Complex Numbers (When and Why)

Question 1

What is the formula for frequency in terms of period?

Answer 1

f = 1/T

Frequency is also expressed as w / 2π.

Question 2

In sinusoidal signals, what does ‘v2 leads v1 by q’ mean?

Answer 2

v2 leads v1 by phase angle q

Question 3

What is phasor notation?

Answer 3

Phasor notation involves representing voltages and currents oscillating with frequency w in polar form

Question 4

How do you convert a time-domain function to phasor form?

Answer 4

Extract the amplitude and phase, and write each in polar form

Question 5

What is the relationship between voltage and current in a resistor in phasor analysis?

Answer 5

V = RI, where current and voltage are in-phase

Question 6

In an inductor, how does the voltage relate to the current?

Answer 6

V = jωL I, where voltage leads current by 90°

Question 7

In a capacitor, how does the current relate to the voltage?

Answer 7

I = jωC V, where current leads voltage by 90°

Question 8

What is impedance?

Answer 8

Impedance, Z, is the ratio of phasor voltage to phasor current for an electrical element or network

Question 9

How does impedance behave with frequency for a resistor?

Answer 9

Impedance is constant and does not depend on frequency

Question 10

What is the impedance of an inductor at higher frequencies?

Answer 10

At higher frequencies, less current is passed for constant voltage

Question 11

What is the impedance of a capacitor at higher frequencies?

Answer 11

At higher frequencies, more current is passed for constant voltage

Question 12

What is the formula for calculating impedance of a capacitor?

Answer 12

Z_C = 1/(jωC)

Question 13

What is the formula for calculating impedance of an inductor?

Answer 13

Z_L = jωL

Question 14

What is the formula for calculating impedance of a resistor?

Answer 14

Z_R = R

Question 15

When converting from phasor to time domain, what must be determined?

Answer 15

The instantaneous value of the voltage or current at a specific time

Question 16

What is the phase angle of a sinusoidal function represented in phasor form?

Answer 16

It is the angle component in polar representation

Question 17

What is the significance of using KCL in phasor domain analysis?

Answer 17

It allows for solving circuit equations in the phasor domain

Question 18

Fill in the blank: The transient response of an RL circuit is negligible after approximately ______.

Answer 18

5t

Question 19

Fill in the blank: For a capacitor, the current ______ leads the voltage by 90°.

Answer 19

always

Question 20

True or False: In phasor analysis, voltage and current are always in phase for a resistor.

Answer 20

True

Question 21

What is the primary use of complex numbers in engineering?

Answer 21

To analyze AC circuits.

Question 22

What do sinusoids become when using complex algebra?

Answer 22

Phasors.

Question 23

What mathematical laws are solved using phasors?

Answer 23

• KCL (Kirchhoff’s Current Law)
• KVL (Kirchhoff’s Voltage Law)
• Ohm’s Law.

Question 24

What is the definition of an imaginary number?

Answer 24

The product of a real number and the operator j.

Question 25

What is a complex number?

Answer 25

The sum of a real number and an imaginary number.

Question 26

What is the rectangular form of a complex number?

Answer 26

It consists of a real part and an imaginary part denoted as z.

Question 27

How is graphical addition and subtraction of complex numbers performed?

Answer 27

Like vector addition ('tip-to-tail').

Question 28

In which form is multiplication of complex numbers easier to perform?

Answer 28

Polar form.

Question 29

What does the exponential form of a complex number assume?

Answer 29

|z| is positive and real. ##footnote## assumes it can be represented as a product of its magnitude (\(r\)) and a rotation factor (\(e^{i\theta }\)), where the angle (\(\theta \)) is in radians. This form relies on Euler's formula, which states that \(e^{i\theta }=\cos \theta +i\sin \theta \). This representation simplifies complex number operations like multiplication and division.  

Question 30

What does polar form serve as in relation to exponential form?

Answer 30

A shorthand for the exponential form.

Question 31

What is the complex conjugate of a complex number z?

Answer 31

The same number except that the imaginary part is negated. ##footnote## gate of a complex number \(z=a+bi\) is \(a-bi\). To find the conjugate, you simply change the sign of the imaginary part while keeping the real part the same. It is denoted by \(\={z}\) or \(z^{*}\).  

Question 32

Graphically, how can the complex conjugate of z be represented?

Answer 32

As the mirror image of z across the Real axis.