🌊 AC Waveforms

In the chapter "Time Domain," we saw how varying quantities can be represented over time through time graphs. Since alternating current (AC) is characterized by a voltage that changes and fluctuates regularly with the passage of time, we use time domain diagrams to illustrate the shapes of AC waves.

When we talk about AC waveforms, we do not mean tangible physical waves like ocean waves; the waveform of alternating current is merely a graph that illustrates how voltage changes over time. Although there are close and specific relationships between AC waveforms and physical waves, AC waves are not things that occupy space or travel through space like ocean waves, sound waves, or radio waves; they are simply a schematic representation of voltage changes.

If we return to the model of the rotary battery for alternating current, we will find that the current remains stable and constant for a moment, then its direction suddenly and instantaneously reverses. This behavior repeats with the rotation of the battery, causing sharp jumps similar to mechanically opening and closing a switch; which results in a "square wave."

 In contrast, other waveforms are characterized by very smooth and gradual changes, the most prominent of which is the sine wave.

The electricity supplied through the public power grid comes in the form of a sine wave; this is the natural physical result of rotating a uniform magnetic field past stationary wire coils. Alternating current is also generated through electronic oscillator circuits to produce low-power waves suitable for various applications, such as sound generators, radio circuits, and digital circuits.

Function generators use these oscillators to produce various waveforms for testing and maintenance purposes, and the most prominent waveforms available are:

1. Sine Wave.
2. The triangle wave: appears in the graph as repeated isosceles triangles.
3. The square wave: It is the direct result of the rotary battery model.
4. Sawtooth Wave: It is the time diagram of the movement of the clock hands.

1. The Sine Wave

تتميز الموجة الجيبية بـ انتقال سلس وتدريجي للغاية عند عكس اتجاهها، وهي أكثر الأشكال الموجية نقاءً في الطبيعة. وفي الواقع الفيزيائي، يمكن تفكيك وتحليل جميع الأشكال الموجية الأخرى وتحويلها إلى مجموعة من الموجات الجيبية المستقلة. ترتبط الموجة الجيبية رياضياً بالدائرة؛ ولذلك تُستخدم بعض المصطلحات والمسميات الخاصة بالدوائر لوصفها وقياسها.

🔄 Rotating Disk Model

To visualize how a sine wave forms, imagine a bright point drawn on the surface of a rotating disk:

• If you look at the disk face-on, you will see the point moving in a regular circular path.
• If you look at the disk from the edge (Edge-on), you will see the point moving up and down only in a straight line.
• If this rotating disk is moved at a constant speed along its horizontal axis of rotation, the point will trace an ideal sine wave in space. In other words, the sine wave appears as a three-dimensional spiral when viewed from the side of the edge.

There is a close relationship between the time graph of the angle of the clock hands and the sine wave; if we were to plot the graph using the sine of the angle ($\sin\theta$) instead of its raw values in degrees, the curve would appear as a deterministic sine wave.

📏 Basic Parameters of the Sine Wave (Parameters)

The sinusoidal wave is measured and its identity is determined geometrically based on four basic parameters:

A. Amplitude

The capacity represents the magnitude and size of the wave, and it can be measured in any suitable physical unit (volt, ampere, watt, etc.). The capacity expresses the voltage in three different ways in the field:

• Peak Voltage - ($V_p$): is the maximum deviation or displacement of the wave from the midpoint (zero) in either direction (positive or negative).
• Peak-to-Peak Voltage ($V_{pp}$): It is the total difference from the maximum positive peak of the wave to its minimum negative peak, and it is always equal to twice the peak voltage ($V_{pp} = 2 \times V_p$).
• Root Mean Square Voltage (RMS Voltage): A complex mathematical calculation that represents the value of direct current (DC) voltage that generates and dissipates the same amount of thermal power as the alternating current (AC) voltage measured when applied to the same resistance.

It is calculated mathematically by taking the Root-Mean-Square (RMS) of instantaneous voltage samples; exclusively in clean sine waves (and not in any other waveform), the RMS voltage simply represents $70.7\%$ of the peak voltage:

$$V_{rms} = \frac{V_p}{\sqrt{2}} \approx 0.707 \times V_p$$

Maintenance Note: Most traditional commercial voltmeters measure the RMS value by sampling the peak voltage and then using an internal voltage divider to display $70.7\%$ of it on the screen; if you need to measure the effective value of a non-sinusoidal waveform (such as a square or modified inverter wave), you must use a meter that supports the "True RMS" feature.

B. Phase

The term "phase" is used in comparison to waves; when displaying a sine wave on an oscilloscope screen at a frequency of 50 Hz (meaning its complete cycle time is 20 milliseconds), we optionally draw it starting from zero volts at time zero, and we align the circular scale so that 0 degrees matches 0 milliseconds.

Based on this distribution:

• The wave reaches its positive peak at 90 degrees (5 milliseconds).
• The zero volt line crosses downward at 180 degrees (10 milliseconds).
• It reaches its minimum negative peak at 270 degrees (15 milliseconds).
• It returns to the starting point to complete the cycle at 360 degrees (20 milliseconds).

If we add a second wave that is exactly the same but delayed by 5 milliseconds, we will notice that every event in the second wave (the red color) occurs 90 degrees later than the corresponding event in the first wave. In this case, we say that the two waves are "90 degrees out of phase."

The phase is used here to compare two waves that have the same frequency but occur at different times, and it is not necessary for them to be equal in amplitude; even if one is larger than the other, the relationship of the angular phase difference between them remains constant and unchanged.

C. Frequency

It is the number of complete cycles that the wave executes and completes in one second. It is measured in hertz (Hz); where 1 hertz equals one cycle per second.

Public electricity networks operate at a stable frequency of 50 Hz or 60 Hz depending on the geographical location. In electronic circuits, however, we deal with massive frequency ranges:

• Audio frequencies: range from 20 Hz to 20,000 Hz (20 kHz).
• Ultrasonic frequencies: start from 20 kHz and above.
• Radio frequencies: range from $50\text{ kHz}$ to billions of cycles per second (Giga-Hz).

D. Period / Periodic Time (Period - T)

It is the time taken in seconds to complete one full cycle. The mathematical relationship between frequency ($f$) and period ($T$) is an inverse relationship with the reciprocal of the value:

$$T = \frac{1}{f} \quad \text{and} \quad f = \frac{1}{T}$$

Accordingly, as the frequency increases, the period of the wave decreases; a wave with a frequency of 60 Hertz has a period of 16.6 milliseconds ($1/60 = 0.0166\text{ s}$), while a wave with a frequency of 120 Hertz takes exactly half that time, which is 8.33 milliseconds ($1/120 = 0.00833\text{ s}$).

🛠️ Second: Disassembling and Analyzing Wave Components

1. Square Waves

The square wave is simply produced by the sharp and instantaneous switching between two levels of voltage (such as quickly turning a switch on and off). Despite the simplicity of its external geometric shape, it is a wave of great complexity in its physical composition.

Physicists discovered (according to Fourier analysis) that the square wave is fundamentally composed of a fundamental sine wave (Fundamental Frequency) plus an infinite series of sine waves with odd harmonic frequencies (Odd Harmonics) - and harmonics are multiples of the fundamental frequency -.

• Composite example: A square wave with a frequency of 1 kilohertz ($1\text{ kHz}$) consists of:
  1. Basic sinusoidal wave is reflected 1 kHz} At full capacity.
  2. A second sine wave with a frequency of $3\text{ kHz}$ (the third harmonic) but with a smaller amplitude.
  3. A third sine wave with a frequency of $5\text{ kHz}$ (the fifth harmonic) with a smaller amplitude, and so on (the seventh, ninth harmonics...).
  4. As we add higher odd harmonics, the edges of the resulting wave become more vertical and its surface straightens; by adding all odd harmonics to infinity, the ideal square wave with sharp edges is created.

🎯 Field Use and Bandwidth Testing:

Square waves are an effective tool for assessing the bandwidth of circuits, as they help determine whether the circuit is capable of passing a wide range of frequencies without distortion. A circuit with a wide bandwidth will display the square wave clearly with sharp edges, while if any distortion occurs in the shape of the wave, we can accurately identify the problem:

• Erosion and distortion of the leading edge: This means that the circuit suffers from poor response and failure to pass high frequencies.
• The slope and distortion of the trailing edge: This means that the circuit suffers from poor response and failure to pass low frequencies (just as occurs when examining high-pass filter circuits that block low frequencies, resulting in a distorted output at the straight edges).

2. Triangle Wave

The triangular wave is characterized by the presence of perfectly symmetrical linear slopes that rise and fall regularly to meet at the peak point.

Like the square wave, the triangular wave is primarily composed of the fundamental sine wave and all the odd harmonics; however, the key difference here is that each harmonic is phase-inverted by 180 degrees from the harmonic that precedes it.

• The third harmonic is completely out of phase with the fundamental wave, which reduces the slope of the sides and pushes the peak higher; the fifth harmonic comes back in phase with the fundamental wave, making the ends geometrically straight; and by alternating the remaining inverted harmonics, the shape of the slopes straightens out to form the pure triangular wave.

3. Sawtooth Wave

The sawtooth wave is essentially a special case of the triangular wave, but it has asymmetrical slopes; where one slope is very steep and vertical while the opposite slope is sloped and gradual (a slow rise followed by a sudden sharp drop).

🛠️ Historical and Industrial Applications:

Ramp waves were heavily utilized in television screens and older computers based on cathode ray tubes (CRT Monitors), where they were used to control the deflection of the scanning electron beam; the slow, gradual rise of the wave directs the electron beam to move leisurely and scan the screen from left to right to draw the image. Once the wave reaches its peak, it suddenly and instantaneously drops to zero, forcing the beam to quickly rebound to the left to safely start drawing the next line.

• Dense harmonic composition: A sawtooth wave consists of the fundamental sine wave plus all odd and even harmonics together, provided that each harmonic is phase-inverted by 180 degrees from the harmonic that precedes it directly. This comprehensive and alternating mix is physically responsible for the formation of this sharp and asymmetric wave diffraction.