# Phasors and Phasor Algebra

“Phasors and Phasor Algebra” seeks to explain the topic thoroughly. The introduction defines the study’s scope and purpose. It introduces phasor fundamentals, applications, and advanced subjects. Introductions prepare readers for further study by familiarizing them with the topic.

Electrical engineering students face a page of complicated exponentials and imaginary values. What dark sorcery is phasor and phasor algebra that’s puzzling you? Phasors aren’t as terrifying as they appear. Phasors are essential to AC circuit analysis, so we’ll explain them in basic English. In 100 words, we’ll explain phasors, why they’re useful, and how phasor algebra simplifies difficult AC network computations. Read on for a simple introduction—no math or electrical engineering skills needed.

## Phasor Definition

This section explains phasors. In electrical engineering and physics, phasors are complex numbers that indicate sinusoidal waveform amplitude and phase. Engineers and scientists can simplify AC system analysis with phasors. The definition of phasors explains how these complex numbers are used to analyze AC circuits and related phenomena.

## Phasor Algebra Importance

Phasor algebra is essential in electrical engineering and circuit analysis. The practical uses and benefits of phasor algebra are discussed in this section. Engineering can simplify difficult AC circuit computations by representing sinusoidal quantities as vectors with phasor algebra. Phasor algebra simplifies AC circuit analysis and design by adding, subtracting, multiplying, and dividing phasors. Its ability to simplify calculations and provide significant insights into AC system performance makes it essential for engineers and researchers.

## What Are Phasors and Phasor Algebra?

When studying AC circuits, you’ll need to comprehend phasors and algebra. Phasors represent sinusoidal waveform amplitude and phase. You can visualize and calculate AC circuit properties.

Vector algebra, or phasor algebra, simplifies phasor representation and manipulation. Complex AC circuits containing resistive and reactive parts like inductors and capacitors can be investigated using phasors and algebra.

### Representing Phasors

A phasor is represented as a vector on an Argand diagram. The length of the vector corresponds to the amplitude or magnitude of the waveform. The angle by which the phasor leads or lags the reference phasor represents the phase angle of the waveform.

For example, in a purely resistive AC circuit, the voltage and current are in phase. So their phasors lie along the same line. In a circuit with inductance, the current lags the voltage by 90 degrees. So the current phasor will be rotated 90 degrees counter-clockwise from the voltage phasor.

Two or more phasors can be added together using the phasor addition rule to determine the resultant phasor. The resultant phasor represents the amplitude and phase of the combined waveforms.

For subtraction, the phasor to be subtracted is rotated by 180 degrees and then added to the other phasor. The magnitude and direction of the resultant phasor gives the difference between the two phasors.

By understanding phasors and applying phasor algebra, you can analyze complex AC circuits with multiple sinusoidal voltages and currents, having different frequencies and phases. Phasor representation provides a simple way to calculate voltages, currents and impedances in these circuits.

## Representing Sinusoidal Quantities as Phasors

Phasors are a way to represent sinusoidal quantities, like voltage, current, and other signals in AC circuits. They provide a visual representation that makes calculations involving these quantities much simpler.

Instead of dealing with the instantaneous values of sine waves, phasors give you a single vector quantity to represent the sine wave. The length of the phasor represents the amplitude or peak value of the sine wave, while the angle of the phasor represents the phase shift of the sine wave.

For example, take a simple sine wave like:

v = 5sin(ωt + π/4)

This represents a sine wave with:

• An amplitude of 5 volts
• An angular frequency of ω radians per second
• A phase shift of π/4 radians

We can represent this sine wave with a phasor that has:

• A length of 5 units (representing the 5V amplitude)
• An angle of π/4 radians (representing the π/4 radian phase shift)

So the phasor for this sine wave would look like:

Using phasors, calculations with multiple sine waves become much more straightforward. You simply add the phasors vectorially, and the resulting phasor will represent the combination of the sine waves. The length and angle of the resulting phasor directly give you the amplitude and phase of the combined waveform.

Phasors provide an intuitive way to visualize and calculate with the sinusoidal signals that are so common in AC circuits and signal processing. By representing the amplitude and phase of sine waves in a simple vector format, phasor algebra allows you to add, subtract, and manipulate these signals with ease.

To add or subtract phasors, you need to consider both magnitude and phase angle. The magnitude (length) of the resultant phasor will be the square root of the sum of the squares of the individual phasors. The phase angle of the resultant phasor will be the arctangent of the imaginary part divided by the real part.

1. Graph the two phasors with the real axis horizontal and the imaginary axis vertical. Place phasors head-to-tail.
2. Finish the parallelogram and measure the diagonal. Resultant phasor magnitude.
3. Measure the resultant phasor-real axis angle. Phase angle of the resulting phasor.
4. Convert the phasor to rectangular (real + imaginary) or polar (magnitude ∠ phase angle) form.

To include phasors 3∠30° and 4∠-60°, draw them, complete the parallelogram, and measure the result as 5∠0°. 5 cos 0° = 5 and 5 sin 0° = 0. The resulting phasor is 5∠0° in polar form.

### Subtracting Phasors

Subtracting two phasors requires similar processes but head-to-tail placement. It will point from the tail of the first phasor to the head of the second. Measure its magnitude and phase angle similarly.

To subtract 4∠45° from 6∠30°, draw the phasors head-to-tail and measure the resultant 5∠15°, with real portion 5 cos 15° = 4.33 and imaginary part 5 sin 15° = 1.25. The resulting phasor is 5∠15° in polar form.

Using phasor addition and subtraction, we can combine sinusoidal signals of the same frequency. The geometric methods for adding and subtracting phasors will become automatic with practice. Have more questions? Let me know!

### Phasor Multiplication/Division

• Phasor multiplication and division require knowledge of vectors. Vectors with constant angular velocity are phasors. Multiplying two phasors yields:
• A magnitude that is the product of the two phasors.
Equal to the sum of the two phasors’ phase angles
• To multiple phasors, multiply magnitudes and add angles.
• For instance, if phasor A has 3 units and a phase angle of 30° and phasor B has 2 units and 45°, the product A x B would have:

Magnitude: 3×2=6.
Phase angle: 75° (30° + 45°).

#### This gives the phasor C = A x B a magnitude of 6 units and a phase angle of 75°.

To divide phasors, reverse operations. Subtract angles from magnitudes. To calculate C ÷ B from the previous example, we would use:

• C (6 units) ÷ B (2 units) = 3 units.
C (75°) – B (45°) = 30° phase angle.

Thus, C ÷ B = A, with a magnitude of 3 units and a phase angle of 30°, as first stated.

Multiply and divide to find the product or ratio of any two phasors. These methods determine consequent phasors when several phasors are merged, which is necessary for studying AC circuits and other phasor mathematics applications.

#### Circuit Analysis Phasor Algebra Applications

Phasor algebra simplifies AC circuit analysis by representing sinusoidal voltages and currents as complex values. Using magnitude and phase angle phasors, we may detect circuit properties without messy sine waves.

Impedance Solving

Use the phasor sum of AC circuit components’ impedances to compute Z. Add the impedance phasors of a resistor and inductor in series to determine their total impedance:

Total = Zresistor + Zinductor

Where Zresistor = R and Zinductor = jωL. The phasor gives the overall impedance magnitude (ohms) and phase angle (degrees).

### Finding Voltage and Current

Use Ohm’s law to compute circuit voltage and current from total impedance. A series circuit:

Itotal x Ztotal = Vtotal

Total voltage and current phasors are Vtotal and Itotal.

Use reciprocity for parallel circuits:

Itotal = V/Z

The magnitudes of the phasors give the voltage and current sine waves’ amplitude, while the phase angles show their separation.

### Power Analysis

AC circuit power is calculated using the complex power equation:

P = VI V and I are the phasor voltage and current, while * is the complex conjugate. Using the complex value, the real component gives you true power in watts and the imaginary part reactive power in var. Calculating power lets you compare circuit energy dissipation to inductor and capacitor energy.

With these phasor algebra rules, you may simplify and comprehend AC circuit behavior. Phasors express and calculate AC quantities intuitively despite their complexity. You’ll quickly analyze impedance, voltage, current, and power with practice!

#### Conclusion

Phasors and algebra simplified. We learned how to encode sinusoidal signals as phasors, plot them on the complex plane, and execute algebraic operations on them. With these basics, you can analyze AC circuits and signals with this powerful method. Remember that Euler’s formula underpins all you study. Practice representing and operating on waveforms as phasors. Soon, phasors will be second nature and an essential engineering analytical tool. You’ll wonder how you survived without them! Now explore the phasor realm.