Alternative Voltage & Phasors

Electrical Engineering — Use these Links to Solve Phasors Problem #ElectricSeries — Episode #02

Hi, Phasor is difficult to grasp…but here are some tips & tricks to solve it in a minute!

Please, to get the context, read the previous episode of this series :)(#ElectricSeries — Episode #01)

GeoGebra Classic - GeoGebra

Let’s cut to the chase:

Determine currents I1, I2, and I3 of the circuit:

Circuit 1. https://www.multisim.com/content/xxcSy56E5U9XQp54tbEkRV/27_phasor_-circuit/open/

As you can see, a complex number is present! (j is i, a complex number in electronics;)

00Step: Download in your Android Phone this calculator:

HiPER Scientific Calculator From HiPER Development Studio

The calculator has up to 100 digits of significand and 9 digits of exponent. It detects repeating decimals and numbers can be also entered as fractions or converted to fractions.

01 Step: Now let’s get down to hard calculation.

First thing first: Make the Equations:

We know that:

V = Vmax.cos(ωt+φ).

As you can see from the circuit, I3 is equal to 5∠0°; this is equal to 5cos(ωt+0) that is 5A.

And 20∠90°V is equal to 20j V. How can we calculate it?

Open you HiPER Scientific Calculator and first, configure it like this:

a) Hit buttons below:

MENU, choose Mode > Expression (Two-line display w/ expression and result) and as DATA RANGE, select Complex, hit OK.

b) Type

20, SHIFT i, 90
=20i

So 20∠90° is equal to 20j!

02 Step: Now let’s apply for MESH 1:

The Second Kirchhoff’s Law:
FOR ANY CLOSED-LOOP Σ V = 0, so:
Start at a, MESH_(1): -8I1 - j10(I1-I3) + j2(I1-I2) = 0

Circuit 1. The circuit again, for your convenience!

03Step: Now open Wolfram|Alpha, and type:

-8x-j*10(x-z)+j*2(x-y)=0
Where:
I1=x
I2=y
I3=z

https://www.wolframalpha.com/input/?i=-8x-j*10%28x-z%29%2Bj*2%28x-y%29%3D0

Go to Expanded form, and you’ll see this:

-8 j x — 2 j y + 10 j z — 8 x = 0

Return I1, I2, and I3 to the equation:

8 j I1 — 2 j I2 + 10 j I3 — 8 I1 = 0

As I3 = 5A, we have:

8 j I1 — 2 j I2 + 10 j * 5 - 8 * I1 = 0
8 j I1 — 8 I1 — 2 j I2 + 50 j = 0
(8 j — 8) I1 - 2 j I2 = - 50 j (First Equation)

04Step: Now let’s apply for MESH 2:

Again, From the Second Kirchhoff’s Law:
FOR ANY CLOSED-LOOP Σ V = 0, so:
Start at b, MESH_(2):  j2 (I2-I1) + j2 (I2-I3) - 4 * I2  - 20 j = 0

Circuit 1. The circuit again, for your convenience!

05 Step: Now another tab for Wolfram|Alpha, and type:

j*2(y-x)+j*2(y-z)-4y-j*20=0
Where:
I1=x
I2=y
I3=z

https://www.wolframalpha.com/input/?i=j*2%28y-x%29%2Bj*2%28y-z%29-4y-j*20%3D0

Go to Expanded form:

-2 j x + 4 j y - 2 j z - 20 j - 4 y = 0

Return I1, I2, and I3 to the equation:

-2 j I1 + 4 j I2 - 2 j I3 - 20 j - 4 I2 = 0

As I3 = 5A, we have:

-2 j I1 + 4 j I2 - 2 j * 5 - 20 j - 4 I2 = 0
-2 j I1 + 4 j I2 - 4 I2 - 30 j = 0
-2 j I1 + (4 j - 4)I2 = 30 j (Second Equation)

06Step: Now open Matrix Calculator:

Type all two-equation at once (if you wish just click the link below):

https://matrixcalc.org/pt/slu.html#solve-using-Cramer%27s-rule%28%7B%7B-8-8%2aj,-j%2a2,0,0,-50%2aj%7D,%7B-2%2aj,4%2aj-4,0,0,30%2aj%7D%7D%29

Hit Solve By Crammer’s rule button:

Answer: 
x1=(35⁢j^2−50⁢j)/(9⁢j^2−8)
x2=(85⁢j^2+60⁢j)/(9⁢j^2−8)

(9⁢j2−8≠0)

07Step: Now Open your HiPER Scientific Calculator; For MESH 1,

And type :

I1 = (35i^2-50i)/(9i^2-8)

Hit More:

Answer:
I1 = 2.0588 + 2.9411 j A (Decimal value)
or
I1 = 35/17 + 50/17 j A (Alternative Fraction)
or
I1 = 3,5901 ∠ 55° A (Polar coordinates)

08 Step: Now Open your HiPER Scientific Calculator; For MESH 2,

And type :

I2 = (85⁢j^2+60⁢j)/(9⁢j^2−8)

Hit More:

Answer:
I2 = 5 - 3.5294 j A (Decimal value)
or
I2 = 5 - 60j/17 A (Alternative Fraction)
or
I2 = 6.1201 ∠ -35° (Polar coordinates)

And finally, All the answers together:

I1= 3,5901 ∠ 55° A
I2= 6.1201 ∠ -35° A
I3= 5 A

We have reached the end.

I hope you can take advantage of these tips & tricks.

I made a point of registering this step by step tutorial because I know that in a few weeks I will not remember anything else … a memory that’s no survival’s memory lasts 3 weeks …

And that is it!

bye!

Credits & References

FELIPE DE AGUIAR NEVES — Bachelor’s degree in Electrical Engineering (emphasis in Electronics) from FEI’s University Center (2008). I received an M.S. degree in microelectronics from FEI University Center, Brazil in 2011, and a Ph.D. degree in microelectronics at the University of Sao Paulo (USP) in joint collaboration with KU Leven / imec, Belgium in 2015.

PRISCILA ERTMANN BOLZAN — Graduated (2015) and Master (2017) in Electrical Engineering from the Federal University of Santa Maria (UFSM)

Book: Fundamentals of Electric Circuits Charles K. Alexander Department of Electrical and Computer Engineering Cleveland State University Matthew N. O. Sadiku Department of Electrical Engineering Prairie View A&M University.

iLectureOnline — A collection of organized lecture lists and search bar make your studies much easier! — Physics, Astronomy, Chemistry, Maths, Engineering.

This is my circuits: https://www.multisim.com/contributors/420092-gilj3/my-circuits/

No great work is ever done in a hurry. To develop a great scientific discovery, to print a great picture, to write an immortal poem, to become a minister, or a famous general — to do anything great requires time, patience, and perseverance. These things are done by degrees, “little by little.” — W. J. Wilmont Buxton

From https://en.wikipedia.org/wiki/Phasor

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01#Episode — #ElectricSeries — Electricity — Classroom Notes — Electricity — Working Out A Solution! — How to Calculate Circuits Efficiently Online.

02#Episode — #ElectricSeries — Electricity —Alternative Voltage & Phasors — Electrical Engineering — Use these Links to Solve Phasors Problem. (this one)