Phasor Conversion and Representation

Core Idea

A phasor is a complex number representing the amplitude and phase of a sinusoid. The transformation v(t) = Re[V̅ e^jωt] converts time-domain sinusoids to frequency-domain phasors V̅ = |V|e^jφ. This greatly simplifies AC circuit analysis by converting differential equations into algebraic equations.

How It's Best Learned

Practice converting between time-domain and phasor domains. Use Euler's formula e^jθ = cos(θ) + j sin(θ) to move between rectangular and polar forms. Verify using circuit simulations.

Common Misconceptions

• Phasors only apply to single-frequency signals. - A phasor magnitude is the same as peak voltage; RMS values are used in phasors. - Phasor addition in the complex plane is vector addition, not scalar addition.

Explainer

In your prerequisite work, you learned that AC circuits in steady state have voltages and currents oscillating at the same frequency as the source — only the amplitudes and phases differ from branch to branch. You also learned Euler's formula: e^jθ = cos(θ) + j·sin(θ), which connects complex exponentials to sinusoids. Phasors bring these two ideas together into one powerful transformation: they let you represent any sinusoidal signal as a single complex number, freezing the time-varying behavior into a static amplitude and phase angle that you can manipulate with algebra instead of calculus.

The conversion starts from the observation that any sinusoid v(t) = Vm·cos(ωt + φ) can be written as the real part of Vm·e^(j(ωt + φ)) = Vm·e^jφ · e^jωt. In a single-frequency circuit, every voltage and current shares the same e^jωt factor — it represents the common rotation in the complex plane at frequency ω. Since this factor is the same everywhere, you can factor it out and set it aside. The phasor V̅ = Vm·e^jφ = Vm∠φ captures everything that distinguishes one sinusoid from another: its peak amplitude and its phase angle. Written in polar form Vm∠φ or in rectangular form Vm·cos(φ) + j·Vm·sin(φ), it's a static complex number you can add, subtract, and multiply using ordinary complex arithmetic.

The algebraic payoff is immediate and substantial. Consider KVL: in the time domain, summing voltages means adding sinusoids with different phases — messy trigonometric manipulations. In the phasor domain, summing voltages means adding complex numbers: V̅_total = V̅_1 + V̅_2, which is just vector addition in the complex plane. Differentiation — needed for inductors (v = L·di/dt) and capacitors (i = C·dv/dt) — becomes multiplication by jω in the phasor domain. This transforms the differential equations governing inductor and capacitor voltages into algebraic equations: V̅_L = jωL·Ī and Ī_C = jωC·V̅_C. The concept of complex impedance follows directly: Z_L = jωL, Z_C = 1/(jωC), and Ohm's law in the phasor domain is simply V̅ = Z·Ī.

A critical boundary condition governs where phasors are valid: they only apply to single-frequency, sinusoidal steady state. If a circuit has two independent sources at different frequencies ω₁ and ω₂, you cannot add their phasors directly — they rotate at different rates. Instead, you solve the circuit twice, once at each frequency (two separate phasor analyses), and then add the resulting time-domain signals using superposition. This is not a workaround but reflects a fundamental principle: phasor analysis is a bijection between one frequency's sinusoidal behavior and the complex plane. Mixed-frequency circuits simply have two such bijections operating independently. Once you internalize this constraint, phasors become the natural language for any single-frequency analysis — impedance networks, AC circuit theorems, power calculations, and frequency response all live in the phasor domain.