# Series Resonance

Add Abstract, Introduction and Conclusion to the Inductors in DC Circuits Lab. Input calculation

Electric Circuits Lab

Series Resonance

I.

Objectives:

After completing this lab experiment using, you should be able to:

1. Observe the effect of frequency on inductive and capacitive reactance.

2. Calculate and verify the resonant frequency.

3. Identify the phase relation between current and voltage in a series RLC circuit.

4. Use a Bode plotter to measure the resonant frequency.

II.

Parts List:

1. Resistor (1) 1 kΩ.

2. Inductor (1) 100 mH.

3. Capacitor (1) 0.22 µF, (1) 47 pF, (1) 0.01 µF

III.

Procedures:

Part I:

1.
Connect the following circuit on the breadboard.

Figure 1: Series RLC Circuit

2.
Set the voltage source amplitude to
2.5 VP; frequency to
300 Hz, sine wave

3.
Calculate the resonant frequency of the circuit using the flowing equation:

Resonant frequency (fr) = ____1.1k____ Hz.

4.
Calculate and
record the inductive reactance, capacitive reactance, impedance, and phase angles for each frequency shown in
Table 1.

Frequency
(in Hz)

Calculated

Measured

XL

XC

Z

Ө

VR

300

188.5 Ω

2.41 k Ω

1.03 V

500

314.2 Ω

1.45 k Ω

1.66

700

439.8 Ω

1.03 k Ω

2.15

1k

628.3 Ω

723.4 k Ω

2.485

Resonant freq. (fr)
(from step 8)

691.2 Ω

657.7 k Ω

2.49

2k

1.26 k Ω

574.2 k Ω

1.85

3k

1.89 k Ω

382.8 k Ω

1.295

5k

3.14k Ω

230.4 k Ω

0.79

Table 1: Calculated and measured values

5.
Connect the Channel A of the oscilloscope across the resistor. The oscilloscope should be set to AC mode.
Measure and
record the peak voltage drop (VR) in
Table 1 at the various frequencies listed.

6.
Draw the frequency response curve from the above results.

(Use Word or Excel to Create Plot)

Plot 1: Frequency Response

7. Now
connect Channel A of your oscilloscope across the capacitor and Channel B across the inductor.

8.
Measure and
record the peak voltage drop across the capacitor (VC ) and the inductor (VL) in
Table 2 at the various frequencies listed.

Frequency (in Hz)

VC (Ch A)

VL (Ch B)

300

2.473

194.518

500

2.389

522.063

700

2.213

947.858

1k

1.792

1.566

Resonant freq. (fr) (from step 8)

1.634

1.728

2k

667.243

2.333

3k

310.901

2.446

5k

113.7

1.485

Table 2: Measured voltage values

9. From the above results the phase angle between VC and VL at the resonance frequency is ____________ degrees.

10.
Draw the impedance phasor on
Plot 2.

(Use Word or Excel to Create Plot)

Plot 2: Impedance Phasor

Part II: Bode Plotter

11.
Remove the oscilloscope from the circuit and
connect the Bode plotter as shown in
Figure 2.

Figure 2: Series RLC Circuit with Bode Plotter

12.
Record the calculated resonant frequency from step 8 in
Table 3.

13.

Use the Bode Plotter to
measure the resonant frequency with C = 0.22 µF and
record the value in
Table 3.

(
Note: To measure the resonant frequency of a series RLC circuit using the Bode plotter, connect the input across the voltage source and the output across the resistor. Run the simulation and use the cursor to find the maximum gain. The frequency where the maximum gain occurs will be the resonant frequency.)

14.
Replace the 0.22 µF capacitor with a 470 pF capacitor.

15.
Calculate and
record the resonant frequency with the new capacitor value in
Table 3.

16. Use the Bode Plotter to
measure the resonant frequency with C = 470 pF and
record the value in
Table 3.

17.
Replace the 470 pF capacitor with a 0.01 µF capacitor.

18.
Calculate and
record the resonant frequency with the new capacitor value in
Table 3.

19. Use the Bode Plotter to
measure the resonant frequency with C = 0.01 µF and record the value in
Table 3.

Resonant Frequency

Capacitor Value

Calculated Frequency

Measured Frequency

C = 0.22 µF

1.07 kHz

1.072 kHz

C = 470 pF

23.215 kHz

23.442 kHz

C = 0.01 µF

5.012 kHz

5.012 kHz

Table 3: Calculated and Measured Resonant Frequency

1

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Electric Circuits Lab

Instructor: ———–

Series Resonance

Student Name(s):

Click or tap here to enter text.

Click or tap here to enter text.

Honor Pledge:

I pledge to support the Honor System of ECPI. I will refrain from any form of academic dishonesty or deception, such as cheating or plagiarism. I am aware that as a member of the academic community, it is my responsibility to turn in all suspected violators of the honor code. I understand that any failure on my part to support the Honor System will be turned over to a Judicial Review Board for determination. I will report to the Judicial Review Board hearing if summoned.

Date: 1/1/2018

Contents
Abstract 3
Introduction 3
Procedures 4
Data Presentation & Analysis

6

Calculations 8
Required Screenshots 8
Conclusion 9
References 10

 Lab Report Instructions: (This instruction box is to be deleted before submission of the Lab report) Before starting on your lab report, please follow the following steps: 1) Follow the instructions listed provided in the lab instructions. 2) Complete this lab report . Upon completion, you will submit this lab report and your working Multisim files to your instructor.

## Abstract

(This instruction box is to be deleted before submission of the Lab report)

What is an Abstract?

This should include a brief description of all parts of the lab. The abstract should be complete in itself. It should summarize the entire lab; what you did, why you did it, the results, and your conclusion. Think of it as a summary to include all work done. It needs to be succinct yet detailed enough for a person to know what this report deals with in its entirety.

Objectives of Week 4 Lab 1:

· Observe the effect of frequency on inductive and capacitive reactance.
· Calculate and verify the resonant frequency.
· Identify the phase relation between current and voltage in a series RLC circuit.
· Use a Bode plotter to measure the resonant frequency.

## Introduction

(This instruction box is to be deleted before submission of the Lab report)

What is an Introduction?

In your own words, explain the reason for performing the experiment and give a concise summary of the theory involved, including any mathematical detail relevant to later discussion in the report. State the objectives of the lab as well as the overall background of the relevant topic.

· What is series resonance? (Give formula)
· How do capacitance and inductance affect resonance?

## Procedures

(This instruction box is to be deleted before submission of the Lab report)
This section should contain the procedures as outlined in the lab instructions.

Part I:

1.
Connect the following circuit on the breadboard.

Figure 1: Series RLC Circuit

2.
Set the voltage source amplitude to
2.5 VP; frequency to
300 Hz, sine wave

3.
Calculate the resonant frequency of the circuit using the flowing equation:

Resonant frequency (fr) = ________ Hz.

4.
Calculate and
record the inductive reactance, capacitive reactance, impedance, and phase angles for each frequency shown in
Table 1.

5.
Connect the Channel A of the oscilloscope across the resistor. The oscilloscope should be set to AC mode.
Measure and
record the peak voltage drop (VR) in
Table 1 at various frequencies listed.

6.
Draw the frequency response curve from the above results.

7. Now
connect Channel A of your oscilloscope across the capacitor and Channel B across the inductor.

8.
Measure and
record the peak voltage drop across the capacitor (VC ) and the inductor (VL) in
Table 2 at various frequencies listed.

9. From the above results the phase angle between VC and VL at the resonance frequency is ____________ degrees.

10.
Draw the impedance phasor on
Plot 2.

Part II: Bode Plotter

11.
Remove the oscilloscope from the circuit and connect the Bode plotter as shown in
Figure 2.

Figure 2: Series RLC Circuit with Bode Plotter

12.
Record the calculated resonant frequency from step 8 in
Table 3.

13.

Use the Bode Plotter to
measure the resonant frequency with C = 0.22 µF and
record the value in
Table 3.

(
Note: To measure the resonant frequency of a series RLC circuit using the Bode plotter, connect the input across the voltage source and the output across the resistor. Run the simulation and use the cursor to find the maximum gain. The frequency where the maximum gain occurs will be the resonant frequency.)

14.
Replace the 0.22 µF capacitor with a 470 pF capacitor.

15.
Calculate the resonant frequency with the new capacitor value in
Table 3.

16. Use the Bode Plotter to
measure the resonant frequency with C = 470 pF and
record the value in
Table 3.

17.
Replace the 470 pF capacitor with a 0.01 µF capacitor.

18.
Calculate and
record the resonant frequency with the new capacitor value in
Table 3.

19. Use the Bode Plotter to
measure the resonant frequency with C = 0.01 µF and
record the value in
Table 3.

## Data Presentation & Analysis

(This instruction box is to be deleted before submission of the Lab report)
This section is the most important section of the report. Data representations and analysis are crucial in the engineering field. This section should include all raw data collected, e.g., voltage and current readings. All results are to be presented in both tabular and graphical forms. All tables must have titles and all figures must have brief captions.

Frequency

(in Hz)

Calculated

Measured

XL

XC

Z

Ө

VR

300

500

700

1k

Resonant freq. (fr)
(from step 8)

2k

3k

5k

Table 1: Calculated and measured values

(Use Word or Excel to Create Plot)

Plot 1: Frequency Response

Frequency (in Hz)

VC (Ch A)

VL (Ch B)

300

500

700

1k

Resonant freq. (fr) (from step 8)

2k

3k

5k

Table 2: Measured voltage values

Resonant Frequency

Capacitor Value

Calculated Frequency

Measured Frequency

C = 0.22 µF

C = 470 pF

C = 0.01 µF

Table 3: Calculated and Measured Resonant Frequency

(Use Word or Excel to Create Plot)

Plot 2: Impedance Phasor

## Calculations

(This instruction box is to be deleted before submission of the Lab report)
Show all of your calculations in this section.

Part 1 Step 3: fr =

Part I Step 4 (f= 300 Hz only):

XL =

XC =

Z =

θ =

Part 2 Step 15:

Part 2 Step 18:

## Required Screenshots

(This instruction box is to be deleted before submission of the Lab report)

Place screenshots of measurements in this section.

Figure 3: Screenshot of Waveforms Part 2 Step 13

Figure 4: Screenshot of Waveforms Part 2 Step 16

Figure 5: Screenshot of Waveforms Part 2 Step 19

## Conclusion

(This instruction box is to be deleted before submission of the Lab report)

What is a Conclusion?

This section should reflect your understanding of the experiment conducted. Important points to include are a brief discussion of your results, and an interpretation of the actual experimental results as they apply to the objectives of the experiment set out in the introduction should be given. Also, discuss any problems encountered and how they were resolved.

· What happens to inductive and capacitive reactance as frequency varies above and below the resonant frequency?
· What is the relationship between capacitive and inductive reactance at resonance?
· Is the output voltage across the resistor maximum or minimum at the resonant frequency? Why?
· What is the relationship between current and voltage in a series RLC circuit?
· How does resonant frequency change with capacitance?

References

(This instruction box is to be deleted before submission of the Lab report)

What is a Reference Section?

This section should list all sources used in the completion of the lab report using APA format. At a minimum, you should include your book and your instructor’s notes and videos. Be sure to list all sources to avoid plagiarism.

Note:

The below reference section contains the reference for your book. Add to it as necessary. The second entry is the way to cite your instructor’s Zoom video.

Floyd, T. L., & Buchla, D. M. (2019).
Principles of Electric Circuits (10th Edition). Pearson Education (US).

https://bookshelf.vitalsource.com/books/9780134880068

Last Name, First initial. Second initial (Date of Video).
Title and Subtitle of Video. Video URL

6

## image7

ELECTRIC CIRCUITS I

METRIC PREFIX TABLE

Metric

Prefix

Symbol

Multiplier

Expo-

nential

Description

Yotta

Y

1,000,000,000,000,000,000,000,000

1024

Septillion

Zetta

Z

1,000,000,000,000,000,000,000

1021

Sextillion

Exa

E

1,000,000,000,000,000,000

1018

Quintillion

Peta

P

1,000,000,000,000,000

1015

Tera

T

1,000,000,000,000

1012

Trillion

Giga

G

1,000,000,000

109

Billion

Mega

M

1,000,000

106

Million

kilo

k

1,000

103

Thousand

hecto

h

100

102

Hundred

deca

da

10

101

Ten

Base

b

1

100

One

deci

d

1/10

10-1

Tenth

centi

c

1/100

10-2

Hundredth

milli

m

1/1,000

10-3

Thousandth

micro

µ

1/1,000,000

10-6

Millionth

nano

n

1/1,000,000,000

10-9

Billionth

pico

p

1/1,000,000,000,000

10-12

Trillionth

femto

f

1/1,000,000,000,000,000

10-15

atto

a

1/1,000,000,000,000,000,000

10-18

Quintillionth

zepto

z

1/1,000,000,000,000,000,000,000

10-21

Sextillionth

yocto

y

1/1,000,000,000,000,000,000,000,000

10-24

Septillionth

4-BAND RESISTOR COLOR CODE TABLE

BAND

COLOR

DIGIT

Band 1: 1st Digit

Band 2: 2nd Digit

Band 3: Multiplier
(# of zeros
following 2nd digit)

Black

0

Brown

1

Red

2

Orange

3

Yellow

4

Green

5

Blue

6

Violet

7

Gray

8

White

9

Band 4: Tolerance

Gold

± 5%

SILVER

± 10%

5-BAND RESISTOR COLOR CODE TABLE

BAND

COLOR

DIGIT

Band 1: 1st Digit

Band 2: 2nd Digit

Band 3: 3rd Digit

Band 4: Multiplier
(# of zeros
following 3rd digit)

Black

0

Brown

1

Red

2

Orange

3

Yellow

4

Green

5

Blue

6

Violet

7

Gray

8

White

9

Gold

0.1

SILVER

0.01

Band 5: Tolerance

Gold

± 5%

SILVER

± 10%

EET Formulas & Tables Sheet

Page
1 of
21

UNIT 1: FUNDAMENTAL CIRCUITS

CHARGE

Where:
Q = Charge in Coulombs (C)
Note:
1 C = Total charge possessed by 6.25×1018 electrons

VOLTAGE

Where:
V = Voltage in Volts (V)
W = Energy in Joules (J)
Q = Charge in Coulombs (C)

CURRENT

Where:
I = Current in Amperes (A)
Q = Charge in Coulombs (C)
t = Time in seconds (s)

OHM’S LAW

Where:
I = Current in Amperes (A)
V = Voltage in Volts (V)
R = Resistance in Ohms (Ω)

RESISTIVITY

Where:
ρ = Resistivity in Circular Mil – Ohm per Foot (CM-Ω/ft)
A = Cross-sectional area in Circular Mils (CM)
R = Resistance in Ohms (Ω)
ɭ = Length in Feet (ft)
Note:
CM: Area of a wire with a 0.001 inch (1 mil) diameter

CONDUCTANCE

Where:
G = Conductance in Siemens (S)
R = Resistance in Ohms (Ω)

CROSS-SECTIONAL AREA

Where:
A = Cross-sectional area in Circular Mils (CM)
d = Diameter in thousandths of an inch (mils)

ENERGY

Where:

W = Energy in Joules (J). Symbol
is an italic
W.

P = Power in Watts (W). Unit
is not an italic W.

t = Time in seconds (s)
Note:
1 W = Amount of power when 1 J of energy
is used in 1 s

POWER

Where:
P = Power in Watts (W)
V
= Voltage in Volts (V)

I = Current in Amperes (A)
Note:
Ptrue = P in a resistor is also called true power

OUTPUT POWER

Where:
POUT = Output power in Watts (W)
PIN = Input power in Watts (W)
PLOSS = Power loss in Watts (W)

POWER SUPPLY EFFICIENCY

Where:
POUT = Output power in Watts (W)
PIN = Input power in Watts (W)
Efficiency = Unitless value
Note:
Efficiency expressed as a percentage:

UNIT 2: SERIES CIRCUITS (R1, R2, , Rn)

TOTAL RESISTANCE

Where:
RT = Total series resistance in Ohms (Ω)
Rn
= Circuit’s last resistor in Ohms (Ω)

KIRCHHOFF’S VOLTAGE LAW

Where:
VS = Voltage source in Volts (V)
Vn = Circuit’s last voltage drop in Volts (V)

VOLTAGE – DIVIDER

Where:
Vx = Voltage drop in Ohms (Ω)
Rx
= Resistance where Vx occurs in Ohms (Ω)

RT = Total series resistance in Ohms (Ω)
VS
= Voltage source in Volts (V)

TOTAL POWER

Where:
PT = Total power in Watts (W)
Pn = Circuit’s last resistor’s power in Watts (W)

UNIT 3: PARALLEL CIRCUITS (R1||R2||||Rn)

TOTAL RESISTANCE

Where:
RT = Total parallel resistance in Ohms (Ω)
Rn
= Circuit’s last resistor in Ohms (Ω)

TOTAL RESISTANCE – TWO RESISTORS IN PARALLEL

Where:
RT = Total parallel resistance in Ohms (Ω)

TOTAL RESISTANCE – EQUAL-VALUE RESISTORS

Where:
RT = Total parallel resistance in Ohms (Ω)
R = Resistor Value in Ohms (Ω)
n = Number of equal value resistors (Unitless)

UNKNOWN RESISTOR

Where:
Rx = Unknown resistance in Ohms (Ω)
RA = Known parallel resistance in Ohms (Ω)
RT = Total parallel resistance in Ohms (Ω)

KIRCHHOFF’S CURRENT LAW

Where:
n = Number of currents into node (Unitless)
m = Number of currents going out of node (Unitless)

CURRENT – DIVIDER

Where:
Ix = Branch “x” current in Amperes (A)
RT = Total parallel resistance in Ohms (Ω)
Rx = Branch “x” resistance in Ohms (Ω)
IT = Total current in Amperes (A)

TWO-BRANCH CURRENT – DIVIDER

Where:
I1 = Branch “1” current in Amperes (A)
R2 = Branch “2” resistance in Ohms (Ω)
R1 = Branch “1” resistance in Ohms (Ω)
IT = Total current in Amperes (A)

TOTAL POWER

Where:
PT = Total power in Watts (W)
Pn = Circuit’s last resistor’s power in Watts (W)

OPEN BRANCH RESISTANCE

Where:
ROpen = Resistance of open branch in Ohms (Ω)
RT(Meas) = Measured resistance in Ohms (Ω)
GT(Calc) = Calculated total conductance in Siemens (S)
GT(Meas) = Measured total conductance in Siemens (S)
Note:
GT(Meas) obtained by measuring total resistance, RT(Meas)

UNIT 4: SERIES – PARALLEL CIRCUITS

BLEEDER CURRENT

Where:
IBLEEDER = Bleeder current in Amperes (A)
IT = Total current in Amperes (A)
IRL1 = Load resistor 1 current in Amperes (A)
IRL2 = Load resistor 2 current in Amperes (A)

THERMISTOR BRIDGE OUTPUT

Where:
= Change in output voltage in Volts (V)
= Change in thermal resistance in Ohms (Ω)
VS = Voltage source in Volts (V)
R = Resistance value in Ohms (Ω)

UNKNOWN RESISTANCE IN A WHEATSTONE BRIDGE

Where:
RX = Unknown resistance in Ohms (Ω)
RV = Variable resistance in Ohms (Ω)
R2 = Resistance 2 in Ohms (Ω)
R4 = Resistance 4 in Ohms (Ω)

UNIT 5: MAGNETISM AND ELECTROMAGNETISM

MAGNETIC FLUX DENSITY

Where:
B = Magnetic flux density in Tesla (T)
= Flux in Weber (Wb)
(Greek letter Phi)
A = Cross-sectional area in square meters (m2)
Note:
Tesla (T) equals a Weber per square meter (Wb/m2)

RELATIVE PERMEABILITY

Where:
= Relative permeability (Unitless)
(Greek letter Mu)
= Permeability in Webers per Ampere-turn · meter
(Wb/At·m)
= Vacuum permeability in Webers per Ampere-
turn · meter (Wb/At·m)
Note:
= Wb/ At·m

RELUCTANCE

Where:
R = Reluctance in Ampere-turn per Weber (At/Wb)
ɭ = Length of magnetic path in meters (m)
µ = Permeability in Weber per Ampere-turn · meter
(Wb/At · m)
A = Cross-sectional area in meters squares (m2)

MAGNETOMOTIVE FORCE

Where:
Fm = Magnetomotive force (mmf) in Ampere-turn (At)
N
= Number of Turns of wire (t)

I = Current in Amperes (A)

MAGNETIC FLUX

Where:
= Flux in Weber (Wb)
Fm = Magnetomotive force in Ampere-turn (At)
R = Reluctance in Ampere-turn per Weber (At/Wb)

MAGNETIC FIELD INTENSITY

Where:
H = Magnetic field intensity in Amperes-turn per
meter (At/m)
Fm = Magnetomotive force in Ampere-turn (At)
ɭ = Length of material in meters (m)

INDUCED VOLTAGE

Where:
vind = Induced voltage in Volts (V)
B = Magnetic flux density in Tesla (T)
ɭ = Length of the conductor exposed to the magnetic
field in meters (m)
v = Relative velocity in meters per second (m/s)
Note:
Tesla (T) equals a Weber per square meter (Wb/m2)

Where:
vind = Induced voltage in Volts (V)
N = Number of turns of wire in the coil (Unitless)
= Rate of change of magnetic field with respect
to the coil in Webers per second (Wb/s)

ELECTRIC CIRCUITS II

UNIT 1: ALTERNATE CURRENT & INDUCTORS

ALTERNATE CURRENT

FREQUENCY & PERIOD

Where:
f = Frequency in Hertz (Hz)
T = Period in Seconds (s)
Note:
1 Hertz = 1 cycle per 1 second

PEAK TO PEAK VOLTAGE

Where:
Vpp = Peak to peak voltage in Volts (V)
Vp = Peak voltage in Volts (V)

ROOT MEAN SQUARE (RMS) VOLTAGE

Where:
Vrms = Root mean square voltage in Volts (V)
Vp = Peak voltage in Volts (V)

HALF-CYCLE AVERAGE VOLTAGE

Where:
Vavg = Half-cycle average voltage in Volts (V)
Vp = Peak voltage in Volts (V)

Where:
Degrees = Number of degrees in Degrees (0)
Note:
= 3.1416 (Greek letter Pi)

GENERATOR OUTPUT FREQUENCY

Where:
f = Frequency in Hertz (Hz)
Number of pole pairs = Number of pole pairs (Unitless)
rps = Revolutions per second in Revolutions per
Second (rps)

PEAK TO PEAK CURRENT

Where:
Ipp = Peak to peak current in Amperes (A)
Ip = Peak current in Amperes (A)

ROOT MEAN SQUARE (RMS) CURRENT

Where:
Irms = Root mean square current in Amperes (A)
Ip = Peak current in Amperes (A)

HALF-CYCLE AVERAGE CURRENT

Where:
Iavg = Half-cycle average current in Amperes (A)
Ip = Peak current in Amperes (A)

SINE WAVE GENERAL FORMULA

Where:
y = Instantaneous voltage or current value
at angle in Volts or Amperes (V or A)
(Greek letter Theta)
A = Maximum voltage or current value in Volts or
Amperes (V or A)
= Angle where given instantaneous voltage or
current value exists

SINE WAVE LAGGING THE REFERENCE

Where:
y = Instantaneous voltage or current value
at angle in Volts or Amperes (V or A)
A = Maximum voltage or current value in Volts or
Amperes (V or A)
= Angle where given instantaneous voltage or
current value exists
= Angle sine wave is shifted right (lagging) of
reference (Greek letter Phi)

ANGULAR VELOCITY

Where:
(Small Greek letter omega)
f = Frequency in Hertz (Hz)
Note:
= 3.1416

SINE WAVE VOLTAGE

Where:
v = Sinusoidal voltage in Volts (V)
Vp = Peak voltage in Volts (V)
f = Frequency in Hertz (Hz)
t = Time in Seconds (s)
Note:
= 3.1416

PULSE WAVEFORM AVERAGE VALUE

Where:
vavg = Pulse waveform average value in Volts (V)
baseline = Baseline in Volts (V)
duty cycle = Percent duty cycle in Percent/100%
(Unitless)
Amplitude = Amplitude in Volts (V)

Where:
y = Instantaneous voltage or current value
at angle in Volts or Amperes (V or A)
A = Maximum voltage or current value in Volts or
Amperes (V or A)
= Angle where given instantaneous voltage or
current value exists
= Angle sine wave is shifted left (leading) of
reference

PHASE ANGLE

Where:
t = Time in Seconds (s)

DUTY CYCLE

Where:
Percent duty cycle = Percent duty cycle in Percentage (%)
tw = Pulse width in Seconds (s)
T = Period in Seconds (s)
F = Frequency in Hertz (Hz)

INDUCTORS

INDUCED VOLTAGE

Where:
vind = Induced voltage in Volts (V)
L = Inductance in Henries (H)
= Time rate of change of the current in Amperes
per second (A/s)

INDUCTANCE OF A COIL

Where:
L = Inductance of a coil in Henries (H)
N = Number of turns of wire (Unitless)
= Permeability in Henries per meter (H/m)
A = Cross-sectional area in Meters squared (m2)
= Core length in Meters (m)
Notes:
Permeability in H/m is equal to Wb/At·m
Non-magnetic core = Permeability of a vacuum, µ0
µ0 = 4 x 10-7 H/m

RL TIME CONSTANT

Where:
= RL time constant in Seconds (s) (Greek letter Tau)
L = Inductance in Henries (H)
R = Resistance in Ohms (Ω)

GENERAL EXPONENTIAL VOLTAGE FORMULA

Where:
v = Instantaneous voltage at time, t, in Volts (V)
VF = Voltage final value in Volts (V)
Vi = Voltage initial value in Volts (V)
R = Resistance in Ohms (Ω)
t = Time in Seconds (s)
L = Inductance in Henries (H)

INDUCTOR ENERGY STORAGE

Where:

W = Energy in Joules (J)

L = Inductance in Henries (H)
I = Current in Amperes (A)

TOTAL INDUCTANCE – SERIES

Where:
LT = Total series inductance in Henries (H)
Ln = Circuit’s last inductor in Henries (H)

TOTAL INDUCTANCE – PARALLEL

Where:
LT = Total parallel inductance in Henries (H)
Ln
= Circuit’s last inductor in Henries (H)

RL CIRCUIT CURRENT INCREASE AND DECREASE

FOR GIVEN NUMBER OF TIME CONSTANTS

# of Time Constants

Approx % of Final Current

Approx % of Initial Charge

1

63

37

2

86

14

3

95

5

4

98

2

5

99
Considered 100%

1
Considered 0%

GENERAL EXPONENTIAL CURRENT FORMULA

Where:
i = Instantaneous current at time, t, in Amperes (A)
IF = Current final value in Amperes (A)
Ii = Current initial value in Amperes (A)
R = Resistance in Ohms (Ω)
t = Time in Seconds (s)
L = Inductance in Henries (H)

INDUCTIVE REACTANCE

Where:
XL = Inductive reactance in Ohms (Ω)
f = Frequency in Hertz (Hz)
L = Inductance in Henries (H)
Note:
= 3.1416 (Greek letter “Pi”)

INDUCTOR REACTIVE POWER

Where:
Pr = Reactive Power in Watts (W)
Vrms = Voltage rms in Volts (V)
Irms = Current rms in Amperes (A)
XL = Inductive reactance in Ohms (Ω)

UNIT 2: RL CIRCUITS

SERIES RL CIRCUIT

IMPEDANCE IN RECTANGULAR FORM

Where:

Z = Impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
Note:
Bold letters represent complete phasor quantities.
For example, “
Z” in the formula above

VOLTAGE IN RECTANGULAR FORM

Where:

Vs = Voltage in Volts (V)

VR = Resistor voltage in Volts (V)
VL = Inductor voltage in Volts (V)

INDUCTOR TRUE POWER

Where:
Ptrue = True Power in Watts (W)
Irms = Current rms in Amperes (A)
RW = Winding resistance in Ohms (Ω)

COIL QUALITY FACTOR

Where:
Q = Coil quality factor (Unitless)
XL = Inductive reactance in Ohms (Ω)
RW = Winding resistance of the coil or the resistance
in series with the coil in Ohms (Ω)
Note:
Circuit Q and the coil Q are the same when the resistance is only the coil winding resistance

IMPEDANCE IN POLAR FORM

Where:

Z = Impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
Note:
= Magnitude
= Phase Angle

VOLTAGE IN POLAR FORM

Where:

Vs = Voltage in Volts (V)

VR = Resistor voltage in Volts (V)
VL = Inductor voltage in Volts (V)

ANGLE BETWEEN VOLTAGE IN & OUT

Where:
= Angle between voltage in and out in Degrees (0)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)

OUTPUT VOLTAGE MAGNITUDE

Where:
Vout = Voltage output in Volts (V)
XL = Inductive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)

LAG CIRCUIT

ANGLE BETWEEN VOLTAGE IN & OUT

Where:
= Angle between voltage in and out in Degrees (0)
XL = Inductive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)

OUTPUT VOLTAGE MAGNITUDE

Where:
Vout = Output voltage in Volts (V)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
Vin = Input voltage in Volts (V)

PARALLEL RL CIRCUIT

TOTAL 2-COMPONENT IMPEDANCE

Where:

Z = Total 2-component impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)

CURRENT IN POLAR FORM

Where:

Itot = Total current in Amperes (A)

IR = Resistor current in Amperes (A)
IL = Inductor current in Amperes (A)

Where:

Y = Total admittance in Siemens (S)

G = Conductance in Siemens (S)
BL = Inductive Susceptance in Siemens (S)
Note:

CURRENT IN RECTANGULAR FORM

Where:

Itot = Total current in Amperes (A)

IR = Resistor current in Amperes (A)
IL = Inductor current in Amperes (A)

PARALLEL TO SERIES FORM CONVERSION

Where:
Req = Resistance in Ohms (Ω)
Z = Impedance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
= Angle where given instantaneous voltage or
current value exists

POWER

RL CIRCUIT REACTIVE POWER

Where:
Pr = Reactive power in Volt-Ampere Reactive (VAR)
Itot = Total current in Amperes (A)
XL = Inductive reactance in Ohms (Ω)

UNIT 3: CAPACITORS

CAPACITANCE

Where:
C = Capacitance in Farads (F)
Q = Charge in Coulombs (C)
V = Voltage in Volts (V)

ENERGY STORED IN A CAPACITOR

Where:

W = Energy in Joules (J)

C = Capacitance in Farads (F)
V = Voltage in Volts (V)

DIELECTRIC CONSTANT (RELATIVE PERMITTIVITY)

Where:
= Dielectric constant (Unitless)
(Greek letter Epsilon)
= Absolute permittivity of a material in Farads per
meter (F/m)
= Absolute permittivity of a vacuum in Farads per
meter (F/m)
Note:
= 8.85 x 10-12 F/m

CAPACITANCE

Where:
C = Capacitance in Farads (F)
A = Plate area in Meters squared (m2)
= Dielectric constant (Unitless)
d = Plate separation in Meters (m)
Note:
If d is in mils, 1 mil = 2.54 x 10-5 meters

SERIES CAPACITORS

TOTAL CHARGE

Where:
QT = Total charge in Coulombs (C)
Qn = Circuit’s last capacitor charge in Coulombs (C)

TOTAL CAPACITANCE

Where:
CT = Total series capacitance in Farads (F)
Cn
= Circuit’s last capacitor’s capacitance in

TOTAL CAPACITANCE – TWO CAPACITORS

Where:
CT = Total series capacitance in Farads (F)

VOLTAGE ACROSS A CAPACITOR

Where:
Vx = Voltage drop in Volts (V)
CT = Total series capacitance in Farads (F)
Cx = Capacitor x’s capacitance in Farads (F)
VT = Total voltage in Volts (V)

TOTAL CAPACITANCE – EQUAL-VALUE CAPACITORS

Where:
CT = Total series capacitance in Farads (F)
n = Number of equal value capacitors (Unitless)

PARALLEL CAPACITORS

TOTAL CHARGE

Where:
QT = Total charge in Coulombs (C)
Qn = Circuit’s last capacitor charge in Coulombs (C)

TOTAL CAPACITANCE – EQUAL-VALUE CAPACITORS

Where:
CT = Total series capacitance in Farads (F)
n = Number of equal value capacitors (Unitless)

CAPACITORS IN DC CIRCUITS

RC TIME CONSTANT

Where:
= Time constant in Seconds (s)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)

TOTAL CAPACITANCE

Where:
CT = Total series capacitance in Farads (F)
Cn
= Circuit’s last capacitor’s capacitance in

RC CIRCUIT CURRENT INCREASE AND DECREASE

FOR GIVEN NUMBER OF TIME CONSTANTS

# of Time Constants

Approx % of Final Current

Approx % of Initial Charge

1

63

37

2

86

14

3

95

5

4

98

2

5

99
Considered 100%

1
Considered 0%

GENERAL EXPONENTIAL VOLTAGE FORMULA

Where:
v = Instantaneous voltage at time, t, in Volts (V)
VF = Voltage final value in Volts (V)
Vi = Voltage initial value in Volts (V)
t = Time in Seconds (s)
= Time constant in Seconds (s)

CHARGING TIME TO A SPECIFIED VOLTAGE

Where:
t = Time in Seconds (s)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
v = Specified voltage level in Volts (V)
VF = Final voltage level in Volts (V)
Note:
Assumes Vi = 0 Volts

GENERAL EXPONENTIAL CURRENT FORMULA

Where:
i = Instantaneous current at time, t, in Amperes (A)
IF = Current final value in Amperes (A)
Ii = Current initial value in Amperes (A)
t = Time in Seconds (s)
= Time constant in Seconds (s)

DISCHARGING TIME TO A SPECIFIED VOLTAGE

Where:
t = Time in Seconds (s)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
v = Specified voltage level in Volts (V)
Vi = Initial voltage level in Volts (V)
Note:
Assumes VF = 0 Volts

CAPACITORS IN AC CIRCUITS

INSTANTANEOUS CAPACITOR CURRENT

Where:
i = Instantaneous current in Amperes (A)
C = Capacitance in Farads (F)
= Instantaneous rate of change of the voltage
across the capacitor in Volts per second (V/s)

CAPACITOR REACTIVE POWER

Where:
Pr = Reactive Power in Volt-Ampere Reactive (VAR)
Vrms = Voltage rms in Volts (V)
Irms = Current rms in Amperes (A)
XC = Capacitive reactance in Ohms (Ω)

CAPACITIVE REACTANCE

Where:
XC = Capacitive reactance in Ohms (Ω)
f = Frequency in Hertz (Hz)
C = Capacitance in Farads (F)
Note:
= 3.1416 (Greek letter “Pi”)

SWITCHED-CAPACITORS CIRCUITS

AVERAGE CURRENT

Where:
I1(avg) = Instantaneous current in Amperes (A)
C = Capacitance in Farads (F)
V1 = Voltage 1 in Volts (V)
V2 = Voltage 2 in Volts (V)
T = Period of time in Seconds (s)

UNIT 4: RC CIRCUITS

RC SERIES CIRCUITS

IMPEDANCE IN RECTANGULAR FORM

Where:

Z = Impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)

OHM’S LAW

Where:

I = Current in Amperes (A)

Z = Impedance in Ohms (Ω)

V = Voltage in Volts (V)

VOLTAGE IN RECTANGULAR FORM

Where:

Vs = Voltage in Volts (V)

VR = Resistor voltage in Volts (V)
VC = Capacitor voltage in Volts (V)

ANGLE BETWEEN VOLTAGE IN & OUT

Where:
= Angle between voltage in and out in Degrees (0)
XC = Capacitive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)

EQUIVALENT RESISTANCE

Where:
R = Equivalent resistance in Ohms (Ω)
T = Period of time in Seconds (s)
C = Capacitance in Farads (F)
f = Frequency in Hertz (Hz)

IMPEDANCE IN POLAR FORM

Where:

Z = Impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)

VOLTAGE IN POLAR FORM

Where:

Vs = Voltage in Volts (V)

VR = Resistor voltage in Volts (V)
VC = Capacitor voltage in Volts (V)

OUTPUT VOLTAGE MAGNITUDE

Where:
Vout = Voltage output in Volts (V)
R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)

LAG CIRCUIT

ANGLE BETWEEN VOLTAGE IN & OUT

Where:

= Angle between voltage in and out in Degrees (0)
R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)

RC PARALLEL CIRCUITS

TOTAL 2-COMPONENT IMPEDANCE

Where:

Z = Total 2-component impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)

OHM’S LAW

Where:

I = Current in Amperes (A)

V = Voltage in Volts (V)

Y = Admittance in Siemens (S)

CURRENT IN RECTANGULAR FORM

Where:

Itot = Total current in Amperes (A)

IR = Resistor current in Amperes (A)
IC = Capacitor current in Amperes (A)

PARALLEL TO SERIES FORM CONVERSION

Where:
Req = Resistance in Ohms (Ω)
Z = Impedance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
= Angle where given instantaneous voltage or
current value exists

OUTPUT VOLTAGE MAGNITUDE

Where:
Vout = Voltage output in Volts (V)
XC = Capacitive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)

Where:

Y = Total admittance in Siemens (S)

G = Conductance in Siemens (S)
BC = Capacitive susceptance in Siemens (S)
Note:

CURRENT IN POLAR FORM

Where:

Itot = Total current in Amperes (A)

IR = Resistor current in Amperes (A)
IC = Capacitor current in Amperes (A)

RC SERIES –PARALLEL CIRCUITS

PHASE ANGLE

Where:
Req = Resistance in Ohms (Ω)
Z = Impedance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
= Angle where given instantaneous voltage or
current value exists

POWER

APPARENT POWER

Where:
Pa = Apparent power in Volt-ampere (VA)
I = Current in Amperes (A)
Z = Impedance in Ohms (Ω)

POWER FACTOR

Where:
PF = Power Factor (Unitless)
= Phase angle in Degrees (0)

OSCILLATOR AND FILTER

OSCILLATOR OUTPUT FREQUENCY

Where:
fr = Output frequency in Hertz (Hz)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
Note:
= 3.1416

UNIT 5: RLC CIRCUITS AND PASSIVE FILTERS

RLC SERIES CIRCUITS

TOTAL REACTANCE

Where:
Xtot = Total reactance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)

TOTAL IMPEDANCE IN POLAR FORM

Where:

Z = Total impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
Xtot = Total reactance in Ohms (Ω)
Note:
When XL > XC, the angle is positive
When XC > XL, the angle is negative

TRUE POWER

Where:
Ptrue = True power in Watts (W)
V = Voltage in Volts (V)
I = Current in Amperes (A)
= Phase angle in Degrees (0)

FILTER CUTOFF FREQUENCY

Where:
fc = Cutoff frequency in Hertz (Hz)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
Note:
= 3.1416

TOTAL IMPEDANCE IN RECTANGULAR FORM

Where:

Z = Total impedance in Ohms (Ω)

R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)

RESONANT FREQUENCY

Where:
fr = Resonant frequency in Hertz (Hz)
L = Inductance in Henries (H)
C = Capacitance in Farads (F)
Note:
At resonance, XL = XC and the j terms cancel
= 3.1416

RLC PARALLEL CIRCUITS

TOTAL CURRENT

Where:

Itot = Total current in Amperes (A)

IR = Resistor current in Amperes (A)
IC = Capacitor current in Amperes (A)
IL = Inductor current in Amperes (A)
ICL = Total current into the L and C branches
in Amperes (A)

RLC PARALLEL RESONANCE

RESONANT FREQUENCY – IDEAL

Where:
fr = Resonant frequency in Hertz (Hz)
L = Inductance in Henries (H)
C = Capacitance in Farads (F)
Note:
At resonance, XL = XC and Zr =
= 3.1416

CURRENT AND PHASE ANGLE

Where:
Itot = Total current in Amperes (A)
VS = Voltage source in Volts (V)
Zr = Impedance at resonance in Ohms (Ω)

RESONANT FREQUENCY – PRECISE

Where:
fr = Resonant frequency in Hertz (Hz)
RW = Winding resistance in Ohms (Ω)
C = Capacitance in Farads (F)
L = Inductance in Henries (H)
Note:
= 3.1416

RLC SERIES – PARALLEL CIRCUITS

SERIES-PARALLEL TO PARALLEL CONVERSION

EQUIVALENT INDUCTANCE

Where:
Leq = Equivalent inductance in Henries (H)
L = Inductance in Henries (H)
Q = Coil quality factor (Unitless)

EQUIVALENT PARALLEL RESISTANCE

Where:
Rp(eq) = Equivalent parallel resistance in Ohms (Ω)
RW = Winding resistance in Ohms (Ω)
Q = Coil quality factor (Unitless)

NON-IDEAL TANK CIRCUIT

TOTAL IMPEDANCE AT RESONANCE

Where:
ZR = Total impedance in Ohms (Ω)
RW = Resistance in Ohms (Ω)
Q = Coil quality factor (Unitless)

SPECIAL TOPICS

RESONANT CIRCUIT BANDWIDTH

BANDWIDTH

Where:
BW = Bandwidth in Hertz (Hz)
f2 = Upper critical frequency at Z=0.707·Zmax
in Hertz (Hz)
f1 = Lower critical frequency at Z=0.707·Zmax
in Hertz (Hz)

BANDWIDTH AND QUALTIY FACTOR

Where:
BW = Bandwidth in Hertz (Hz)
fr = Center (resonant) frequency in Hertz (Hz)
Q = Coil quality factor (Unitless)

PASSIVE FILTERS

POWER RATIO IN DECIBELS

Where:
dB = Power ratio in decibels (dB)
Pout = Output power in Watts (W)
Pin = Input power in Watts (W)

OVERALL QUALITY FACTOR WITH AN EXTERNAL LOAD

Where:
QO = Overall quality factor (Unitless)
Rp(tot)= Total parallel equivalent resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)

CENTER (RESONANT) FREQUENCY

Where:
fr = Center (resonant) frequency in Hertz (Hz)
f1 = Lower critical frequency at Z=0.707·Zmax
in Hertz (Hz)
f2 = Upper critical frequency at Z=0.707·Zmax
in Hertz (Hz)

VOLTAGE RATIO IN DECIBELS

Where:
dB = Power ratio in decibels (dB)
Vout = Output voltage in Volts (V)
Vin = Input voltage in Volts (V)

LOW-PASS & HIGH-PASS FILTERS

RC FILTERS

Where:
fC = Filter critical frequency in Hertz (Hz)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
Note:
= 3.1416
At fC, Vout = (0.707)·Vin

SERIES RESONANT BAND-PASS FILTER

Where:
BW = Bandwidth in Hertz (Hz)
f0 = Center frequency in Hertz (Hz)
Q = Coil quality factor (Unitless)

RL FILTERS

Where:
fc = Filter critical frequency in Hertz (Hz)
L = Inductance in Henries (H)
R = Resistance in Ohms (Ω)
Note:
= 3.1416
At fC, Vout = (0.707)·Vin

GENERAL INFORMATION

AREA AND VOLUMES

AREAS

CIRCLE AREA

Where:
A = Circle area in meters squared (m2)
r = Radius in meters (m)
Note:
= 3.1416

RECTANGULAR AND POLAR FORMS

RECTANGULAR FORM

Where:
A = Coordinate value on real axis (Horizontal Plane)
j = j operator
B = Coordinate value on imaginary axis (Vertical Plan)
Note:
“j operator” prefix indicates designated coordinate value is on imaginary axis.

COMPLEX PLANE AND RECTANGULAR FORM PHASOR

+A
-A
+jB
-jB
(A + jB)
(A – jB)
(-A + jB)
(-A – jB)
00/3600
1800
900
2700

POLAR FORM

Where:
C = Phasor magnitude
= Phasor angle relative to the positive real axis

COMPLEX PLANE AND POLAR FORM PHASOR

Real Axis
+j
-j
Length = Magnitude

+

RECTANGULAR TO POLAR CONVERSION

Where:
A = Coordinate value on real axis (Horizontal Plane)
j = j operator
B = Coordinate value on imaginary axis (Vertical Plan)
C = Phasor magnitude
= Phasor angle relative to the positive real axis
Note:
To calculate C:

To calculate in Quadrants 1 and 4 (A is positive):

Use +B for +B values, -B for –B values

To calculate in Quadrants 2 and 3 (A is negative):

Use for +B values
Use for –B values

POLAR TO RECTANGULAR CONVERSION

Where:
C = Phasor magnitude
= Phasor angle relative to the positive real axis
A = Coordinate value on real axis (Horizontal Plane)
j = j operator
B = Coordinate value on imaginary axis (Vertical Plan)
Note:
To calculate A:

To calculate B:

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