AC Signals and Sinusoidal Waveforms

Summary

This chapter introduces alternating current (AC) and the sinusoidal waveforms that are fundamental to power systems and signal processing. Students will learn to characterize sinusoids by their amplitude, frequency, period, and phase, and understand the relationships between these parameters. The chapter covers important measurement quantities including peak, peak-to-peak, RMS, and average values. Complex numbers are introduced as the mathematical foundation for phasor analysis in the next chapter.

Concepts Covered

This chapter covers the following 18 concepts from the learning graph:

Alternating Current
Sinusoidal Waveform
Complex Numbers
Rectangular Form
Polar Form
Euler's Formula
Signal
Periodic Signal
Aperiodic Signal
Time Domain
Frequency Domain
DC Component
AC Component
Signal Amplitude
Crest Factor
Form Factor
Voltage Gain
Current Gain

Prerequisites

This chapter builds on concepts from:

title: AC Signals and Sinusoidal Waveforms description: Master alternating current, sinusoidal analysis, complex numbers, and signal characterization generated_by: chapter-content-generator v0.03 date: 2026-01-30

Introduction: Why the World Runs on Waves

Walk into any room, flip a light switch, and you're tapping into a global network of energy that flows back and forth, back and forth—60 times per second in the Americas, 50 times per second in most of the rest of the world. This isn't a design flaw or a compromise; it's a deliberate choice that makes modern electrical power possible.

Welcome to the world of alternating current (AC), where voltages and currents don't just sit there—they dance. They follow a graceful sinusoidal pattern, the smoothest curve mathematics can produce, and this simple waveform turns out to be absurdly useful for everything from power transmission to radio communication to music recording.

In this chapter, you'll learn to describe these dancing signals mathematically, extract meaningful measurements from them, and prepare for the powerful analytical tools (phasors!) that make AC circuit analysis surprisingly elegant. You'll also meet complex numbers—not because we want to make your life complicated, but because they make this math beautifully simple.

Alternating Current: The Back-and-Forth Flow

Alternating current (AC) is electric current that periodically reverses direction. Unlike direct current (DC), where electrons flow steadily in one direction like a river, AC electrons shuffle back and forth like commuters in a subway car.

Why does AC dominate power systems?

Transformers work with AC - Voltage can be stepped up for efficient long-distance transmission, then stepped down for safe household use
Generation is natural - Rotating generators naturally produce sinusoidal AC
Motors are simpler - AC induction motors have no brushes or commutators to wear out
Switching is easier - AC crosses through zero every half-cycle, making circuit breakers more effective

Characteristic	DC	AC
Direction	Constant	Reverses periodically
Voltage transformation	Difficult	Easy (transformers)
Long-distance transmission	High losses	Low losses (high voltage)
Storage	Batteries	Requires conversion
Generation	Requires rectification	Natural (rotating machines)

The great AC/DC debate: Thomas Edison championed DC, while Nikola Tesla and George Westinghouse promoted AC. AC won for power distribution because transformers allowed high-voltage transmission with low losses. DC has made a comeback in some applications (data centers, solar panels, electric vehicles) but AC still reigns for the grid.

Sinusoidal Waveforms: Nature's Favorite Curve

A sinusoidal waveform (or sinusoid) is a mathematical curve describing smooth, repetitive oscillation. It's defined by the sine or cosine function:

\[v(t) = V_m \sin(\omega t + \phi)\]

Or equivalently:

\[v(t) = V_m \cos(\omega t + \phi - 90°)\]

Where:

\(V_m\) = Peak amplitude (maximum value)
\(\omega\) = Angular frequency in rad/s (\(\omega = 2\pi f\))
\(t\) = Time in seconds
\(\phi\) = Phase angle in radians (or degrees)

Why sinusoids are special:

Derivatives and integrals are sinusoids - The derivative of a sine is a cosine, and vice versa. This makes differential equations much easier!
Sinusoids through linear systems remain sinusoids - Only the amplitude and phase change, not the shape
Any periodic signal can be built from sinusoids - Fourier showed that any repeating waveform is just a sum of sinusoids at different frequencies

Parameter	Symbol	Formula	Units
Peak amplitude	\(V_m\), \(I_m\)	Maximum value	V, A
Angular frequency	\(\omega\)	\(2\pi f\)	rad/s
Frequency	\(f\)	\(1/T\)	Hz
Period	\(T\)	\(1/f\)	s
Phase	\(\phi\)	Angle offset	rad or °

Diagram: Sinusoidal Waveform Anatomy

Sinusoidal Waveform Anatomy

Type: microsim Bloom Level: Understand (L2) Bloom Verb: explain Learning Objective: Students will explain the relationship between sinusoidal parameters (amplitude, frequency, phase) and the resulting waveform shape. Visual elements: - Main plot: Sinusoidal waveform v(t) vs time - Amplitude markers: Peak-to-peak, peak, RMS shown as horizontal lines - Period marker: Arrow spanning one complete cycle - Phase reference: Vertical line at t=0 showing phase offset - Mathematical formula displayed and updated in real-time - Zero crossings highlighted Interactive controls: - Slider: Amplitude Vm (1V to 10V) - Slider: Frequency f (1Hz to 100Hz, or period T) - Slider: Phase φ (-180° to +180°) - Toggle: Show/hide RMS level - Toggle: Show sine or cosine reference - Display: Current formula with values Annotations: - "Period T = 1/f" spanning one cycle - "Peak Vm" at maximum point - "Phase shift" showing offset from reference Default parameters: - Vm = 5V - f = 10Hz - φ = 0° Canvas layout: - Plot area: 600 × 350 pixels - Control area: 100 pixels below Implementation: p5.js

Amplitude Measurements

Different applications require different ways of characterizing amplitude:

Peak Value (\(V_p\) or \(V_m\)): The maximum instantaneous value from zero [V_p = V_m]

Peak-to-Peak Value (\(V_{pp}\)): The total swing from minimum to maximum [V_{pp} = 2V_m]

RMS Value (\(V_{rms}\)): The "equivalent DC" value for power calculations [V_{rms} = \frac{V_m}{\sqrt{2}} \approx 0.707 \times V_m]

Average Value (\(V_{avg}\)): The mean value over one half-cycle (full-cycle average is zero!) [V_{avg} = \frac{2V_m}{\pi} \approx 0.637 \times V_m]

Measurement	Formula	For 120V RMS Wall Outlet
RMS	\(V_{rms}\)	120 V
Peak	\(V_m = \sqrt{2} \times V_{rms}\)	170 V
Peak-to-Peak	\(V_{pp} = 2V_m\)	340 V
Average (half-cycle)	\(V_{avg} = 2V_m/\pi\)	108 V

Why RMS Matters

RMS (Root Mean Square) is the most important AC measurement because it directly relates to power. A 120V RMS AC source delivers the same heating power to a resistor as a 120V DC source. This is why your wall outlet is rated at 120V RMS—that's the effective voltage for power delivery.

Frequency and Period

Frequency (\(f\)) tells you how many complete cycles occur per second: [f = \frac{1}{T} \text{ Hz}]

Period (\(T\)) is the time for one complete cycle: [T = \frac{1}{f} \text{ seconds}]

Angular frequency (\(\omega\)) converts frequency to radians: [\omega = 2\pi f \text{ rad/s}]

Standard	Frequency	Period	Angular Frequency
US Power	60 Hz	16.67 ms	377 rad/s
EU Power	50 Hz	20 ms	314 rad/s
Audio (A4)	440 Hz	2.27 ms	2,765 rad/s
AM Radio	1 MHz	1 μs	6.28 × 10⁶ rad/s
WiFi	2.4 GHz	0.42 ns	1.51 × 10¹⁰ rad/s

Phase: The Starting Point

Phase (\(\phi\)) describes where the waveform starts at time \(t = 0\). Two signals with the same frequency but different phases are said to be "out of phase."

Phase relationships:

In phase (\(\phi = 0°\)): Peaks align
90° out of phase (quadrature): One peaks when the other crosses zero
180° out of phase (antiphase): Peaks of one align with troughs of the other

[v_1(t) = V_m \sin(\omega t)] [v_2(t) = V_m \sin(\omega t + 90°) = V_m \cos(\omega t)]

When we say "\(v_2\) leads \(v_1\) by 90°," we mean \(v_2\) reaches its peak earlier in time.

Diagram: Phase Relationship Visualizer

Phase Relationship Visualizer

Type: microsim Bloom Level: Understand (L2) Bloom Verb: compare Learning Objective: Students will compare two sinusoidal signals with different phase relationships and interpret leading/lagging terminology. Visual elements: - Dual waveform plot showing two signals (blue and red) - Reference signal (blue): v₁(t) = Vm sin(ωt) - Phase-shifted signal (red): v₂(t) = Vm sin(ωt + φ) - Phase angle marked between corresponding points - Phasor diagram showing both signals as rotating vectors - "Leads" or "Lags" label indicating relationship Interactive controls: - Slider: Phase difference φ (-180° to +180°) - Button: Quick select: 0°, 90°, 180°, -90° - Toggle: Show/hide phasor diagram - Toggle: Animate phasor rotation - Display: Phase relationship description Visual feedback: - φ > 0: "Signal 2 LEADS signal 1 by φ°" - φ < 0: "Signal 2 LAGS signal 1 by |φ|°" - φ = 0: "Signals are IN PHASE" - φ = ±180°: "Signals are in ANTIPHASE" Default parameters: - φ = 45° (signal 2 leads) - Same amplitude for both signals - f = 5 Hz for visible animation Canvas layout: - Waveform plot: 400 × 300 pixels - Phasor diagram: 200 × 300 pixels (side) - Control area: 100 pixels below Implementation: p5.js

Signals: The Language of Information

A signal is any quantity that varies with time and conveys information. Signals are everywhere:

Voltage from a microphone (audio signal)
Current through a sensor (measurement signal)
Digital bits on a network cable (data signal)
Your heartbeat as recorded by an ECG (biosignal)

Periodic and Aperiodic Signals

Periodic signal: A signal that repeats exactly after a fixed time interval (the period T). [x(t) = x(t + T) \text{ for all } t]

Examples: Power line voltage, clock signals, musical tones

Aperiodic signal: A signal that doesn't repeat (or only exists for a finite duration).

Examples: Single pulse, speech, transients, noise

Signal Type	Mathematical Property	Examples
Periodic	\(x(t) = x(t+T)\)	Sine wave, square wave, clock
Aperiodic	No repetition	Pulse, noise, speech
Continuous	Defined for all t	Analog audio
Discrete	Defined at sample points	Digital audio

DC and AC Components

Any signal can be decomposed into:

DC Component: The average (constant) value of the signal [V_{DC} = \frac{1}{T}\int_0^T v(t) \, dt]

AC Component: The time-varying part that oscillates around the DC level [v_{AC}(t) = v(t) - V_{DC}]

A pure sinusoid centered at zero has no DC component. Add a constant offset, and you've introduced DC:

\[v(t) = V_{DC} + V_m \sin(\omega t)\]

This decomposition is crucial in electronics:

Power supplies filter out AC ripple to produce clean DC
Coupling capacitors block DC while passing AC signals
Bias circuits add DC offsets to center signals in their operating range

Diagram: DC and AC Component Separator

DC and AC Component Separator

Type: microsim Bloom Level: Understand (L2) Bloom Verb: classify Learning Objective: Students will classify signal components as DC or AC by observing how a composite signal can be separated into its constant and varying parts. Visual elements: - Top plot: Original composite signal v(t) = V_DC + Vm sin(ωt) - Middle plot: DC component only (horizontal line at V_DC) - Bottom plot: AC component only (sinusoid centered at zero) - Visual arrows showing extraction - Coupling capacitor icon showing "blocks DC, passes AC" Interactive controls: - Slider: DC offset V_DC (0 to 5V) - Slider: AC amplitude Vm (0 to 5V) - Slider: Frequency f (1Hz to 20Hz) - Toggle: Show/hide component labels - Display: Calculated values for DC and AC RMS Annotations: - "Average value = DC component" - "Oscillation around average = AC component" - Formula showing v(t) = V_DC + v_AC(t) Default parameters: - V_DC = 2.5V - Vm = 2V - f = 5Hz Canvas layout: - Three stacked plot areas: 600 × 120 pixels each - Control area: 100 pixels below Implementation: p5.js

Time Domain and Frequency Domain: Two Perspectives

Every signal can be viewed from two complementary perspectives:

Time Domain: Shows how the signal varies over time. The x-axis is time, and you see the actual waveform shape.

Frequency Domain: Shows what frequencies are present in the signal. The x-axis is frequency, and you see the spectrum—which sinusoids combine to create the signal.

Think of it like describing a musical chord:

Time domain: The actual sound wave shape—complicated!
Frequency domain: A list of notes being played—C, E, G (a C major chord)

The transformation between these views is called the Fourier Transform, and it's one of the most powerful tools in engineering. We'll explore this more in Chapter 14, but for now, understand that:

A pure sinusoid appears as a single spike in the frequency domain
A complex wave appears as multiple spikes (multiple frequency components)
The frequency domain reveals information hidden in the time domain

Domain	X-axis	Y-axis	Shows
Time	Time (s)	Amplitude (V)	Waveform shape, timing
Frequency	Frequency (Hz)	Amplitude (V)	Spectral content, harmonics

Diagram: Time Domain vs Frequency Domain

Time Domain vs Frequency Domain

Type: microsim Bloom Level: Understand (L2) Bloom Verb: explain Learning Objective: Students will explain how time domain waveforms correspond to frequency domain spectra by observing simple sinusoids and their spectral representations. Visual elements: - Left panel: Time domain plot v(t) vs t - Right panel: Frequency domain spectrum |V(f)| vs f - For composite signals: show individual components in different colors - Spectral lines with amplitude markers - Animation showing how adding sinusoids builds complex waves Step-through mode: - Stage 1: Single sinusoid → single spectral line - Stage 2: Two sinusoids → two spectral lines - Stage 3: Three sinusoids → three lines, complex time waveform - Stage 4: Many sinusoids → approaching square wave Interactive controls: - Dropdown: Signal type (single sine, two sines, square wave approximation) - Slider: Fundamental frequency f₁ (1-20 Hz) - Slider: Number of harmonics (for square wave) - Button: Step through / Animate - Toggle: Show individual components Annotations: - "Time domain shows SHAPE" - "Frequency domain shows CONTENT" Default parameters: - Single sinusoid at 5 Hz - Step-through mode active Canvas layout: - Two side-by-side plots: 300 × 350 pixels each - Control area: 100 pixels below Instructional Rationale: Step-through helps students see the one-to-one correspondence between time and frequency representations. Implementation: p5.js

Complex Numbers: Your New Best Friend

If you've been dreading complex numbers, prepare for a surprise: they're going to make your life easier. Complex numbers provide a compact way to represent both magnitude and phase in a single mathematical object.

Definition: A complex number \(z\) has a real part and an imaginary part: [z = a + jb]

Where:

\(a\) = Real part (\(\text{Re}(z)\))
\(b\) = Imaginary part (\(\text{Im}(z)\))
\(j = \sqrt{-1}\) (Engineers use \(j\) because \(i\) is reserved for current!)

Why complex numbers for AC circuits?

Sinusoids at the same frequency can be represented as complex numbers called phasors
Adding sinusoids becomes adding complex numbers (simple!)
Differentiation and integration become multiplication and division (even simpler!)
Circuit analysis with complex impedances follows all the DC rules you already know

Rectangular Form

Rectangular form (or Cartesian form) expresses a complex number as: [z = a + jb]

This form is best for:

Addition and subtraction
Visualizing real and imaginary parts separately

Operations in rectangular form:

Addition: \((a_1 + jb_1) + (a_2 + jb_2) = (a_1 + a_2) + j(b_1 + b_2)\)

Subtraction: \((a_1 + jb_1) - (a_2 + jb_2) = (a_1 - a_2) + j(b_1 - b_2)\)

Multiplication: \((a_1 + jb_1)(a_2 + jb_2) = (a_1 a_2 - b_1 b_2) + j(a_1 b_2 + a_2 b_1)\)

Polar Form

Polar form expresses a complex number using magnitude and angle: [z = r \angle \theta = r(\cos\theta + j\sin\theta)]

Where:

\(r = |z| = \sqrt{a^2 + b^2}\) (magnitude)
\(\theta = \tan^{-1}(b/a)\) (angle or argument)

This form is best for:

Multiplication and division
Representing phasors (sinusoid magnitude and phase)

Operations in polar form:

Multiplication: \(r_1 \angle \theta_1 \times r_2 \angle \theta_2 = r_1 r_2 \angle (\theta_1 + \theta_2)\)

Division: \(\frac{r_1 \angle \theta_1}{r_2 \angle \theta_2} = \frac{r_1}{r_2} \angle (\theta_1 - \theta_2)\)

Converting between forms:

From	To	Formulas
Rectangular	Polar	\(r = \sqrt{a^2+b^2}\), \(\theta = \tan^{-1}(b/a)\)
Polar	Rectangular	\(a = r\cos\theta\), \(b = r\sin\theta\)

Choosing the Right Form

Use rectangular form for addition/subtraction, polar form for multiplication/division. Your calculator can switch between them, but understanding both builds intuition for what complex numbers represent.

Diagram: Complex Number Visualizer

Complex Number Visualizer

Type: microsim Bloom Level: Apply (L3) Bloom Verb: calculate Learning Objective: Students will calculate complex number conversions between rectangular and polar forms using the complex plane visualization. Visual elements: - Complex plane with real (x) and imaginary (y) axes - Point representing complex number z - Vector from origin to point showing magnitude - Angle arc showing θ - Projections onto real and imaginary axes - Both rectangular and polar forms displayed Interactive controls: - Mode toggle: "Enter rectangular" vs "Enter polar" - Rectangular mode: Sliders for a (-5 to 5) and b (-5 to 5) - Polar mode: Sliders for r (0 to 7) and θ (-180° to 180°) - Button: Generate random number (quiz mode) - Display: Both forms shown with live updates Conversion formulas shown: - r = √(a² + b²) with calculation - θ = tan⁻¹(b/a) with quadrant adjustment - a = r cos(θ) with calculation - b = r sin(θ) with calculation Visual feedback: - Dotted lines showing projections - Color coding: real part (blue), imaginary part (green) - Magnitude shown as red vector Default parameters: - z = 3 + j4 (classic 3-4-5 triangle) - Shows r = 5, θ = 53.13° Canvas layout: - Complex plane: 400 × 400 pixels - Control/display panel: 200 × 400 pixels (side) - Control area: 100 pixels below Implementation: p5.js

Euler's Formula: The Bridge Between Worlds

Euler's formula is one of the most beautiful equations in mathematics:

\[e^{j\theta} = \cos\theta + j\sin\theta\]

This connects exponentials with trigonometry, allowing us to write complex numbers as:

\[z = re^{j\theta}\]

Why this matters for circuits:

The sinusoid \(v(t) = V_m \cos(\omega t + \phi)\) can be written as:

\[v(t) = \text{Re}\{V_m e^{j(\omega t + \phi)}\}\]

This exponential form makes differentiation trivial:

\[\frac{d}{dt}e^{j\omega t} = j\omega e^{j\omega t}\]

Differentiation just multiplies by \(j\omega\)! This is why complex exponentials (and phasors) are so powerful for AC analysis.

Euler's identity (the "most beautiful equation"): [e^{j\pi} + 1 = 0]

This single equation connects five fundamental constants: \(e\), \(j\), \(\pi\), 1, and 0.

Form	Expression	Best for
Rectangular	\(a + jb\)	Addition, subtraction
Polar	\(r \angle \theta\)	Multiplication, division
Exponential	\(re^{j\theta}\)	Calculus, phasor analysis

Diagram: Euler's Formula on the Unit Circle

Euler's Formula on the Unit Circle

Type: microsim Bloom Level: Understand (L2) Bloom Verb: explain Learning Objective: Students will explain how Euler's formula relates rotating phasors to sinusoidal projections by observing the unit circle animation. Visual elements: - Unit circle on complex plane - Rotating phasor e^(jωt) as animated vector - Projection onto real axis: cos(ωt) - Projection onto imaginary axis: sin(ωt) - Side plots showing cos and sin waveforms being traced - Current angle θ displayed Step-through or animation mode: - Step through key angles: 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360° - Or continuous rotation at adjustable speed Interactive controls: - Button: Step / Animate - Slider: Rotation speed (when animating) - Button: Reset to 0° - Toggle: Show/hide projection waveforms - Display: e^(jθ) = cos(θ) + j·sin(θ) with values Visual annotations: - "Real part = cos(θ)" on horizontal projection - "Imaginary part = sin(θ)" on vertical projection - Formula: e^(jθ) = cos(θ) + j sin(θ) Default parameters: - Step mode starting at θ = 0° - Unit magnitude (r = 1) Canvas layout: - Unit circle: 350 × 350 pixels - Side waveform plots: 250 × 175 pixels (stacked) - Control area: 100 pixels below Instructional Rationale: Step-through reveals the fundamental connection between rotating vectors and sinusoidal waveforms. Implementation: p5.js

Signal Amplitude: Beyond Peak Values

We've already introduced peak, peak-to-peak, RMS, and average values. Now let's look at two ratios that characterize waveform shape:

Crest Factor

Crest factor (or peak factor) is the ratio of peak value to RMS value: [CF = \frac{V_{peak}}{V_{RMS}}]

For a pure sinusoid: [CF = \frac{V_m}{V_m/\sqrt{2}} = \sqrt{2} \approx 1.414]

Crest factor tells you how "peaky" a waveform is:

Waveform	Crest Factor
DC	1.00
Sine wave	1.414
Square wave	1.00
Triangle wave	1.73
Pulse (10% duty)	3.16

Why crest factor matters:

High crest factor signals stress components more than their RMS value suggests
Amplifiers must handle peak values without clipping
Power supplies must provide headroom for peaks

Form Factor

Form factor is the ratio of RMS value to average (rectified) value: [FF = \frac{V_{RMS}}{V_{avg}}]

For a pure sinusoid: [FF = \frac{V_m/\sqrt{2}}{2V_m/\pi} = \frac{\pi}{2\sqrt{2}} \approx 1.111]

Waveform	Form Factor
Sine wave	1.111
Square wave	1.00
Triangle wave	1.155
Sawtooth wave	1.155

Form factor describes the "fullness" of a waveform. A square wave fills every moment at maximum, giving it a form factor of 1. A sine wave has "wasted" area near zero crossings.

Voltage Gain and Current Gain

Gain measures how much a circuit amplifies (or attenuates) a signal:

Voltage gain: [A_V = \frac{V_{out}}{V_{in}}]

Current gain: [A_I = \frac{I_{out}}{I_{in}}]

Power gain: [A_P = \frac{P_{out}}{P_{in}} = A_V \times A_I]

Gain can be:

Greater than 1: Amplification
Equal to 1: Unity gain (buffer)
Less than 1: Attenuation
Negative: Signal inversion (180° phase shift)

Decibel notation is often used because:

Large gain ranges become manageable numbers
Cascaded gains add instead of multiply

[A_V(dB) = 20\log_{10}|A_V|] [A_P(dB) = 10\log_{10}|A_P|]

Voltage Gain	Decibels	Description
1000	60 dB	High amplification
100	40 dB	Strong amplification
10	20 dB	Moderate amplification
1	0 dB	Unity
0.1	-20 dB	Moderate attenuation
0.01	-40 dB	Strong attenuation

Diagram: Gain and Decibel Converter

Gain and Decibel Converter

Type: microsim Bloom Level: Apply (L3) Bloom Verb: calculate Learning Objective: Students will calculate conversions between linear gain and decibel notation for voltage and power gains. Visual elements: - Input signal representation (fixed amplitude) - Output signal representation (scaled by gain) - Visual comparison of input vs output amplitudes - Logarithmic scale showing dB relationship - Bidirectional conversion display Interactive controls: - Toggle: Enter as "Linear gain" or "Decibels" - Slider: Gain value (0.001 to 1000, logarithmic) - Toggle: Voltage gain or Power gain mode - Display: Both linear and dB values Calculations shown: - Voltage: dB = 20 log₁₀(A_V) - Power: dB = 10 log₁₀(A_P) - Inverse formulas for conversion Quick reference displayed: - ×2 = +6 dB (voltage), +3 dB (power) - ×10 = +20 dB (voltage), +10 dB (power) - ×0.5 = -6 dB (voltage), -3 dB (power) Default parameters: - Voltage gain mode - A_V = 10 (20 dB) Canvas layout: - Signal visualization: 300 × 200 pixels - Conversion display: 300 × 200 pixels - Control area: 100 pixels below Implementation: p5.js

Worked Example: Characterizing an AC Signal

Let's put it all together with a practical example.

Problem: A voltage signal is measured as: [v(t) = 3 + 5\sin(2\pi \cdot 60 \cdot t + 30°) \text{ V}]

Find: DC component, AC amplitude, frequency, period, RMS value (total), peak value (total), and crest factor.

Solution:

DC Component: [V_{DC} = 3 \text{ V}]

AC Amplitude (peak): [V_m = 5 \text{ V}]

Frequency: The argument of sine is \(2\pi \cdot 60 \cdot t\), so \(\omega = 2\pi \cdot 60\) rad/s [f = 60 \text{ Hz}]

Period: [T = \frac{1}{f} = \frac{1}{60} = 16.67 \text{ ms}]

RMS of AC component only: [V_{AC,rms} = \frac{V_m}{\sqrt{2}} = \frac{5}{\sqrt{2}} = 3.54 \text{ V}]

Total RMS value (DC + AC): [V_{rms,total} = \sqrt{V_{DC}^2 + V_{AC,rms}^2} = \sqrt{3^2 + 3.54^2} = \sqrt{9 + 12.5} = 4.64 \text{ V}]

Peak value (total): [V_{peak} = V_{DC} + V_m = 3 + 5 = 8 \text{ V}]

Minimum value: [V_{min} = V_{DC} - V_m = 3 - 5 = -2 \text{ V}]

Crest factor: [CF = \frac{V_{peak}}{V_{rms,total}} = \frac{8}{4.64} = 1.72]

Note that the crest factor is higher than a pure sinusoid (1.414) because the DC offset pushes the peak higher relative to the RMS value.

Complex Number Operations: Practice

Example 1: Addition (use rectangular form)

Add \(z_1 = 3 + j4\) and \(z_2 = 2 - j5\):

\[z_1 + z_2 = (3 + 2) + j(4 - 5) = 5 - j1\]

Example 2: Multiplication (use polar form)

Multiply \(z_1 = 5\angle 53.13°\) and \(z_2 = 3\angle -30°\):

\[z_1 \times z_2 = (5 \times 3)\angle(53.13° - 30°) = 15\angle 23.13°\]

Example 3: Division (use polar form)

Divide \(z_1 = 10\angle 60°\) by \(z_2 = 2\angle 15°\):

\[\frac{z_1}{z_2} = \frac{10}{2}\angle(60° - 15°) = 5\angle 45°\]

Example 4: Convert and multiply

Multiply \(z_1 = 3 + j4\) and \(z_2 = 1 + j1\):

First convert to polar: - \(z_1 = 5\angle 53.13°\) - \(z_2 = \sqrt{2}\angle 45°\)

Then multiply: [z_1 \times z_2 = 5\sqrt{2}\angle 98.13° = 7.07\angle 98.13°]

Convert back to rectangular if needed: [z = 7.07(\cos 98.13° + j\sin 98.13°) = -1 + j7]

You can verify: \((3 + j4)(1 + j1) = 3 + j3 + j4 + j^2 4 = 3 + j7 - 4 = -1 + j7\) ✓

Self-Check Questions

1. A sinusoidal voltage has a peak value of 170V and a frequency of 60 Hz. Write its time-domain expression (assuming zero phase) and calculate its RMS value.

Time-domain expression: [v(t) = 170\sin(2\pi \cdot 60 \cdot t) = 170\sin(377t) \text{ V}]

RMS value: [V_{rms} = \frac{V_m}{\sqrt{2}} = \frac{170}{\sqrt{2}} = 120.2 \text{ V}]

This is actually the standard US wall outlet specification: 120V RMS with 170V peak at 60 Hz!

2. Convert z = 5 - j12 to polar form.

Magnitude: [r = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13]

Angle (note: the point is in the fourth quadrant): [\theta = \tan^{-1}\left(\frac{-12}{5}\right) = \tan^{-1}(-2.4) = -67.38°]

Polar form: [z = 13\angle -67.38°]

Or in exponential form: [z = 13e^{-j67.38°}]

3. Two voltage sources are v₁(t) = 10sin(ωt) and v₂(t) = 10sin(ωt + 90°). What is the phase relationship between them?

Signal v₂ has a phase of +90° relative to v₁.

Since v₂ has a positive phase angle (leading angle), v₂ leads v₁ by 90°.

Equivalently, v₁ lags v₂ by 90°.

At any instant, v₂ reaches its peak 1/4 of a period earlier than v₁. When v₁ is at zero (crossing positive), v₂ is at its positive peak.

Note: v₂(t) = 10sin(ωt + 90°) = 10cos(ωt), so this is the classic sine-cosine relationship.

4. An amplifier has a voltage gain of 40 dB. What is the linear voltage gain? If the input is 10 mV RMS, what is the output?

Convert dB to linear gain: [A_V = 10^{(40/20)} = 10^2 = 100]

Output voltage: [V_{out} = A_V \times V_{in} = 100 \times 10 \text{ mV} = 1000 \text{ mV} = 1 \text{ V RMS}]

The amplifier increases the signal by a factor of 100, turning a 10 mV signal into 1 V.

Summary

This chapter introduced the mathematical language of AC circuits:

Alternating current reverses periodically - AC dominates power systems because transformers enable efficient voltage transformation and transmission
Sinusoids are characterized by amplitude, frequency, and phase - These three parameters completely describe any pure sinusoidal signal
RMS is the most important amplitude measure - It directly relates to power delivery and equals \(V_m/\sqrt{2}\) for sinusoids
Signals can be viewed in time domain or frequency domain - Time shows waveform shape, frequency shows spectral content
DC and AC components can be separated - Any signal is the sum of its constant (DC) and varying (AC) parts
Complex numbers simplify AC analysis - Rectangular form for addition, polar form for multiplication
Euler's formula bridges trigonometry and exponentials - \(e^{j\theta} = \cos\theta + j\sin\theta\) enables phasor analysis
Gain measures amplification - Expressed as linear ratio or in decibels; voltage dB = 20 log(A_V)

With these foundations in place, you're ready for the next chapter where we'll transform sinusoids into phasors—stationary complex numbers that represent rotating signals. This transformation will make AC circuit analysis as straightforward as DC analysis with Ohm's law. The math of complex numbers that might seem like extra work now will pay enormous dividends in simplifying circuit calculations.