Show all steps leading to the final answer, where applicable . See attachemnt
NEED DONE ASAP
College of Technology
2022
ESET 111
Electric Circuits II
Midterm Exam
Name: __________________________ Date: ____________________ |
Exam time: 3 Hours
Show all steps leading to the final answer, where applicable, for partial credit.
1. A certain sine wave has a frequency of 2 kHz and a peak value of V = 10 VP. Assuming a given cycle begins at t = 0 s (zero crossing).
a. What is the change in voltage from t1 = 0 µs to t2 = 125 µs.
2. Initially, the capacitors in the following circuit are uncharged. Calculate the following values.
a. After the switch is closed, how much charge is supplied by the source?
b. What is the voltage across each capacitor?
3. For the circuit shown below, calculate:
a. The total circuit current.
b. The branch currents through L2 and L3
c. The voltage across each inductor.
4. For the circuit shown below, perform the following tasks.
a. Find the circuit impedance in both rectangular and polar coordinates.
b. Find the total circuit current.
c. Draw the phasor diagram showing the circuit voltage and current and the phase angle.
5. For the circuit below, calculate the following:
a. Determine the circuit impedance.
b. Determine the total circuit current.
c. Find the voltage magnitude across each circuit element.
Rev September 2022
Page 1
ESET 111 Week 3: Capacitors and RC Circuits
Chapter 12 Objectives:
Describe characteristics of a capacitor
Analyze series and parallel capacitors
Analyze capacitors in DC circuits
Analyze capacitors in AC circuits
Chapter 15 Objectives:
Determine relationship between current and voltage in an RC circuit
Determine impedance of series, parallel, and series-parallel RC circuits
Analyze series, parallel, and series-parallel RC circuits
Weekly Assignments:
3.1 Discussion: Application of RC Circuits
3.2 Review Assignment: Capacitors and RC Circuits
3.3 Quiz: Capacitors and RC Circuits (Practice)
3.4 Exam: Midterm
3.1 Discussion: Applications of RL Circuits
Capacitive Touch Screens
Run and Start Capacitors
Myth Buster: Capacitors
Capacitor Discharging
Advantages and Disadvantages: Capacitors
Supercapacitors
Troubleshooting Capacitors
Volatile Digital Memory
3.2 Review Assignment: Inductors and RL Circuits
12-1 The Basic Capacitor
12-2 Types of Capacitors
12-3 Series Capacitors
12-4 Parallel Capacitors
12-5 Capacitors in DC Circuits
12-6 Capacitors in AC Circuits
12-7 Capacitor Applications
15-1 The Complex Number System
15-2 Sinusoidal Response of Series RC Circuits
15-3 Impedance of Series RC Circuits
15-4 Analysis of Series RC Circuits
15-5 Impedance and Admittance of Parallel RC Circuits
15-6 Analysis of Parallel RC Circuits
15-7 Analysis of Series-Parallel RC Circuits
15-8 Power in RC Circuits
15-9 Basic Applications
15-10 Troubleshooting
Chapter 15: The Complex Number System
Complex Numbers allow us to do mathematical calculations on phasor quantities in out AC circuits. Numbers are plotted on the complex plane. Numbers one the complex plane can be represented in either polar or rectangular format.
A complex number in rectangular coordinates is written as Re + j Im
A complex number in polar coordinates is written as
4
Chapter 15: Rectangular to Polar Conversion
General
Convert rectangular to coordinates as follows:
The evaluation of the inverse tangent depends upon the quadrant of the angle.
Tan-1 (the principal arctangent) is only defined for -90° to 90°.
If the resultant angle is in the 2nd quadrant, you must add 180° to the result from your calculator.
If the resultant angle is in the 3rd quadrant, you must subtract 180° from the results of your calculator.
We like to express our angles from -180° to 180°
5
Chapter 15: Rectangular to Polar Conversion
First Quadrant
Convert the following number to rectangular coordinates:
Given:
10 + j 500
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
510
78.1°
6
Y-Values 0 100 0 0 500 Column1 0 100 0 Column2 0 100 0
Chapter 15: Rectangular to Polar Conversion
Second Quadrant
Convert the following number to rectangular coordinates:
Given:
-122 + j 340
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
361
109.7°
7
Y-Values 0 -122 0 340 Column1 0 -122 Column2 0 -122
Chapter 15: Rectangular to Polar Conversion
Third Quadrant
Convert the following number to rectangular coordinates:
Given:
-222 – j 230
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
320
-134°
8
Y-Values 0 -222 0 -230 Column1 0 -222 Column2 0 -222
Chapter 15: Rectangular to Polar Conversion
Fourth Quadrant
Convert the following number to rectangular coordinates:
Given:
416 – j 450
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
613
-47.2°
9
Y-Values 0 416 0 -450 Column1 0 416 Column2 0 416
Chapter 15: Polar to Rectangular Conversion
General
Convert rectangular to coordinates as follows:
Drop a perpendicular from the endpoint of the vector to the real axis. This forms a right triangle
Using trigonometry (soh cah toh):
Similarly:
Mag
θ
10
Y1 0 0 0 2 2 0 0 0 0 2 Y2 0 0 0 2 2 0 0 0 Y3 0 0 0 2 2 0 0 2
Real Axis
Imaginary Axis
Chapter 15: Convert Radians to Degrees
Convert the following angular value from radians to degrees.
Given:
θ = ½ π radians
Find:
θ (in degrees)
The relationship between degrees and radians can be determined as follows:
Therefore, we can calculate the angle as:
11
Chapter 15: Convert Degrees to Radians
Convert the following angular value from radians to degrees.
Given:
θ = 260°
Find:
θ (in radians)
The relationship between degrees and radians can be determined as follows:
Therefore, we can calculate the angle as:
12
Chapter 15: Adding Complex Number
Add the following complex numbers.
Given:
A = 4 + j 3
B = 7 – j 1
Find:
A + B
Complex numbers must be added in rectangular coordinates. To add complex numbers, add the real parts and add the imaginary parts. Note how the negative sign is included with the number.
13
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 12 + j 15
B = 6 – j 5
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
Polar Multiplication: To multiply numbers in polar coordinates, convert all numbers to polar form, then multiply the magnitudes and add the phase angles.
14
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 12 + j 15
B = 6 – j 5
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
Rectangular Multiplication: To multiply numbers in rectangular coordinates, use FOIL. Recall that j2 = -1
15
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 9 + j (-4)
B = 27 40°
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
For mixed coordinate multiplication, convert both numbers to the same coordinate system and follow the procedures for multiplication, My preference is to multiply in polar coordinates.
16
Chapter 12: Energy in a Capacitor
What is the energy stored in a 7.2 µF capacitor with a voltage of 8.2 V.
Given:
C = 7.2 µF (capacitance)
V = 8.2 V (voltage)
Find:
W = ½ C V2 (energy)
This formula can be derived by integrating the power over time.
Energy stored in the electric field of a capacitor is found as follows:
17
Chapter 12: Calculate Series Capacitors
What is the total series capacitance for the following circuit?
Given:
C1 = 5.7 µF
C2 = 11.9 µF
C3 = 17.1 µF
Find:
CT (Total capacitance)
The total capacitance of series capacitors is calculated as follows:
Or
C=3.15 µF
18
Chapter 12: Voltage Across Series Capacitors Given Charge
Find the Voltage across C2.
Given:
C1 = 7.1 µF
C2 = 10.3 µF
C3 = 9.2 µF
QT = 51 µC
Find:
V2 (voltage across C2)
When a voltage (potential difference) is applied across series capacitors, each capacitor takes on the same charge. In addition, this is the same charge as that across all capacitors:
We also know that for capacitors:
19
Chapter 15: Voltage Across Series Capacitors Given Vs
Find the Voltage across C2.
Given:
C1 = 17.5 µF
C2 = 8.5 µF
C3 = 10.5 µF
VS = 13.1 V
Find:
VC1 (voltage across C1)
Charge across series capacitors:
We also know that for capacitors:
Therefore:
20
Chapter 12: Voltage Across Series Capacitors Given Vs
Find the Voltage across C2.
Given:
C1 = 17.5 µF
C2 = 8.5 µF
C3 = 10.5 µF
VS = 13.1 V
Find:
VC1 (voltage across C1)
From the previous slide:
Find CT:
21
Chapter 12: Calculate Parallel Capacitance
What is the total parallel capacitance for the following circuit?
Given:
C1 = 13.4 µF
C2 = 7.0 µF
C3 = 13.5 µF
Find:
CT (Total capacitance)
The total capacitance of parallel capacitors is calculated as follows:
C=33.9 µF
22
Chapter 12: Calculate Series-Parallel Capacitor Voltage
What is the voltage between nodes A and B?
Given:
C1 = 147 pF
C2 = 147 pF
C3 = 1470 pF
C4 = 565 pF
C5 = 1470 pF
C6 = 565 pF
VS = 10.3 V
Find:
VAB
First calculate the total capacitance for the voltage divider:
Now perform voltage divider between C5 and C6:
23
Chapter 12: RC Time Constant
The following circuit shows a capacitor and a resistor in DC circuit. What is the time constant, , for the circuit?
Given:
R = 1.2 kΩ
C = 0.13 µF
Find:
(time constant)
The circuit time constant for and RC circuit determines the rate at which voltage changes in the circuit.
Find the time constant using the following equation:
24
Chapter 12: Capacitor Charging Value
A charging capacitor will reach what percent of its final value in 0.7 time-constants? Assume the capacitor is initially uncharged.
The capacitor instantaneous voltage is found using:
Where,
VF is the final voltage across the capacitor
vC is the capacitor voltage
is the RC time constant
Start with the capacitor instantaneous voltage:
In our problem
Therefore,
The capacitor reaches 50.3% of its final value in 0.7 time-constants.
25
Chapter 12: Capacitor Instantaneous Voltage
Find the voltage across the capacitor 22 µs after the switch is closed.
Given:
VS = 39 V
R = 9.5 kΩ
L = 1.8 µF
Find:
VC (22 µs)
First find the time constant using the following equation:
Next determine the final capacitor voltage after the transient response:
Finally, calculate the instantaneous voltage using the following equation:
26
Chapter 12: Time to Reach Full Charge
How long does it take fort the capacitor to reach full charge once the switch closes? Assume the capacitor is initially uncharged.
Given:
R = 1.4 kΩ
C = 0.19 µF
Find:
Time to full charge
Find the time constant using the following equation:
From the universal exponential curves,
27
Chapter 12: Calculate Reactance Given Capacitance
What is the value of reactance, XC, for the following circuit given the frequency and capacitance.
Given:
C= 0.039 µH
f = 2 kHz
Find:
XC (reactance)
Capacitive reactance is the opposition to sinusoidal current, expressed in Ohms. The equation for inductive reactance is:
In our problem,
28
Chapter 12: Series-Parallel Capacitive Voltage
Find the voltage between nodes A and B?
Given:
C1 = 13 µF
C2 = 18 µF
C3 = 9.2 µF
C4 = 9.7 µF
Find:
CAB (Total inductance)
First calculate the series capacitance of C1 and C2:
Next find the parallel combination of C3 and C4:
29
Chapter 12: Series-Parallel Capacitive Voltage
Find the voltage between nodes A and B?
Given:
C1 = 13 µF
C2 = 18 µF
C3 = 9.2 µF
C4 = 9.7 µF
Find:
CAB (Total inductance)
From the previous slide:
Now find the total capacitance:
Use voltage divider for capacitors,
30
Chapter 15: Calculate Circuit Impedance
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 130 Ω
XC = 220 Ω
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
In Rectangular Coordinates:
In Polar Coordinates:
31
Chapter 15: Calculate Circuit Impedance in Rectangular Coordinates
What is the impedance for the following circuit in both rectangular coordinates?
Given:
R = 47 kΩ
C = 2.2 nF
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
Find XC:
f = 100 Hz
f = 500 Hz
f = 2.5 kHz
Impedance for an RL circuit is given by:
Find XC:
f = 100 Hz
f = 500 Hz
f = 2.5 kHz
32
Chapter 15: Calculate Circuit Impedance –
Series Capacitors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 110 kΩ
R2 = 65 kΩ
C1 = 0.047 µF
C2 = 0.047 µF
f = 225 kHz
Find:
Z (Impedance)
Impedance for an RC circuit is given by:
First, find the total series resistance and capacitance:
33
Chapter 15: Calculate Circuit Impedance –
Series Capacitors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 110 kΩ
R2 = 65 kΩ
C1 = 0.047 µF
C2 = 0.047 µF
f = 225 Hz
Find:
Z (Impedance)
Impedance for an RC circuit is given by:
Next find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
34
Chapter 15: Calculate Circuit Impedance –
Parallel Capacitors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 40 kΩ
C1 = 100 pF
C2 = 470 pF
f = 20 kHz
Find:
Z (Impedance)
Impedance for an RC circuit is given by:
First, find the total parallel capacitance:
Next find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
35
Chapter 15: Calculate Impedance of
Parallel Circuits
Determine the impedance magnitude and phase angle in polar coordinates for the following circuit.
Given:
R = 2.2 kΩ
XC = 2.0 kΩ
Find:
Z (Impedance)
To find the impedance of a parallel circuit, start by finding the Admittance:
Where:
First, find G and :
Next find Y in rectangular coordinates:
36
Chapter 15: Calculate Impedance of
Parallel Circuits
Determine the impedance magnitude and phase angle in polar coordinates for the following circuit.
Given:
R = 2.2 kΩ
XC = 2.0 kΩ
Find:
Z (Impedance)
From the previous slide:
Convert to polar coordinates:
Find Z in polar coordinates:
You can also convert back to rectangular coordinates:
37
Chapter 15: Circuit Analysis Series RC Circuit
Determine the total current for the following circuit in polar coordinates?
Given:
R1 = 110 kΩ
R2 = 55 kΩ
C1 = 0.01 µF
C2 = 0.047 µF
f = 200 Hz
V =
Find:
I (Current)
Start by finding the total series resistance and series capacitance: :
Next find the capacitive reactance:
Then find the circuit impedance:
38
Chapter 15: Circuit Analysis Series RC Circuit
Determine the total current for the following circuit in polar coordinates?
Given:
R1 = 110 kΩ
R2 = 55 kΩ
C1 = 0.01 µF
C2 = 0.047 µF
f = 200 Hz
V =
Find:
I (Current)
From the previous slide:
Use Ohm’s Law to find the current:
39
Chapter 15: Circuit Parameters and Voltages
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 47 Ω
C = 100 µF
f = 20 Hz
V =
Find:
Impedance for an RC circuit is given by:
First, find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
Use Ohm’s Law to find the circuit current:
The voltage across the resistor is given by:
The voltage across the capacitors is given by:
40
Chapter 15: Circuit Current and Component Voltages
Determine the total current and the voltage across the resistor and capacitors for the following circuit in polar coordinates?
Given:
R = 10 kΩ
C1 = 470 pF
C2 = 220 pF
f = 10 kHz
V =
Find:
Impedance for an RC circuit is given by:
First, find the total parallel capacitance:
Next find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
Use Ohm’s Law to find the circuit current:
41
Chapter 15: Circuit Current and Component Voltages
Determine the total current and the voltage across the resistor and capacitors for the following circuit in polar coordinates?
Given:
R = 10 kΩ
C1 = 470 pF
C2 = 220 pF
f = 10 kHz
V =
Find:
From the previous slide:
The voltage across the resistor is given by:
The voltage across the capacitors is given by:
Notice that :
42
Chapter 15: RC Lag Circuit Phase Lag
Determine the phase shift between the input and the output voltage for the following RC lag circuit.
Given:
R = 1.5 kΩ
C = 0.27 µF
VS =
f = 250 kHz
Find:
(Phase lag of RC circuit)
The phase lag of an RC circuit is given by the following equation:
Notice the similarity and difference with the phase angle which is given by:
Find capacitive reactance:
Phase Lead:
Phasor diagram showing Vout
Leading Vin (Floyd, 2019)
43
Chapter 15: Analysis of Parallel RC Circuit
Determine the total circuit current as well as the resistor and capacitor voltages and currents.
Given:
R = 1.5 kΩ
XC = 1.6 kΩ
VS =
Find:
VR, IR, VC, IC, IS
There are at least two approaches to this problem. I will start by finding the branch currents and then adding them:
Find VR and VC:
Use Ohm’s Law to find IR and IL :
44
Chapter 15: Analysis of Parallel RC Circuit
Determine the total circuit current as well as the resistor and inductor voltages and currents.
Given:
R = 1.5 kΩ
XL = 1.6 kΩ
VS =
Find:
VR, IR, VC, IC, IS
From the last slide:
Use Ohm’s Law to find IL :
Now add the two:
Convert to polar coordinates:
45
Chapter 16: Power in an RC Circuit
Determine the total, true, and reactive power for the following circuit.
Given:
R = 130Ω
XC = 220 Ω
VS = 10 V
Find:
PTrue, PReactive, PApparent,
Power in an RC circuit is given by:
First find Z:
Next, find I (magnitude only):
46
Chapter 16: Power in an RC Circuit
Determine the total, true, and reactive power for the following circuit.
Given:
R = 130Ω
XC = 220 Ω
VS = 10 V
Find:
PTrue, PReactive, PApparent,
Find true power:
Find reactive power:
Find apparent power:
From the previous slide:
Power in an RL circuit is given by:
47
Calculus Based Problem: Derive the Equation for Capacitive Reactance
Derive the expression for capacitive reactance.
Taking derivative:
From this:
Finally,
Start with the equation for inductor voltage:
Voltage is:
Substituting:
48
Calculus Based Problem: Circuit Analysis in the Time Domain
In the following circuit, determine the current through the resistor (VR) and the current through the capacitor (VC) as well as the total circuit current.
Given:
C = 6.8 µF
R = 220 kΩ
V
Find iR:
Find iS:
Find iC:
49
3.3 Quiz: Inductors and RC Circuits
Chapter 12 Series Capacitance
Chapter 12 Parallel Capacitance
Chapter 15 Circuit Current and Component Voltages
Chapter 15 Series RC Circuit Analysis
Chapter 15 Parallel RC Circuit Analysis
References
Floyd, Thomas, L. and David M. Buchla. Principles of Electric Circuits. Available from: VitalSource Bookshelf, (10th Edition). Pearson Education (US), 2019.
image1
image2
image3
image4
image5
image6
image7
image8
image9
image10
image11
image12
image13
image14
image15
image16
image17
image18
image19
image20
image21
image22
image23
image28
image24
image25
image26
image27
image29
image30
image31
image32
image33
image34
image35
image36
image37
image38
image39
image40
image41
image42
image43
image44
image45
image46
image47
image48
image49
image50
image51
image52
image53
image54
image55
image56
image57
image58
image59
image60
image61
image62
image63
image64
image65
image66
image67
image68
image69
image70
image71
image72
image73
image74
image75
image76
image77
image78
image760
image770
image79
image80
image81
image810
image82
image83
image84
image85
image87
image86
image88
image89
image91
image92
image90
image94
image93
image96
image97
image98
image99
image100
image101
image102
image103
image95
image105
ESET 111 Week 2: Inductors and RL Circuits
Chapter 13 Objectives:
Describe characteristics of an inductor
Analyze series and parallel inductors
Analyze inductors in DC circuits
Analyze inductors in AC circuits
Chapter 15 Objective:
The Complex Number System
Chapter 16 Objectives:
Determine relationship between current and voltage in an RL circuit
Determine impedance of series, parallel, and series-parallel RL circuits
Analyze series, parallel, and series-parallel RL circuits
Weekly Assignments:
2.1 Discussion: Applications of RL Circuits
2.2 Review Assignment: Inductors and RL Circuits
2.3 Quiz: Inductors and RL Circuits
2.1 Discussion: Applications of RL Circuits
Fluorescent Light Ballast
Inductor Loop Circuit
Myth Buster: Inductors
Induction Cooktop
Advantages and Disadvantages: Inductors
Transformers
Troubleshooting Inductors
Inductive Pass Filters
2.2 Review Assignment: Inductors and RL Circuits
13-1 The Basic Inductor
13-2 Types of Inductors
13-3 Series and Parallel Inductors
13-4 Inductors in DC Circuits
13-5 Inductors in AC Circuits
13-6 Inductor Applications
15-1 The Complex Number System
16-1 Sinusoidal Response of Series RL Circuits
16-2 Impedance of Series RL Circuits
16-3 Analysis of Series RL Circuits
16-4 Impedance and Admittance of Parallel RL Circuits
16-5 Analysis of Parallel RL Circuits
16-6 Analysis of Series-Parallel RL Circuits
16-7 Power in RL Circuits
16-8 Basic Applications
16-9 Troubleshooting
Chapter 15: The Complex Number System
Complex Numbers allow us to do mathematical calculations on phasor quantities in out AC circuits. Numbers are plotted on the complex plane. Numbers one the complex plane can be represented in either polar or rectangular format.
A complex number in rectangular coordinates is written as Re + j Im
A complex number in polar coordinates is written as
4
Chapter 15: Rectangular to Polar Conversion
General
Convert rectangular to coordinates as follows:
The evaluation of the inverse tangent depends upon the quadrant of the angle.
Tan-1 (the principal arctangent) is only defined for -90° to 90°.
If the resultant angle is in the 2nd quadrant, you must add 180° to the result from your calculator.
If the resultant angle is in the 3rd quadrant, you must subtract 180° from the results of your calculator.
We like to express our angles from -180° to 180°
5
Chapter 15: Rectangular to Polar Conversion
First Quadrant
Convert the following number to rectangular coordinates:
Given:
100 + j 500
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
510
78.1°
6
Y-Values 0 100 0 0 500 Column1 0 100 0 Column2 0 100 0
Chapter 15: Rectangular to Polar Conversion
Second Quadrant
Convert the following number to rectangular coordinates:
Given:
-122 + j 340
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
361
109.7°
7
Y-Values 0 -122 0 340 Column1 0 -122 Column2 0 -122
Chapter 15: Rectangular to Polar Conversion
Third Quadrant
Convert the following number to rectangular coordinates:
Given:
-222 – j 230
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
320
-134°
8
Y-Values 0 -222 0 -230 Column1 0 -222 Column2 0 -222
Chapter 15: Rectangular to Polar Conversion
Fourth Quadrant
Convert the following number to rectangular coordinates:
Given:
416 – j 450
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
613
-47.2°
9
Y-Values 0 416 0 -450 Column1 0 416 Column2 0 416
Chapter 15: Polar to Rectangular Conversion
General
Convert rectangular to coordinates as follows:
Drop a perpendicular from the endpoint of the vector to the real axis. This forms a right triangle
Using trigonometry (soh cah toh):
Similarly:
Mag
θ
10
Y1 0 0 0 2 2 0 0 0 0 2 Y2 0 0 0 2 2 0 0 0 Y3 0 0 0 2 2 0 0 2
Real Axis
Imaginary Axis
Chapter 15: Convert Radians to Degrees
Convert the following angular value from radians to degrees.
Given:
θ = ½ π radians
Find:
θ (in degrees)
The relationship between degrees and radians can be determined as follows:
Therefore, we can calculate the angle as:
11
Chapter 15: Convert Degrees to Radians
Convert the following angular value from radians to degrees.
Given:
θ = 260°
Find:
θ (in radians)
The relationship between degrees and radians can be determined as follows:
Therefore, we can calculate the angle as:
12
Chapter 15: Adding Complex Number
Add the following complex numbers.
Given:
A = 4 + j 3
B = 7 – j 1
Find:
A + B
Complex numbers must be added in rectangular coordinates. To add complex numbers, add the real parts and add the imaginary parts. Note how the negative sign is included with the number.
13
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 12 + j 15
B = 6 – j 5
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
Polar Multiplication: To multiply numbers in polar coordinates, convert all numbers to polar form, then multiply the magnitudes and add the phase angles.
14
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 12 + j 15
B = 6 – j 5
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
Rectangular Multiplication: To multiply numbers in rectangular coordinates, use FOIL. Recall that j2 = -1
15
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 9 + j (-4)
B = 27 40°
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
For mixed coordinate multiplication, convert both numbers to the same coordinate system and follow the procedures for multiplication, My preference is to multiply in polar coordinates.
16
Chapter 13: Induced Voltage
What is the voltage induced across a 4.9 H inductor when the current changes at a rate of 4.2 A/s.
Given:
L = 4.9 H (inductance)
I = 42 A (current)
Find:
(induced voltage)
Inductance is the measure of a coils ability to establish an induced voltage as a result of a change in its current. This is an application of Faraday’s Law.
The induced voltage i:
17
Chapter 13: Energy in an Inductor
What is the energy stored in a 1.1 mH inductor with a current of 2 A.
Given:
L = 1.1 mH (inductance)
I = 2 A (current)
Find:
W = ½ L I2 (energy)
This formula can be derived by integrating the power over time.
Energy stored in the magnetic field of an inductor is found as follows:
18
Chapter 13: Calculate Series Inductance
What is the total series inductance for the following circuit?
Given:
L1 = 5 mH
L2 = 21 mH
L3 = 7 mH
Find:
LT (Total inductance)
The total inductance of series inductors is calculated as follows:
L= 33 mH
19
Chapter 13: Calculate Series Inductance – Mixed Units
What is the total series inductance for the following circuit?
Given:
L1 = 21 mH
L2 = 21 mH
L3 = 1900 µH
Find:
LT (Total inductance)
The total inductance of series inductors is calculated as follows:
L= 43.9 mH
20
Chapter 13: Calculate Parallel Inductace
What is the total parallel inductance for the following circuit?
Given:
L1 = 1 mH
L2 = 5 mH
L3 = 4 mH
Find:
LT (Total inductance)
The total inductance of parallel inductors is calculated as follows:
Or
L=690 µH
21
Chapter 13: Calculate Series-Parallel Inductance
What is the total inductance for the following circuit?
Given:
L1 = 110 mH
L2 = 48 mH
L3 = 39 mH
Find:
LT (Total inductance)
First calculate the series inductance of L2 and L3:
Next find the parallel combination of L1 and L2-3
L= 48.58 µH
22
Chapter 13: Calculate Series-Parallel Inductance
What is the total inductance for the following circuit?
Given:
L1 = 103 mH
L2 = 27 mH
L3 = 30 mH
Find:
LT (Total inductance)
First calculate the parallel inductance of L2 and L3:
Next find the series combination of L1 and L2-3
23
Chapter 13: Calculate Series-Parallel Inductance
What is the total inductance for the following circuit?
Given:
L1 = 797 µH
L2 = 502 µH
L3 = 696 µH
L4 = 743 µH
Find:
LT (Total inductance)
First calculate the parallel inductance of L1 and L2:
First calculate the parallel inductance of L3 and L4:
Next find the series combination of L1-2 and L3-4
=
24
Chapter 13: RL Time Constant
The following circuit shows an inductor and a resistor in DC circuit. What is the time constant, , for the circuit?
Given:
R = 1.2 kΩ
L = 16 mH
Find:
(time constant)
The circuit time constant for and RL circuit determines the rate at which current changes in the circuit.
Find the time constant using the following equation:
25
Chapter 13: Current in a DC Circuit
The following circuit shows an inductor and a resistor in DC circuit. What is the value of the current 14.71 µs after the switch is closed? The initial current is zero.
Given:
VS = 13 V
R = 3.4 kΩ
L = 25 mH
Find:
I (14.71 µs)
First find the time constant using the following equation:
Next determine the final current after the transient response:
Finally, calculate the current using the following equation:
26
Chapter 13: Resistor Instantaneous Voltage
The following circuit shows an inductor and a resistor in DC circuit. What is the time constant of the circuit in µs? What is the instantaneous voltage across the resistor after 210.52 µs?
Given:
VS = 14 V
R = 380 Ω
L = 20 mH
Find:
VR (210.52 µs)
First find the time constant using the following equation:
Next determine the final resistor voltage after the transient response:
Finally, calculate the instantaneous voltage using the following equation:
The inductor instantaneous voltage would be found using:
27
Chapter 13: Calculate Reactance Given Inductance
What is the value of reactance, XL for the following circuit given the frequency and inductance.
Given:
L= 32.6 mH
f = 100 Hz
Find:
XL (inductance)
Inductive reactance is the opposition to sinusoidal current, expressed in Ohms. The equation for inductive reactance is:
In our problem,
28
Chapter 13: Calculate Inductance Given Reactance
What is the value of inductance, L for the following circuit given the frequency and reactance.
Given:
XL = 43.7 kΩ
f = 70 kHz
Find:
L (inductance)
The equation for inductive reactance is:
Solve this equation for L:
29
Chapter 16: Calculate Circuit Impedance
Changing Coordinate Systems
Determine R and XL given the following value of impedance:
Z = 930 Ω + j 300 Ω
R = 930 Ω
XL= 300 Ω
Determine R and XL given the following value of impedance:
30
Chapter 16: Calculate Circuit Impedance
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 440 Ω
L = 6 mH
f = 9 kHz
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
First find the inductive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
31
Chapter 16: Calculate Circuit Impedance –
Series Inductors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 590 Ω
R2 = 180 Ω
L1 = 2 mH
L2 = 10 mH
f = 12 kHz
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
First, find the total series resistance and inductance:
Next find the inductive reactance:
In Rectangular Coordinates:
32
Chapter 16: Calculate Circuit Impedance –
Series Inductors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 590 Ω
R2 = 180 Ω
L1 = 2 mH
L2 = 10 mH
f = 12 kHz
Find:
Z (Impedance)
From the previous slide:
In Polar Coordinates:
33
Chapter 16: Calculate Circuit Impedance –
Parallel Inductors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 380 Ω
L1 = 25 mH
L2 = 47 mH
f = 45 kHz
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
First, find the total parallel inductance:
Next find the inductive reactance:
In Rectangular Coordinates:
34
Chapter 16: Calculate Circuit Impedance –
Parallel Inductors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 380 Ω
L1 = 25 mH
L2 = 47 mH
f = 45 kHz
Find:
Z (Impedance)
From the previous slide:
In Polar Coordinates:
35
Chapter 16: Calculate Circuit Impedance –
Series-Parallel Inductors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 310 Ω
R2 = 910 Ω
R3 = 870 Ω
L1 = 9 mH
L2 = 7 mH
L3 = 3 mH
f = 35 kHz
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
First, find the total series-parallel resistance and inductance:
36
Chapter 16: Calculate Circuit Impedance –
Series-Parallel Inductors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 310 Ω
R2 = 910 Ω
R3 = 870 Ω
L1 = 9 mH
L2 = 7 mH
L3 = 3 mH
f = 35 kHz
Find:
Z (Impedance)
Next find the inductive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
37
Chapter 16: Calculate Impedance of
Parallel Circuits
Determine the impedance magnitude and phase angle in polar coordinates for the following circuit.
Given:
R = 1.1 kΩ
XL = 2.7 kΩ
Find:
Z (Impedance)
To find the impedance of a parallel circuit, start by finding the Admittance:
Where:
First, find G and :
Next find Y in rectangular coordinates:
38
Chapter 16: Calculate Impedance of
Parallel Circuits
Determine the impedance magnitude and phase angle in polar coordinates for the following circuit.
Given:
R = 1.1 kΩ
XL = 2.7 kΩ
Find:
Z (Impedance)
From the previous slide:
Convert to polar coordinates:
Find Z in polar coordinates:
You can also convert back to rectangular coordinates:
39
Chapter 16: Circuit Analysis Simple RL Circuit
Find the current and the magnitude of the resistor and inductor voltage for the following circuit.
Given:
R = 400 Ω
L = 11 mH
VS =
f = 8 kHz
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
Next find the inductive reactance:
In Rectangular Coordinates:
In polar coordinates:
40
Chapter 16: Circuit Analysis Simple RL Circuit
Find the current and the magnitude of the resistor and inductor voltage for the following circuit.
Given:
R = 400 Ω
L = 11 mH
VS =
f = 8 kHz
Find:
Z (Impedance)
From the previous slide:
Calculate current using Ohm’s Law:
Find the resistor and inductor voltages:
41
Chapter 16: Circuit Analysis Series Inductors
Find the current and the magnitude of the resistor and inductor voltages for the following circuit.
Given:
R1 = 100 Ω
R2 = 170 Ω
L1 = 6 mH
L2 = 16 mH
VS =
f = 5 kHz
Find:
Z (Impedance)
Impedance in polar coordinates:
Use Ohm’s Law to find current:
Calculate the resistor voltages using Ohm’s Law:
42
Chapter 16: Circuit Analysis Series Inductors
Find the current and the magnitude of the resistor and inductor voltages for the following circuit.
Given:
R1 = 100 Ω
R2 = 170 Ω
L1 = 6 mH
L2 = 16 mH
VS =
f = 5 kHz
Find:
Z (Impedance)
From the previous slide:
Find inductive reactance:
Calculate the inductor voltages using Ohm’s Law:
43
Chapter 16: Circuit Analysis Parallel Inductors
What is the total current for the following circuit?
Given:
R = 520 Ω
L1 = 14 mH
L2 = 22 mH
VS = 5 V
f = 6 kHz
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
First, find the total parallel inductance:
Next find the inductive reactance:
In Rectangular Coordinates:
44
Chapter 16: Circuit Analysis Parallel Inductors
What is the total current for the following circuit?
Given:
R = 520 Ω
L1 = 14 mH
L2 = 22 mH
VS =
f = 6 kHz
Find:
Z (Impedance)
In Rectangular Coordinates:
In Polar coordinates:
Use Ohm’s Law to find the current:
45
Chapter 16: RL Lead Circuit Phase Lead
Determine the phase shif between the input and the output voltage for the following RL lead circuit.
Given:
R = 1.2 kΩ
L = 6.8 mH
VS =
f = 400 kHz
Find:
(Phase lead of RL circuit)
The phase lead of an RL circuit is given by the following equation:
Notice the similarity and difference with the phase angle which is given by:
Find inductive reactance:
Phase Lead:
Phasor diagram showing Vout
Leading Vin (Floyd, 2019)
46
Chapter 16: Analysis of Parallel RL Circuit
Determine the total circuit current as well as the resistor and inductor voltages and currents.
Given:
R = 1.8 kΩ
XL = 2.2 kΩ
VS =
Find:
VR, IR, VL, IL, IS
There are at least two approaches to this problem. I will start by finding the branch currents and then adding them:
Find VR and VL:
Use Ohm’s Law to find IR and IL :
47
Chapter 16: Analysis of Parallel RL Circuit
Determine the total circuit current as well as the resistor and inductor voltages and currents.
Given:
R = 1.8 kΩ
XL = 2.2 kΩ
VS =
Find:
VR, IR, VL, IL, IS
From the last slide:
Use Ohm’s Law to find IL :
Now add the two:
Convert to polar coordinates:
48
Chapter 16: Power in an RL Circuit
Determine the total, true, and reactive power for the following circuit.
Given:
R = 2.2 kΩ
L = 47 mH
VS = 10 V
f = 7 kHz
Find:
PTrue, PReactive, PApparent,
Power in an RL circuit is given by:
Find the inductive reactance:
Then find Z:
Next, find I (magnitude only):
49
Chapter 16: Power in an RL Circuit
Determine the total, true, and reactive power for the following circuit.
Given:
R = 2.2 kΩ
L = 47 mH
VS = 10 V
f = 7 kHz
Find:
PTrue, PReactive, PApparent,
Find true power:
Find reactive power:
Find apparent power:
From the previous slide:
Power in an RL circuit is given by:
50
Calculus Based Problem: Derive the Equation for Inductive Reactance
Derive the expression for inductive reactance.
Taking derivative:
From this:
Finally,
Start with the equation for inductor voltage:
Current is:
Substituting:
51
Calculus Based Problem: Circuit Analysis in the Time Domain
In the following circuit, determine the voltage across the resistor (VR) and the voltage across the inductor (VL). The total voltage and the energy stored in the inductor.
Given:
L = 2.7 H
R = 1.5 Ω
A
Find VR:
Find VL:
Find VS:
Find WL:
52
2.3 Quiz: Inductors and RL Circuits
Chapter 13 Inductance and Reactance
Chapter 13 Inductor in a DC Circuit
Chapter 16 Circuit Analysis Parallel Inductors
Chapter 16 Circuit Analysis Series Inductors
Chapter 16 Power in a Series RL Circuit
References
Floyd, Thomas, L. and David M. Buchla. Principles of Electric Circuits. Available from: VitalSource Bookshelf, (10th Edition). Pearson Education (US), 2019.
image1
image2
image3
image4
image5
image6
image7
image8
image9
image10
image11
image12
image13
image14
image15
image16
image17
image18
image19
image20
image21
image22
image23
image28
image24
image25
image26
image27
image29
image30
image31
image32
image33
image34
image35
image36
image37
image38
image370
image39
image400
image40
image42
image41
image43
image44
image45
image46
image47
image48
image49
image50
image51
image52
image54
image53
image56
image55
image57
image58
image59
image60
image61
image62
image63
image64
image65
image66
image67
image68
image69
image70
image71
image72
image73
image74
image75
image76
image77
image78
image79
image80
image81
image82
image83
image84
image85
image86
image87
image88
image89
image90
image91
image92
image93
image94
image95
image96
image950
image960
image97
image98
image99
image100
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:
Select your paper details and see how much our professional writing services will cost.
Our custom human-written papers from top essay writers are always free from plagiarism.
Your data and payment info stay secured every time you get our help from an essay writer.
Your money is safe with us. If your plans change, you can get it sent back to your card.
We offer more than just hand-crafted papers customized for you. Here are more of our greatest perks.
Get instant answers to the questions that students ask most often.
See full FAQWe complete each paper from scratch, and in order to make you feel safe regarding its authenticity, we check our content for plagiarism before its delivery. To do that, we use our in-house software, which can find not only copy-pasted fragments, but even paraphrased pieces of text. Unlike popular plagiarism-detection systems, which are used by most universities (e.g. Turnitin.com), we do not report to any public databases—therefore, such checking is safe.
We provide a plagiarism-free guarantee that ensures your paper is always checked for its uniqueness. Please note that it is possible for a writing company to guarantee an absence of plagiarism against open Internet sources and a number of certain databases, but there is no technology (except for turnitin.com itself) that could guarantee no plagiarism against all sources that are indexed by turnitin. If you want to be 100% sure of your paper’s originality, we suggest you check it using the WriteCheck service from turnitin.com and send us the report.
Yes. You can have a free revision during 7 days after you’ve approved the paper. To apply for a free revision, please press the revision request button on your personal order page. You can also apply for another writer to make a revision of your paper, but in such a case, we can ask you for an additional 12 hours, as we might need some time to find another writer to work on your order.
After the 7-day period, free revisions become unavailable, and we will be able to propose only the paid option of a minor or major revision of your paper. These options are mentioned on your personal order page.