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
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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 |
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.
Address the following items in your Introduction:
· 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.
Address the following in your conclusions:
· 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
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ELECTRIC CIRCUITS I
METRIC PREFIX TABLE
Metric
Prefix
Symbol
Multiplier
(Traditional Notation)
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
Quadrillion
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
Quadrillionth
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)
FARADAY’S LAW
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)
RADIAN & DEGREE CONVERSION
Where:
Rad = Number of radians in Rad (rad)
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:
= Angular velocity in Radians per second (rad/s)
(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)
SINE WAVE LEADING 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 left (leading) of
reference
PHASE ANGLE
Where:
= Angle sine wave is shifted in Radians (rad)
= Angular velocity in Radians per second (rad/s)
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)
LEAD CIRCUIT
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)
TOTAL ADMITTANCE
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
Farads (F)
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
Farads (F)
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)
LEAD CIRCUIT
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)
TOTAL ADMITTANCE
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
Quadrant 1
Quadrant 3
Quadrant 4
-A
+jB
-jB
(A + jB)
(A – jB)
(-A + jB)
(-A – jB)
Quadrant 2
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
Quadrant 1
Quadrant 3
Quadrant 4
+j
-j
Length = Magnitude
–
Quadrant 2
+
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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