The j operator.pptx

The Math Used in AC Circuits

• _{Study of AC circuits rely heavily on two areas of}

math:

• _{Sine and cosine functions} • _{Complex numbers}

• _{We’ll review the math after introducing some}

Radians

• _{Recall that the radian (rad) is the SI unit for measuring angle.} • _{It is related to degrees by}

 radians = 180

• _{We’ll often need to convert between radians and degrees:}

Angular Frequency

• _{The quantity 2}_{f, which appears in many equations, is called the}

angular frequency.

• _{Its symbol is} , and its unit is rad/s:

Peak-to-Peak Value

• _{A waveform’s peak-to-peak value is its total height from its lowest}

value to its highest value.

• _{Many waveforms are symmetric about the horizontal axis. In such}

Instantaneous Value

• _{The waveform’s instantaneous value is its value at a specific time.} • _{A waveform’s instantaneous value constantly changes, unlike the}

previous parameters (period, frequency, angular frequency,

Mathematical Expression For a

Sinusoid

• _{The mathematical expression for a sinusoidal voltage looks like this:}

v(t) = V_mcos(t + )

where V_m is the amplitude,  is the angular frequency, and  is the phase angle (relative to some reference).

• Example:

Caution: Radians and Degrees

• _{In the expression for a sinusoid,}

v(t) = V_mcos(t + )

 is usually given in degrees, but  is always given in radians per second.

• _{Recommendation: To compute a sinusoid’s instantaneous value, leave}

Mathematical Review: Complex Numbers

• The system of complex numbers is based

on the so-called imaginary unit, which is equal to the square root of 1.

• Mathematicians use the symbol i for this number, but electrical engineers use j:

or

1





Rectangular versus Polar Form

• Any complex number can be expressed in

three forms:

• Rectangular form

• Example: 3 + j 4

• Polar form

• Example: 5  53.1

• Exponential form

Rectangular Form

• In rectangular form, a complex number z is written as the sum of a real part x and an

imaginary part y:

• We often represent complex numbers as points in the complex

plane, with the real part plotted along the

horizontal axis (or “real axis”) and the

imaginary part plotted along the vertical axis (or “imaginary axis”).

Polar Form

• In polar form, a complex number z is written as a magnitude r at an angle :

z = r

• The angle  is

Converting Between Rectangular and

Polar Forms

• We will very often have to convert from

rectangular form to polar form, or vice versa. • This is easy to do if

you remember a bit of right-angle

• Given a complex number z with real part x and imaginary part y, its magnitude is given by

and its angle is given by

Converting from Rectangular Form to

Polar Form

2 2

_y

x

r





•

Given a complex number

z

with

magnitude

r

and angle



, its real part is

given by

and its imaginary part is

given by

Addition

• Adding complex numbers is easiest if the numbers are in rectangular form.

• _{Suppose z1 = x1+jy1 and z2 = x2+jy2}

Then z₁ + z₂ = (x₁+x₂) + j(y₁+y₂)

• In words: to add two complex numbers in

Subtraction

• Subtracting complex numbers is also easiest if the numbers are in rectangular form.

• _{Suppose z1 = x1+jy1 and z2 = x2+jy2}

Then z₁  z₂ = (x₁x₂) + j(y₁y₂)

• In words: to subtract two complex numbers in rectangular form, subtract their real parts to get the real part of the result, and subtract

Multiplication

• Multiplying complex numbers is easiest if the numbers are in polar form.

• _{Suppose z1 = r1 1 and z2 = r2 2}

Then z₁  z₂ = (r₁r₂)  (₁+ ₂)

Division

• Dividing complex numbers is also easiest if the numbers are in polar form.

• _{Suppose z1 = r1 1 and z2 = r2 2}

Then z₁ ÷ z₂ = (r₁ ÷ r₂)  (₁  ₂)

• In words: to divide two complex numbers in polar form, divide their magnitudes to get the magnitude of the result, and subtract their

Complex Conjugate

• Given a complex number in rectangular

form,

z = x + jy

its complex conjugate is simply z* = x  jy

• Given a complex number in polar form,

z = r



Adding Sinusoids (Continued)

v₁ = 10 cos(200t + 30) V

+ v₂ = 4 cos(200t + 90) V

we’re guaranteed to get another sinusoid of the same angular frequency, 200 rad/s:

v₁ + v₂ = V_m cos(200t + ) V

• _{But how do we figure out the resulting sinusoid’s}

Adding Sinusoids:

A Common

Mistake

• _{Here’s a common beginner mistake that is sure to}

give you the wrong answer:

v₁ = 10 cos(200t + 30) V + v₂ = 4 cos(200t + 90) V

Phasors

• _{A phasor is a complex number that represents the amplitude and}

phase angle of a sinusoidal voltage or current.

• _{The phasor’s magnitude r is equal to the sinusoid’s amplitude.} • _{The phasor’s angle}  is equal to the sinusoid’s phase angle.

• _{Example: To represent the sinusoid}

v(t) = 10 cos(200t + 30) V

Time Domain and Phasor Domain

• _{Some fancy terms:}

• _{We call an expression like}

10 cos(200

t

+ 30



) V

the time-domain representation of a sinusoid.

• _{We call}

10



30



V

Example of Using Phasors to Add

Sinusoids

• _{To add v1 = 10 cos(200t + 30) V}

+ v₂ = 4 cos(200t + 90) V

• _{Transform from time domain to phasor domain:}

V₁ = 1030 V and V₂ = 490 V

• _{Add the phasors:}

V₁ + V₂ = 1030 + 490 = 12.4946.1 V

• _{Transform from phasor domain back to time}

domain:

• _{A p.d. of 282.84 sin (314t + π/6) V is applied to two branches connected in} parallel. The currents in the respective branches are 28.28 sin (314t + π/3) A

and 56.57 sin (314t + π/6) A. Find:

• _{the circuit current in a form similar to the branch currents,}

• _{the complex power of the circuit and hence the apparent power (in kVA) in}

the main network,

• _{the total impedance in the main network and express it in the form} A + jB.

• _{A single-phase network consists of three parallel branches, the currents in the} respective branches being represented by:

• _{i1 = 20 sin 314t amperes;}

• _{i2 = 30 sin (314t – π/4) amperes;} • _{i3= 18 sin (314t + π/2) amperes.}

• _{The supply voltage for the network is also represented by 200 sin314t volts.} • _Calculate: