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Phasors — why we “cheat” (and why it’s allowed)

This is a practical chapter on phasors: why time-domain sine functions become unreadable, how phasors compress sinusoidal steady state into simple algebra, and why RMS and power factor (\(\cos\varphi\)) show up everywhere once you start caring about real-world power.

sinusoidal steady state RMS • impedance • complex power three-phase friendly

1) The problem we’re pretending not to have

AC systems are time-dependent. Voltage and current are functions of time:

\[ v(t) = V_\text{peak}\sin(\omega t + \theta) \]

That’s fine for a single waveform on a napkin. It becomes a swamp when you have:

  • multiple phases (120° separation)
  • inductors and capacitors (phase shifts)
  • power (averages, products, and “why is there a double-frequency term?”)
Phasors are not magic. They are a compression technique for one special (but extremely common) case: sinusoidal steady state.

2) Why we don’t “just use sine functions directly”

In theory, you can solve everything in the time domain with trig + calculus. In practice, it’s repetitive, fragile, and hard to audit.

Inductors and capacitors drag calculus into everything

For an inductor: \(v(t) = L\frac{di(t)}{dt}\). For a capacitor: \(i(t) = C\frac{dv(t)}{dt}\). So even a “simple” circuit wants differentiation/integration immediately.

The system is linear. Our first approach is… melodramatic.

3) An intentionally ugly example

Take a simple RL load driven by a sine:

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

The steady-state current is:

\[ i(t) = \frac{V_m}{\sqrt{R^2 + (\omega L)^2}} \sin\left(\omega t - \tan^{-1}\left(\frac{\omega L}{R}\right)\right) \]

That’s already “fine but annoying”. Now compute instantaneous power:

\[ p(t)=v(t)i(t) \]

You’ll get a constant term (average power) plus a term oscillating at \(2\omega\) (the “double-frequency wiggle”). And that’s before you do three phases. 

// Time-domain reality check (not code you want to maintain):
v(t) = Vm * sin(ωt)
i(t) = (Vm / |Z|) * sin(ωt - φ)
p(t) = v(t) * i(t)  // expands into DC + cos(2ωt - φ) terms
The ugly part isn’t that it’s impossible. The ugly part is that you have to keep doing it.

4) The key realization: steady state doesn’t care about time

In sinusoidal steady state, everything oscillates at the same frequency \(\omega\). Only amplitude and phase differ.

So we factor out the shared time dependence and keep the meaningful parts.

5) Phasors: time removed, meaning retained

We represent a sinusoid by its RMS magnitude and phase:

\[ v(t) = V_\text{peak}\sin(\omega t + \theta) \quad\Longrightarrow\quad \underline{V} = V_\text{RMS}\angle\theta \]

Under the hood, this works because differentiation and integration turn into simple operations on complex exponentials. In phasor land:

\[ \frac{d}{dt}\;\longleftrightarrow\; j\omega \qquad\qquad \int \; dt\;\longleftrightarrow\;\frac{1}{j\omega} \]

Impedance becomes algebra

\[ Z_R = R,\quad Z_L = j\omega L,\quad Z_C = \frac{1}{j\omega C} \]

Now Ohm’s law is still Ohm’s law:

\[ \underline{V} = \underline{I}\,Z \]

φ V I phasor view (steady-state)
A phasor diagram: voltage as reference, current lagging by \(\varphi\) for an inductive load.
P Q S φ power triangle
Complex power geometry: \(S\) (apparent), \(P\) (real), \(Q\) (reactive), with \(\cos\varphi = P/|S|\).

6) Why RMS is the amplitude that survives contact with reality

Peak values are visually dramatic. They are also not what power cares about. Power is energy per time, and for periodic signals we care about an average effect.

RMS (root mean square) is defined as:

\[ V_\text{RMS} = \sqrt{\frac{1}{T}\int_0^T v^2(t)\,dt} \]

RMS of a sinusoid (proof sketch)

Let \(v(t)=V_m\sin(\omega t)\). Then:

\[ V_\text{RMS} = \sqrt{\frac{1}{T}\int_0^T V_m^2\sin^2(\omega t)\,dt} = V_m\sqrt{\frac{1}{T}\int_0^T \sin^2(\omega t)\,dt} \]

Over a full period, the average of \(\sin^2\) is \(1/2\), so:

\[ V_\text{RMS} = \frac{V_m}{\sqrt{2}} \]

RMS is the “DC-equivalent heating value.” That’s why meters use it, why phasors use it, and why power math stays sane.

7) Power in phasor form (where things finally get clean)

Using RMS phasors, complex power is:

\[ \underline{S} = \underline{V}\,\underline{I}^* \]

From this:

  • \(P = \Re\{\underline{S}\}\) is real power (watts)
  • \(Q = \Im\{\underline{S}\}\) is reactive power (var)
  • \(|S|\) is apparent power (VA)

Power factor follows the triangle:

\[ \cos\varphi = \frac{P}{|S|} \]

Example: cos φ = 0.7

If \(\cos\varphi=0.7\), then \(\varphi \approx \cos^{-1}(0.7) \approx 45.57^\circ\). If we choose \(|S| = 10\text{ kVA}\) for round numbers:

\[ P = |S|\cos\varphi = 10\cdot 0.7 = 7\text{ kW} \]

\[ Q = |S|\sin\varphi \approx 10\cdot \sqrt{1-0.7^2} \approx 7.14\text{ kVAr} \]

You can pick different \(|S|\) and the geometry stays the same. That’s the point: phasors preserve structure.

8) Three-phase: the real payoff

Three-phase systems are where phasors stop being “nice” and become “mandatory”. Balanced systems collapse to one phase, and the 120° separation is naturally represented as angles.

Teaser: Δ and Y connections

In a balanced system:

  • Line-to-line voltages in Δ are separated by 120°.
  • Relationships like \(\sqrt{3}\) and 30° shifts appear as simple phasor geometry.
A follow-up chapter can go deep on Δ vs Y: phase voltage vs line voltage, line current vs phase current, and the classic \(\sqrt{3}\) relationships (plus where students usually get stabbed by sign conventions).

9) What phasors are not

  • They do not handle transients well (switching, startup, non-sinusoidal waveforms).
  • They assume a single frequency steady state.
  • They hide time dependence; they don’t deny it.
Phasors are a tool. Powerful tools are dangerous only when you pretend they’re universal.

10) Why this page exists

Because the smooth wave on the front page looks simple. It isn’t.

Phasors are the reason engineers can design AC systems without drowning in trig identities. They are not a trick. They’re a mercy.

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