Complex Trigonometric Functions
Complex trigonometric functions involve the use of complex numbers in trigonometric expressions. They extend the concept of trigonometric functions to complex numbers, allowing for the representation of oscillatory phenomena in engineering and technology. These functions are essential in analyzing alternating current circuits, signal processing, and wave propagation, providing a powerful tool for understanding and manipulating periodic phenomena in various engineering applications.
5 Key excerpts on "Complex Trigonometric Functions"
eBook - ePubNoise and Vibration Analysis
Signal Analysis and Experimental Procedures
- Anders Brandt(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...Here, c is written as (A.8) where the symbol ∠ is read ‘angle.’ When we use complex numbers in signal analysis, there are mainly two operations of interest. The first is a summation of two complex numbers, say c 1 = a 1 + j b 1 and c 2 = a 2 + j b 2. An example of this case is when we have two sound waves with a certain common frequency, and the two sounds are added together at a certain point. Since the sound information contains both amplitude and phase, it becomes a complex addition, see also below where we describe how complex numbers are used to describe sinusoids. With the addition of two complex numbers, the rectangular form is most suitable and the sum, c, of the two numbers is (A.9) that is, the real and imaginary parts are summed separately. This is equivalent to vector addition. The other important operation is multiplication of two complex numbers. An example of this is if we let a sinusoidal force excite a structure for which we know the frequency response between force and response at a certain point. The response at this point may be obtained by multiplying the complex sinusoid by the (complex) value of the frequency response at the frequency of the sinusoid. When we multiply two complex numbers, we prefer to use the polar form of Equation (A.7) and the product then becomes (A.10) that is, with multiplication, the amplitudes are multiplied and the phase angles are summed. The most important reason for using complex numbers in signal analysis (noise and vibration analysis) is that when we have sinusoids, it is quite effective to replace them with their complex analogs...
eBook - ePubDigital Signal Processing 101
Everything You Need to Know to Get Started
- Michael Parker(Author)
- 2017(Publication Date)
- Newnes(Publisher)
...Chapter 2 Complex Numbers and Exponentials Abstract This chapter introduces the importance of complex numbers and exponentials which are an integral part of digital communications and digital signal processing (DSP). The reason of delving deep into this area leads the readers to two-dimensional number plane, which help to understand DSP. The operators such as addition, subtraction, and multiplication take people through the complex side of numbers which are explained in two dimensions. The use of complex conjugate and complex exponential—in the number plane—allows readers understand the relationship of basic characteristics of numbers and its treatment. The chapter further looks into measuring angles in radians, which leads to the use of pi and the equivalence of the degrees and radians in measuring angles. DSP therefore uses this concept in later stages to understand the clockwise and anticlockwise movement around the circle. Keywords Complex conjugate; Complex exponential; Digital signal processing; Euler equation; Polar representation Complex numbers are one of those things many of us were taught a long time ago and have long since forgotten. Unfortunately, they are important in digital communications and digital signal processing (DSP), so we need to resurrect them. What we were taught and some of us vaguely remember is that a complex number has a “real” and “imaginary” part, and the imaginary part is the square root of a negative number, which is really a nonexistent number. This right away sounds fishy, and while it's technically true, there is a much more intuitive way of looking at it. The whole reason for “complex numbers” is that we are going to need a two dimensional number plane to understand DSP. The traditional number line extends from plus infinity to minus infinity, along a single line. To represent many of the concepts in DSP, we need two dimensions. This requires two orthogonal axes, like a North–South line and an East–West line...
eBook - ePubIntroduction to Energy, Renewable Energy and Electrical Engineering
Essentials for Engineering Science (STEM) Professionals and Students
- Ewald F. Fuchs, Heidi A. Fuchs(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
...sinusoidal and cosinusoidal functions), the concept of a phasor is based on complex number relations. Indeed, a phasor represents a simplified complex number by ignoring the term e jωt. Phasor relations for resistor, inductor, and capacitor and the concept of impedance as well as admittance are specified. Also presented are Δ‐Y transformation, solutions based on Kirchhoff’s laws, nodal analysis, mesh/loop analysis, superposition, source transformation/exchange, Thévenin’s (principle of constant voltage source) and Norton’s (principle of constant current source) theorems, and steady‐state power analysis. Various power definitions are reviewed, such as instantaneous power p (t), average or real power P, imaginary power Q, apparent power S, and complex power. Amplitude excursions of voltages (V M) and current (I M) are defined, as well as root mean square (rms) error and effective (eff) values of signals. Lastly, nonsinusoidal steady‐state responses are treated with Fourier analysis, with its trigonometric and exponential forms. Problems 4.1 Complex numbers: conversion from rectangular to polar form and vice versa. Given three complex numbers:,, and. Note 1 radian ≡ 57.2958° or 1° ≡ π/180 radians = 0.017 45 rad. Find. Find in rectangular form, where * denotes the complex conjugate. Evaluate and place the results in polar form. Solve for if and express in polar form. Evaluate the following expressions and put your answer in polar form. What is the rms phasor for v (t) = 5.2 V(100 t − 38°)? What time function is represented by the phasor, if the frequency is f = 400 Hz? Determine the frequency of the following two currents and the phase angle between them: In the circuit of Figure P4.1.1, i s (t) =2 cos(377 t + 60°)A find, where “s” stands for source. Figure P4.1.1 Capacitor supplied by current i s (t). Find the equivalent impedance shown in Figure P4.1.2, where the angular frequency is ω = 377 rad/s. Figure P4.1.2 Impedance calculation. Find the...
- Patrick F. Dunn(Author)
- 2019(Publication Date)
- CRC Press(Publisher)
...So, B n = 4 A { sin n π − sin 0 (2 π n / T) 2 + 0.5 cos n π − 0 (2 π n / T) } = (4 A) (− 0.5 cos n π (2 π n / T)) = − A π n cos n π. Thus, y (t) = A (1 4 + ∑ n[--=PLGO-SEPARAT. OR=--]= 1 ∞ { (cos n π − 1) π 2 n 2 cos 2 π n t T − cos n π π n sin 2 π n t T }) = A (1 4 + ∑ n = 1 ∞ 1 n π { (− 1 + (− 1) n) π n cos 2 π n t T − (− 1) n sin 2[--=PLGO-SEPARATOR. =--]π n t T }) This y(t) is shown (with A = 1) for three different partial sums in Figure 9.5. 9.3 Complex Numbers and Waves Complex numbers can be used to simplify waveform notation. Waves, such as electromagnetic waves that are all around us, also can be expressed using complex notation. FIGURE 9.5 Three partial Fourier series sums for a ramp function. The complex exponential function is defined as exp (z) = e z = e (x + i y) = e x e i y ≡ e x (cos y + i sin y), (9.13) where z = x + iy, with the complex number i ≡ − 1 and x and y as real numbers. The complex conjugate of z, denoted by z*, which is defined as x – iy. The modulus or absolute value of z is given by | z | = z z ∗ = (x + i y) (x − i y) = x 2 + y 2,, which is a real number. Using Equation 9.13, the Euler formula results, e i θ = cos θ + i sin θ, (9.14) which also leads to e − i θ = cos θ − i sin θ. (9.15) The complex expressions for the sine and cosine functions can be found from Equations 9.14 and 9.15. These are cos θ = 1 2 [ e i θ + e − i θ ] (9.16) and sin θ = 1 2 i [ e i θ − e − i θ ]. (9.17) A wave can be represented by sine and cosine functions...
eBook - ePubSTEM Education by Design
Opening Horizons of Possibility
- Brent Davis, Krista Francis, Sharon Friesen(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
...To appreciate this point, it’s useful to pause to think about things in life that repeat themselves in regular cycles. In everyday life, trigonometric functions are probably the most important examples of periodic functions, and perhaps the most familiar of those is the smoothly undulating sine curve, variations of which appear in many of the margins in this chapter. As illustrated in these margin figures, the sine curve can be used to model a great many phenomena, including changes in daylight hours throughout the year and across different latitudes, moon phases and position on the horizon, ocean waves, planetary motion, sound waves, and electromagnetic radiation. Like other mathematics concepts studied in school, this one is important not because it can be used to answer a textbook question about how high a 5 meter ladder will reach if it forms an 80° angle with the ground (although that’s entirely useful!), but because it affords insight into how, for example, making a sound is related to rocking in a boat, playing on a swing, or decreased daylight in winter. The phrasing here is important. Notice that we didn’t say that the sorts of phenomena highlighted in these margins are “examples of the sine function.” Because they aren’t. They are phenomena that can be modeled with the sine function – and there’s a big difference. The sine function isn’t lurking in planetary orbits or sound waves. It is a concept that enables humans to recognize, cluster, and study a particular sort of regularity in the universe. It is a modeling tool. Above and below are several familiar situations that are often modeled using the concept of multiplication. The vital point here is that mathematics is about humanity’s engagement with the world. Concepts are not mined from a mysterious, ideal realm, but are distilled from encounters with many different forms and events. Consider the more familiar concept of multiplication, for example...
