Notes - Fisika - Alternating Current

f with hook equals 1 over T

y equals A sine omega t

E equals E sub m sine omega t equals N times B times A times omega times sine omega t

Where:

• f with hook is frequency

• T is period

• A is amplitude

• omega is phase(?)

Oscilloscope measures maximum voltage, voltmeter measures effective voltage.

V sub cap E equals the fraction with numerator V sub max and denominator the square root of 2

I sub cap E equals the fraction with numerator I sub max and denominator the square root of 2

Effective voltage is basically the amount of DC voltage needed to match the AC voltage.

How do you read an oscilloscope?

The vertical axis uses a scale based on volts per centimeter.

V sub max equals s sub y times A

The horizontal axis uses a scale based on miliseconds per centimeter.

T equals s sub x times lamda

Simple Examples

Problem 1

V sub max equals 80 times 3 equals 240

T equals 2.5 times 8 equals 20 milliseconds equals 20 over 1000 equals 1 over 50 (in seconds)

f with hook equals the fraction with numerator 1 and denominator 1 over 50 equals 50 Hz

(B)

Problem 2

V sub max equals 25 times 2 equals 50

V sub cap E equals the fraction with numerator 50 and denominator the square root of 2 equals 25 the square root of 2

T equals 4 times one tenth equals two fifths equals the fraction with numerator 2 and denominator 5 times 1000 equals 1 over 2500 (in seconds)

f with hook equals 2500 Hz

(D)

Problem 3

Subproblem 3.1

V sub max equals 4 times 10 equals 40

V sub cap E equals the fraction with numerator 40 and denominator the square root of 2 equals 20 the square root of 2

Subproblem 3.2

T equals 8 times 2 equals 16 milliseconds equals 16 over 1000 equals 2 over 125 (in seconds)

f with hook equals the fraction with numerator 1 and denominator 2 over 125 equals 125 over 2 (Hz)

Subproblem 3.3

Amplitudonya jadi 4 times 10 over 20 equals 2 kotak

Subproblem 3.4

Periodenya jadi 8 times two fifths equals sixteen fifths equals 3.2 kotak

Rangkaian AC

I equals I sub max sine omega t

V equals V sub max sine omega t

Resistors still resist.

V equals L times d I over d t equals L times d over d t times I sub max times sine omega t equals L times I sub max times omega times cosine omega t

V equals V sub max times cosine omega t

Inductors resist. omega L is the resistance x sub L. omega equals 2 pi f with hook

Recall that I equals I sub max sine omega t. This means that inductors resist and move the phase of the voltage. The voltage starts 90 degrees earlier.

I equals I sub max times sine omega t

V equals V sub max times the sine of open paren omega t plus 90 degrees close paren

If you draw the phasor diagram, the voltage will always be more anticlockwise.

Rangkaian R-L

V equals the square root of V sub R squared plus V sub L squared

open paren I Z close paren squared equals open paren I R close paren squared plus open paren I x sub L close paren squared

Z equals the square root of R squared plus x sub L squared

I equals the fraction with numerator V sub R and denominator R equals the fraction with numerator V sub L and denominator x sub L equals V over Z

tangent phi equals the fraction with numerator V sub L and denominator V sub R equals the fraction with numerator x sub L and denominator R

I equals I sub max sine omega t

V equals V sub max the sine of open paren omega t plus phi close paren

Circuit Examples

Problem 1

Subproblem 1.1

x sub L equals omega L equals 50 times 0.1 equals 5 ohms

Subproblem 1.2

Z equals the square root of 12 squared plus 5 squared equals 13 ohms

Subproblem 1.3 and 1.4

tangent phi equals the fraction with numerator x sub L and denominator R equals 5 over 12

I equals 260 sine 50 t over 13 equals 20 the sine of open paren 50 t minus 22.6198649 degrees close paren

I sub max equals 20

I sub cap E equals the fraction with numerator 20 and denominator the square root of 2 equals 10 the square root of 2

Problem 2

Subproblem 2.1

omega equals 2 pi 50 equals 100 pi

V equals 100 sine 100 pi t

10 equals 100 over R comma R equals 10 ohms

8 equals the fraction with numerator 100 and denominator x sub L comma x sub L equals 12.5 ohms

x sub L equals 40 over 50 times 12.5 equals 10 ohms

Resistance of inductors is affected by the frequency.

Z equals the square root of 10 squared plus 10 squared equals 10 the square root of 2

I equals V over Z equals 150 over 10 the square root of 2 equals fifteen halves the square root of 2

Subproblem 2.2

V sub R equals fifteen halves the square root of 2 times 10 equals 75 the square root of 2

V sub L equals fifteen halves the square root of 2 times 10 equals 75 the square root of 2

Problem 3

Oh my god bruh.

R equals 60 over 2 equals 30 ohms

Z equals 100 over 2 equals 50 ohms

50 squared equals 30 squared plus x sub L squared

x sub L equals 40 ohms equals 2 pi times 50 times L

L equals 2 over 5 pi

Problem 4

Subproblem 4.1

tangent 45 degrees equals the fraction with numerator x sub L and denominator 25

x sub L equals 25 ohms equals omega L equals 50 times L

L equals one half

Subproblem 4.2

Z equals the square root of 25 squared plus 25 squared equals 25 the square root of 2

2 the sine of open paren 50 t minus 45 degrees close paren equals V over 25 the square root of 2

V equals 50 the square root of 2 the sine of open paren 50 t minus 45 degrees close paren

But don't forget about the phase shift of positive 45 degrees.

V equals 50 the square root of 2 sine 50 t

Capacitors

q equals C times V

d q equals C d V

d q over d t equals C times d V over d t

I sub cap E equals C times d V over d t

d V equals 1 over C times I d t

I equals I sub max sine omega t

the integral of d V equals V equals 1 over C times the integral of I d t equals 1 over C the integral of I sub max sine omega t d t equals negative 1 over C the fraction with numerator I sub max and denominator omega cosine omega t

V equals the fraction with numerator I sub max and denominator omega C times negative cosine omega t equals V sub max times negative cosine omega t equals V sub max times the sine of open paren omega t minus 90 degrees close paren

x sub C equals 1 over omega C

q adalah muatan, Coulomb

C adalah capacitance, Farad

x sub C adalah reaktansi kapasitif

Rangkaian R-C

it's the same, but with different symbols.

Ex. 1

Subproblem 1.1

250 times 10 to the negative 6 power equals C times 250

C equals 1 over 1000000 equals 10 to the negative 6 power

x sub C equals the fraction with numerator 1 and denominator 100 times 250 times 10 to the negative 6 power equals 40 ohms

Subproblem 1.2

Z equals the square root of 40 squared plus 30 squared equals 50

Subproblem 1.3

I equals 250 sine 100 t over 50 equals 5 sine 100 t

therefore I sub max equals 5 amperes

Subproblem 1.4

tangent theta equals 40 over 30 equals four thirds equals 53 degrees

I equals 5 the sine of open paren 100 t minus 53 degrees close paren

Ex. 2

62.5 coulomb per volt

omega equals 50

V equals 200 sine 50 t

Subproblem 2.1

x sub C equals the fraction with numerator 1 and denominator 50 times 62.5 times 10 to the negative 6 power equals 320 ohms

Z equals the square root of 320 squared plus 240 squared equals 400 ohms

Subproblem 2.2

I equals 200 sine 50 t over 400 equals one half sine 50 t

tangent theta equals 320 over 240 equals four thirds

theta equals 53 degrees

I equals one half the sine of open paren 50 t minus 53 degrees close paren

Subproblem 2.3

V sub R equals 240 times one half equals 120

V sub C equals 320 times one half equals 160

Ex. 3

V sub R equals 15

V sub max equals 25

Subproblem 3.1

I sub max equals the fraction with numerator V sub R and denominator R equals 15 over 30 equals one half

Subproblem 3.2

25 equals the square root of 15 squared plus V sub C squared

V sub C equals 20 equals I x sub C

20 equals one half x sub C

x sub C equals 40 ohms equals 1 over omega C equals 1 over 100 C

40 C equals 1 over 100

C equals 1 over 4000 equals 2.5 times 10 to the negative 4 power

Rangkaian R-L-C

In the phasor diagram, V sub R and I have an angle of 0 degrees. V sub L has an angle of 90 degrees and V sub C has an angle of 270 degrees. This means that V sub L and V sub C have opposing directions. This is essential to understanding the formula below, making sure to simplify it instead of using vectors directly.

V equals the square root of V sub R squared plus open paren V sub L minus V sub C close paren squared

I times Z equals the square root of open paren I R close paren squared plus open paren I x sub L minus I x sub C close paren squared

Z equals the square root of R squared plus open paren x sub L minus x sub C close paren squared

I equals V over Z equals the fraction with numerator V sub L and denominator x sub L equals the fraction with numerator V sub R and denominator R equals the fraction with numerator V sub C and denominator x sub C

tangent theta equals the fraction with numerator x sub L minus x sub C and denominator R

x sub L is greater than x sub C right arrow V sub L is greater than V sub C bersifat induktif, V mendahului sebesartheta

x sub C is greater than x sub L right arrow V sub C is greater than V sub L bersifat kapasitif, V tertinggal sebesar theta

x sub L equals x sub C right arrow V sub C equals V sub L terjadi resonansi, maka f with hook squared equals 1 over 4 pi squared L C right arrow f with hook equals 1 over 2 pi the square root of L C

Ex. 1

omega equals 100

x sub C equals the fraction with numerator 1 and denominator 100 times 125 times 10 to the negative 6 power equals 80 ohms

x sub L equals 100 times 1.6 equals 160 ohms

Z equals the square root of 80 squared plus open paren 160 minus 80 close paren squared equals 80 the square root of 2 (a)

160 sine 100 t equals I times 80 the square root of 2

I equals the square root of 2 sine 100 t

tangent theta equals 80 over 80 equals 1

therefore theta equals 45 degrees

It's inductive btw, so use the minus sign.

I equals the square root of 2 the sine of open paren 100 t minus 45 degrees close paren (b)

I sub E equals the fraction with numerator the square root of 2 and denominator the square root of 2 equals 1

P equals 1 squared times 80 equals 80 (c)

f with hook equals 1 over 2 pi the square root of 1.6 times 125 times 10 to the negative 6 power equals 25 the square root of 2 over pi (d)

Ex. 2

wow

V sub R equals 80

80 equals I times 160

I equals 0.5

V sub R L equals 100 equals the square root of V sub R squared plus V sub L squared equals the square root of 80 squared plus V sub L squared

V sub L squared equals 100 squared minus 80 squared

V sub L equals 60

x sub L equals the fraction with numerator V sub L and denominator I equals 60 over 0.5 equals 120 ohms

Yippee

100 equals the square root of 80 squared plus open paren V sub L minus V sub C close paren squared

100 squared equals 80 squared plus open paren V sub L minus V sub C close paren squared

open paren V sub L minus V sub C close paren squared equals 60 squared

V sub L minus V sub C equals 60

V sub C minus V sub L equals 60

V sub C equals 120

x sub C equals 120 over 0.5 equals 240 ohms

Ex. 3

V equals 100 the square root of 2 sine 100 t

omega equals 100

x sub L equals 2 times 100 equals 200 ohms

x sub C equals the fraction with numerator 1 and denominator 10 times 100 times 10 to the negative 6 power equals 1000 ohms

tangent theta equals the fraction with numerator 200 minus 1000 and denominator 600 equals negative four thirds

theta equals negative 53 degrees (a)

Sifat rangkaian kapasitif karena x sub L is less than x sub C (b)

Z equals the square root of 600 squared plus open paren 1000 minus 200 close paren squared equals 1000 ohms

V sub max equals I sub max Z

100 the square root of 2 equals I sub max times 1000

I sub max equals the fraction with numerator the square root of 2 and denominator 10

therefore I equals the fraction with numerator the square root of 2 and denominator 10 the sine of open paren 100 t plus 53 degrees close paren (c)

I sub cap E equals the fraction with numerator the square root of 2 and denominator 10 times the fraction with numerator 1 and denominator the square root of 2 equals one tenth

V sub L equals one tenth times 200 equals 20 (d)

V sub C equals one tenth times 1000 equals 100 (d)

V sub R equals one tenth times 600 equals 60 (d)

P equals the fraction with numerator 1 and denominator 10 squared times 600 equals 6 watts (e)