# Electric Circuits II

Show all steps leading to the final answer, where applicable . See attachemnt

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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

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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

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
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
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
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
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:
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:

The relationship between degrees and radians can be determined as follows:

Therefore, we can calculate the angle as:

12

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

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:

Phasor diagram showing Vout

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 :

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,

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.

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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

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
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
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
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
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:
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:

The relationship between degrees and radians can be determined as follows:

Therefore, we can calculate the angle as:

12

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

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:
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:

Phasor diagram showing Vout

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 :

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,

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.

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ELECTRIC CIRCUITS I

METRIC PREFIX TABLE

Metric

Prefix

Symbol

Multiplier

Expo-

nential

Description

Yotta

Y

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

1024

Septillion

Zetta

Z

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

1021

Sextillion

Exa

E

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

1018

Quintillion

Peta

P

1,000,000,000,000,000

1015

Tera

T

1,000,000,000,000

1012

Trillion

Giga

G

1,000,000,000

109

Billion

Mega

M

1,000,000

106

Million

kilo

k

1,000

103

Thousand

hecto

h

100

102

Hundred

deca

da

10

101

Ten

Base

b

1

100

One

deci

d

1/10

10-1

Tenth

centi

c

1/100

10-2

Hundredth

milli

m

1/1,000

10-3

Thousandth

micro

µ

1/1,000,000

10-6

Millionth

nano

n

1/1,000,000,000

10-9

Billionth

pico

p

1/1,000,000,000,000

10-12

Trillionth

femto

f

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

10-15

atto

a

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

10-18

Quintillionth

zepto

z

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

10-21

Sextillionth

yocto

y

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

10-24

Septillionth

4-BAND RESISTOR COLOR CODE TABLE

BAND

COLOR

DIGIT

Band 1: 1st Digit

Band 2: 2nd Digit

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

Black

0

Brown

1

Red

2

Orange

3

Yellow

4

Green

5

Blue

6

Violet

7

Gray

8

White

9

Band 4: Tolerance

Gold

± 5%

SILVER

± 10%

5-BAND RESISTOR COLOR CODE TABLE

BAND

COLOR

DIGIT

Band 1: 1st Digit

Band 2: 2nd Digit

Band 3: 3rd Digit

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

Black

0

Brown

1

Red

2

Orange

3

Yellow

4

Green

5

Blue

6

Violet

7

Gray

8

White

9

Gold

0.1

SILVER

0.01

Band 5: Tolerance

Gold

± 5%

SILVER

± 10%

EET Formulas & Tables Sheet

Page
1 of
21

UNIT 1: FUNDAMENTAL CIRCUITS

CHARGE

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

VOLTAGE

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

CURRENT

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

OHM’S LAW

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

RESISTIVITY

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

CONDUCTANCE

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

CROSS-SECTIONAL AREA

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

ENERGY

Where:

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

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

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

POWER

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

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

OUTPUT POWER

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

POWER SUPPLY EFFICIENCY

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

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

TOTAL RESISTANCE

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

KIRCHHOFF’S VOLTAGE LAW

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

VOLTAGE – DIVIDER

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

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

TOTAL POWER

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

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

TOTAL RESISTANCE

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

TOTAL RESISTANCE – TWO RESISTORS IN PARALLEL

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

TOTAL RESISTANCE – EQUAL-VALUE RESISTORS

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

UNKNOWN RESISTOR

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

KIRCHHOFF’S CURRENT LAW

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

CURRENT – DIVIDER

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

TWO-BRANCH CURRENT – DIVIDER

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

TOTAL POWER

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

OPEN BRANCH RESISTANCE

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

UNIT 4: SERIES – PARALLEL CIRCUITS

BLEEDER CURRENT

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

THERMISTOR BRIDGE OUTPUT

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

UNKNOWN RESISTANCE IN A WHEATSTONE BRIDGE

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

UNIT 5: MAGNETISM AND ELECTROMAGNETISM

MAGNETIC FLUX DENSITY

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

RELATIVE PERMEABILITY

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

RELUCTANCE

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

MAGNETOMOTIVE FORCE

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

I = Current in Amperes (A)

MAGNETIC FLUX

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

MAGNETIC FIELD INTENSITY

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

INDUCED VOLTAGE

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

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

ELECTRIC CIRCUITS II

UNIT 1: ALTERNATE CURRENT & INDUCTORS

ALTERNATE CURRENT

FREQUENCY & PERIOD

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

PEAK TO PEAK VOLTAGE

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

ROOT MEAN SQUARE (RMS) VOLTAGE

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

HALF-CYCLE AVERAGE VOLTAGE

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

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

GENERATOR OUTPUT FREQUENCY

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

PEAK TO PEAK CURRENT

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

ROOT MEAN SQUARE (RMS) CURRENT

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

HALF-CYCLE AVERAGE CURRENT

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

SINE WAVE GENERAL FORMULA

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

SINE WAVE LAGGING THE REFERENCE

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

ANGULAR VELOCITY

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

SINE WAVE VOLTAGE

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

PULSE WAVEFORM AVERAGE VALUE

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

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

PHASE ANGLE

Where:
t = Time in Seconds (s)

DUTY CYCLE

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

INDUCTORS

INDUCED VOLTAGE

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

INDUCTANCE OF A COIL

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

RL TIME CONSTANT

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

GENERAL EXPONENTIAL VOLTAGE FORMULA

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

INDUCTOR ENERGY STORAGE

Where:

W = Energy in Joules (J)

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

TOTAL INDUCTANCE – SERIES

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

TOTAL INDUCTANCE – PARALLEL

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

RL CIRCUIT CURRENT INCREASE AND DECREASE

FOR GIVEN NUMBER OF TIME CONSTANTS

# of Time Constants

Approx % of Final Current

Approx % of Initial Charge

1

63

37

2

86

14

3

95

5

4

98

2

5

99
Considered 100%

1
Considered 0%

GENERAL EXPONENTIAL CURRENT FORMULA

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

INDUCTIVE REACTANCE

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

INDUCTOR REACTIVE POWER

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

UNIT 2: RL CIRCUITS

SERIES RL CIRCUIT

IMPEDANCE IN RECTANGULAR FORM

Where:

Z = Impedance in Ohms (Ω)

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

VOLTAGE IN RECTANGULAR FORM

Where:

Vs = Voltage in Volts (V)

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

INDUCTOR TRUE POWER

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

COIL QUALITY FACTOR

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

IMPEDANCE IN POLAR FORM

Where:

Z = Impedance in Ohms (Ω)

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

VOLTAGE IN POLAR FORM

Where:

Vs = Voltage in Volts (V)

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

ANGLE BETWEEN VOLTAGE IN & OUT

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

OUTPUT VOLTAGE MAGNITUDE

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

LAG CIRCUIT

ANGLE BETWEEN VOLTAGE IN & OUT

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

OUTPUT VOLTAGE MAGNITUDE

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

PARALLEL RL CIRCUIT

TOTAL 2-COMPONENT IMPEDANCE

Where:

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

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

CURRENT IN POLAR FORM

Where:

Itot = Total current in Amperes (A)

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

Where:

Y = Total admittance in Siemens (S)

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

CURRENT IN RECTANGULAR FORM

Where:

Itot = Total current in Amperes (A)

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

PARALLEL TO SERIES FORM CONVERSION

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

POWER

RL CIRCUIT REACTIVE POWER

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

UNIT 3: CAPACITORS

CAPACITANCE

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

ENERGY STORED IN A CAPACITOR

Where:

W = Energy in Joules (J)

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

DIELECTRIC CONSTANT (RELATIVE PERMITTIVITY)

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

CAPACITANCE

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

SERIES CAPACITORS

TOTAL CHARGE

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

TOTAL CAPACITANCE

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

TOTAL CAPACITANCE – TWO CAPACITORS

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

VOLTAGE ACROSS A CAPACITOR

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

TOTAL CAPACITANCE – EQUAL-VALUE CAPACITORS

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

PARALLEL CAPACITORS

TOTAL CHARGE

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

TOTAL CAPACITANCE – EQUAL-VALUE CAPACITORS

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

CAPACITORS IN DC CIRCUITS

RC TIME CONSTANT

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

TOTAL CAPACITANCE

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

RC CIRCUIT CURRENT INCREASE AND DECREASE

FOR GIVEN NUMBER OF TIME CONSTANTS

# of Time Constants

Approx % of Final Current

Approx % of Initial Charge

1

63

37

2

86

14

3

95

5

4

98

2

5

99
Considered 100%

1
Considered 0%

GENERAL EXPONENTIAL VOLTAGE FORMULA

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

CHARGING TIME TO A SPECIFIED VOLTAGE

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

GENERAL EXPONENTIAL CURRENT FORMULA

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

DISCHARGING TIME TO A SPECIFIED VOLTAGE

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

CAPACITORS IN AC CIRCUITS

INSTANTANEOUS CAPACITOR CURRENT

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

CAPACITOR REACTIVE POWER

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

CAPACITIVE REACTANCE

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

SWITCHED-CAPACITORS CIRCUITS

AVERAGE CURRENT

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

UNIT 4: RC CIRCUITS

RC SERIES CIRCUITS

IMPEDANCE IN RECTANGULAR FORM

Where:

Z = Impedance in Ohms (Ω)

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

OHM’S LAW

Where:

I = Current in Amperes (A)

Z = Impedance in Ohms (Ω)

V = Voltage in Volts (V)

VOLTAGE IN RECTANGULAR FORM

Where:

Vs = Voltage in Volts (V)

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

ANGLE BETWEEN VOLTAGE IN & OUT

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

EQUIVALENT RESISTANCE

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

IMPEDANCE IN POLAR FORM

Where:

Z = Impedance in Ohms (Ω)

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

VOLTAGE IN POLAR FORM

Where:

Vs = Voltage in Volts (V)

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

OUTPUT VOLTAGE MAGNITUDE

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

LAG CIRCUIT

ANGLE BETWEEN VOLTAGE IN & OUT

Where:

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

RC PARALLEL CIRCUITS

TOTAL 2-COMPONENT IMPEDANCE

Where:

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

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

OHM’S LAW

Where:

I = Current in Amperes (A)

V = Voltage in Volts (V)

Y = Admittance in Siemens (S)

CURRENT IN RECTANGULAR FORM

Where:

Itot = Total current in Amperes (A)

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

PARALLEL TO SERIES FORM CONVERSION

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

OUTPUT VOLTAGE MAGNITUDE

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

Where:

Y = Total admittance in Siemens (S)

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

CURRENT IN POLAR FORM

Where:

Itot = Total current in Amperes (A)

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

RC SERIES –PARALLEL CIRCUITS

PHASE ANGLE

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

POWER

APPARENT POWER

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

POWER FACTOR

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

OSCILLATOR AND FILTER

OSCILLATOR OUTPUT FREQUENCY

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

UNIT 5: RLC CIRCUITS AND PASSIVE FILTERS

RLC SERIES CIRCUITS

TOTAL REACTANCE

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

TOTAL IMPEDANCE IN POLAR FORM

Where:

Z = Total impedance in Ohms (Ω)

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

TRUE POWER

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

FILTER CUTOFF FREQUENCY

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

TOTAL IMPEDANCE IN RECTANGULAR FORM

Where:

Z = Total impedance in Ohms (Ω)

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

RESONANT FREQUENCY

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

RLC PARALLEL CIRCUITS

TOTAL CURRENT

Where:

Itot = Total current in Amperes (A)

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

RLC PARALLEL RESONANCE

RESONANT FREQUENCY – IDEAL

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

CURRENT AND PHASE ANGLE

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

RESONANT FREQUENCY – PRECISE

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

RLC SERIES – PARALLEL CIRCUITS

SERIES-PARALLEL TO PARALLEL CONVERSION

EQUIVALENT INDUCTANCE

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

EQUIVALENT PARALLEL RESISTANCE

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

NON-IDEAL TANK CIRCUIT

TOTAL IMPEDANCE AT RESONANCE

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

SPECIAL TOPICS

RESONANT CIRCUIT BANDWIDTH

BANDWIDTH

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

BANDWIDTH AND QUALTIY FACTOR

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

PASSIVE FILTERS

POWER RATIO IN DECIBELS

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

OVERALL QUALITY FACTOR WITH AN EXTERNAL LOAD

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

CENTER (RESONANT) FREQUENCY

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

VOLTAGE RATIO IN DECIBELS

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

LOW-PASS & HIGH-PASS FILTERS

RC FILTERS

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

SERIES RESONANT BAND-PASS FILTER

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

RL FILTERS

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

GENERAL INFORMATION

AREA AND VOLUMES

AREAS

CIRCLE AREA

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

RECTANGULAR AND POLAR FORMS

RECTANGULAR FORM

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

COMPLEX PLANE AND RECTANGULAR FORM PHASOR

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

POLAR FORM

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

COMPLEX PLANE AND POLAR FORM PHASOR

Real Axis
+j
-j
Length = Magnitude

+

RECTANGULAR TO POLAR CONVERSION

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

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

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

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

Use for +B values
Use for –B values

POLAR TO RECTANGULAR CONVERSION

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

To calculate B:

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