ

•Actual value of Fr = 1.78 MHz

•Just estimate freely and pick the closest answer

Q

• Q is a quality factor for a circuit for low resistive losses

• Q is the ratio of reactance to resistance

• Q = X • R

Resonant Circuit Bandwidth

•Bandwidth is the freqency range within 3 dB below peak response

•Delta f = bandwidth = fr

• Q

Question E5A13 Half Power BW

• Half Power BW of a parallel resonant circuit:

• Resonant Frequency = 14.25 MHz• Q = 187

• BW = 14.25 MHz = 15 MHz = 100 kHz• 187 150

• Actual answer: 76.2 kHz

Encouraging Thoughts

• We want you to pass the Extra Exam

• Try to focus on simplifying methods

• True mastery of this material will take a long time

• Commit to taking the test at end of class despite whether feeling ready

In Class Practice PRoblems

• Here is a list of possible in class seminar questions

Chapter 4

E5C11 What do the two numbers represent that are used to define a point on a graph using rectangular coordinates?•The sine and cosine values•The tangent and cotangent values•The coordinate values along the horizontal and vertical axes•The magnitude and phase of the point

46

Chapter 4

E5D05 What is a magnetic field?•The space between the plates of a charged capacitor, through which a magnetic force acts•The force that drives current through a resistor•The region surrounding a magnet through which a magnetic force acts•Electric current through the space around a permanent magnet 47

Chapter 4

E5B05 (A) How long does it take for an initial charge of 20 V DC to decrease to 7.36 V DC in a 0.01-microfarad capacitor when a 2-megohm resistor is connected across it?•0.02 seconds•0.04 seconds•20 seconds•40 seconds 48

Chapter 4

E5D13 (B) How many watts are consumed in a circuit having a power factor of 0.2 if the input is 100-V AC at 4 amperes?•2000 watts• 400 watts• 80 watts• 50 watts

49

Chapter 4

E5C19 (B) Which point on Figure E5-2 best represents that impedance of a series circuit consisting of a 400 ohm resistor and a 38 picofarad capacitor at 14 MHz?•Point 6•Point 5•Point 4•Point 2 50

Chapter 4

Figure E5-2 refers to question E5C19

51

Chapter 4

• E5A12 What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 3.7 MHz and a Q of 118?

• 15.7 kHz• 31.4 kHz• 218.3 kHz• 436.6 kHz

52

Next Class Session

• Study Chapter 5 “Components and Building Blocks”

• Study the Chapter 5 Question Pool questions found in the Blue Boxes

• Prepare your Chapter 5 Study Guide materials

53

Chapter 4

E5C11 What do the two numbers represent that are used to define a point on a graph using rectangular coordinates?•The sine and cosine values•The tangent and cotangent values•The coordinate values along the horizontal and vertical axes•The magnitude and phase of the point

54

Chapter 4

E5D05 What is a magnetic field?•The space between the plates of a charged capacitor, through which a magnetic force acts•The force that drives current through a resistor•The region surrounding a magnet through which a magnetic force acts•Electric current through the space around a permanent magnet

55

Chapter 4

E5B05 (A) How long does it take for an initial charge of 20 V DC to decrease to 7.36 V DC in a 0.01-microfarad capacitor when a 2-megohm resistor is connected across it?•0.02 seconds•0.04 seconds•20 seconds•40 seconds 56

Chapter 4

E5D13 (B) How many watts are consumed in a circuit having a power factor of 0.2 if the input is 100-V AC at 4 amperes?•2000 watts• 400 watts• 80 watts• 50 watts

57

Chapter 4

E5C19 (B) Which point on Figure E5-2 best represents that impedance of a series circuit consisting of a 400 ohm resistor and a 38 picofarad capacitor at 14 MHz?•Point 6•Point 5•Point 4•Point 2 58

Chapter 4

Figure E5-2 refers to question E5C19

59

Chapter 4

• E5A12 (B) What is the half-power bandwidth of a parallel resonant circuit that has a resonant frequency of 3.7 MHz and a Q of 118?

• 15.7 kHz• 31.4 kHz• 218.3 kHz• 436.6 kHz

60

Next Class Session

• Study Chapter 5 “Components and Building Blocks”

• Study the Chapter 5 Question Pool questions found in the Blue Boxes

• Prepare your Chapter 5 Study Guide materials

61