8.1 Sinusoidal sources and phasors

Sinusoidal Waveform Characteristics

Sinusoidal sources are the foundation of AC circuit analysis. Every AC voltage or current can be described by a sine or cosine function, and learning to read and manipulate these functions is the first step toward analyzing real circuits. Phasors then give you a way to convert time-domain sinusoids into simple complex numbers, which makes steady-state AC math far more manageable.

A general sinusoidal voltage looks like this:

$v(t) = A \cos(\omega t + \phi)$

Every sinusoid is fully described by three quantities: amplitude, angular frequency, and phase angle.

Sinusoidal Waveform Properties

• Sinusoidal waveforms are periodic signals that oscillate smoothly between a maximum and minimum value, described by sine or cosine functions in the time domain.
• Angular frequency ($\omega$) tells you how fast the waveform oscillates, measured in radians per second (rad/s). It relates to ordinary frequency ($f$, in hertz) by:

$\omega = 2\pi f$

So a 60 Hz wall outlet has $\omega = 2\pi(60) = 377 \text{ rad/s}$. The period is the time for one full cycle: $T = 1/f$.

Phase and Amplitude Characteristics

• Amplitude ($A$) is the peak value of the waveform. For a 120 V (peak) source, the signal swings between +120 V and −120 V. The peak-to-peak value is $2A$, so 240 V in this case. Amplitude units depend on what you're measuring (volts for voltage, amperes for current).
• Phase angle ($\phi$) sets the horizontal shift of the waveform. A positive $\phi$ shifts the cosine to the left (the waveform "leads"), while a negative $\phi$ shifts it to the right (the waveform "lags"). Phase is measured in degrees or radians and is how you express time differences between two sinusoids of the same frequency.

For example, if $v(t) = 10\cos(377t + 30°)$, the amplitude is 10 V, the angular frequency is 377 rad/s, and the phase is +30°, meaning this waveform leads a pure cosine reference by 30°.

Phasor Representation

Complex Numbers and Phasors

A phasor is a complex number that captures the amplitude and phase of a sinusoid while dropping the time-varying part. This works because in steady-state analysis every signal in the circuit has the same frequency $\omega$, so you only need to track amplitude and phase.

The conversion from a time-domain sinusoid to a phasor:

1. Start with the signal in cosine form: $v(t) = A\cos(\omega t + \phi)$.
2. Drop the $\cos(\omega t + \ldots)$ part.
3. Write the phasor as $\mathbf{V} = A\angle\phi$ (polar form).

So $v(t) = 10\cos(377t + 30°)$ becomes $\mathbf{V} = 10\angle 30°$.

Complex numbers have two equivalent forms you'll use constantly:

• Rectangular form: $a + jb$, where $a$ is the real part, $b$ is the imaginary part, and $j$ is the imaginary unit ($j^2 = -1$). Electrical engineering uses $j$ instead of $i$ to avoid confusion with current.
• Polar form: $A\angle\phi$, where $A = \sqrt{a^2 + b^2}$ and $\phi = \arctan(b/a)$.

Rectangular form is easier for addition and subtraction. Polar form is easier for multiplication and division.

Euler's Formula

Euler's formula is the bridge between exponentials and trig functions:

$e^{j\theta} = \cos(\theta) + j\sin(\theta)$

This means you can write a phasor in exponential form as $\mathbf{V} = Ae^{j\phi}$, which is equivalent to $A\angle\phi$. Euler's formula is why phasors work: multiplying and dividing complex exponentials is simple algebra, and you can always convert back to sines and cosines at the end.

Domain Analysis

Time Domain Representation

Time-domain analysis describes signals as functions of time, like $v(t) = 10\cos(377t + 30°)$. You can directly read off characteristics such as:

• Peak values and peak-to-peak range
• The period and how the waveform evolves over time
• Time delays between two signals (related to their phase difference by $\Delta t = \phi / \omega$)

Time-domain representation is essential for understanding transient behavior (what happens when you flip a switch), but for steady-state AC analysis it's cumbersome because you'd constantly be working with trig identities.

Frequency Domain Representation

Frequency-domain analysis represents signals by their frequency content rather than their time behavior. For a single sinusoid, the frequency domain is just its phasor at frequency $\omega$. For more complex signals, you decompose them into sinusoidal components.

• Fourier series breaks a periodic signal into a sum of harmonically related sinusoids (fundamental frequency plus integer multiples).
• Fourier transforms extend this idea to non-periodic signals, giving a continuous frequency spectrum.

In this unit, you're mostly working with single-frequency sinusoidal sources, so the phasor approach is all you need. The frequency domain becomes more important later when you study filters, frequency response, and communication systems. The key takeaway for now: converting to phasors moves you from the time domain to the frequency domain, turning differential equations into algebra.