Phasor Representation of Sinusoidal Signals

Analyzing circuits with sinusoidal sources using differential equations is a tedious and time-consuming process. The phasor method transforms this complex, time-domain problem into a simpler, frequency-domain one using complex number arithmetic. By representing a sinusoidal signal's amplitude and phase as a single complex number, you can convert integro-differential equations into algebraic equations, radically simplifying the analysis of AC circuits, filters, and power systems.

The Sinusoidal Signal and Its Complex Representation

A sinusoidal signal, such as a voltage or current in an AC circuit, is mathematically described by the function $v(t)=V_m\cos (\omega t+\varphi )$. Here, $V_m$ is the peak amplitude, $\omega$ is the angular frequency in radians per second, and $\varphi$ is the phase angle which determines the signal's shift relative to a reference cosine wave at $t=0$.

The core insight behind phasor analysis is Euler's formula: $e^{j\theta }=\cos \theta +j\sin \theta$. Using this, we can express our cosine function as the real part of a complex exponential: $$v(t)=V_m\cos (\omega t+\varphi )=\Re \{V_m{e}^{j(\omega t+\varphi )}\}=\Re \{V_m{e}^{j\varphi }{e}^{j\omega t}\}.$$ Notice that the term $e^{j\omega t}$ is common to all sinusoids of the same frequency in a linear, time-invariant circuit. The unique information for our specific signal—its amplitude $V_m$ and phase $\varphi$—is contained entirely within the complex constant $V_m{e}^{j\varphi }$.

Defining and Working with Phasors

A phasor is this complex constant that encodes the amplitude and phase of a sinusoid, with the implicit understanding of a common frequency $\omega$. It is not a function of time. The phasor representation $\mathbf{V}$ of the time-domain voltage $v(t)=V_m\cos (\omega t+\varphi )$ is defined as: $$\mathbf{V}=V_m\angle \varphi =V_m{e}^{j\varphi }.$$ The polar form ($V_m\angle \varphi$) is most common, but the rectangular form ($V_m\cos \varphi +j{V}_m\sin \varphi$) is also used. The process is reversible: given a phasor $\mathbf{V}=V_m\angle \varphi$, you recover the time-domain signal by multiplying by $e^{j\omega t}$ and taking the real part: $v(t)=\Re \{\mathbf{V}{e}^{j\omega t}\}$.

Converting between time and phasor domains requires careful attention to the reference waveform. By convention, the phasor is defined relative to a cosine function. If your signal is a sine, you must first convert it to a cosine by subtracting $90^{\circ }$ ($\pi /2$ radians): $$i(t)=I_m\sin (\omega t+\theta )=I_m\cos (\omega t+\theta -90^{\circ })\Rightarrow \mathbf{I}=I_m\angle (\theta -90^{\circ }).$$

Phasor Arithmetic and Circuit Laws

The true power of phasors emerges when you perform operations on sinusoidal signals. Phasor addition and subtraction follow the rules of complex number arithmetic. If you have two voltages $v_1(t)=3\cos (\omega t+10^{\circ })$ V and $v_2(t)=4\cos (\omega t+55^{\circ })$ V, finding their sum $v_s(t)=v_1(t)+v_2(t)$ in the time domain would require messy trigonometric identities. In the phasor domain, it becomes simple vector addition: $${\mathbf{V}}_1=3\angle 10^{\circ },{\mathbf{V}}_2=4\angle 55^{\circ },{\mathbf{V}}_s={\mathbf{V}}_1+{\mathbf{V}}_2.$$ You typically convert to rectangular form to add: ${\mathbf{V}}_1=2.954+j0.521$, ${\mathbf{V}}_2=2.294+j3.277$, so ${\mathbf{V}}_s=5.248+j3.798=6.49\angle 35.9^{\circ }$ V. Therefore, $v_s(t)=6.49\cos (\omega t+35.9^{\circ })$ V.

Critically, Kirchhoff's laws hold in the phasor domain. Kirchhoff's Voltage Law (KVL) states that the phasor sum of voltages around any closed loop is zero: $\sum \mathbf{V}=0$. Kirchhoff's Current Law (KCL) states that the phasor sum of currents entering any node is zero: $\sum \mathbf{I}=0$. This allows you to write circuit equations directly in the frequency domain.

Impedance and Ohm's Law in the Phasor Domain

This is the cornerstone of sinusoidal steady-state circuit analysis. Impedance, denoted $\mathbf{Z}$, is the phasor-domain extension of resistance. It is defined as the ratio of the voltage phasor across a two-terminal element to the current phasor through it: $\mathbf{Z}=\mathbf{V}/\mathbf{I}$. Its unit is the ohm ($\Omega$).

Each passive circuit element has a specific impedance:

• Resistor (R): ${\mathbf{Z}}_R=R$. The voltage and current are in phase.
• Inductor (L): ${\mathbf{Z}}_L=j\omega L$. The voltage leads the current by $90^{\circ }$.
• Capacitor (C): ${\mathbf{Z}}_C=\frac{1}{j\omega C}=-j\frac{1}{\omega C}$. The voltage lags the current by $90^{\circ }$.

With impedance defined, Ohm's Law generalizes directly to the phasor domain: $\mathbf{V}=\mathbf{ZI}$. This allows you to analyze AC circuits using all the techniques learned for DC resistor circuits—series/parallel combinations, voltage/current division, nodal analysis, mesh analysis, and circuit theorems—as long as you use complex-number arithmetic.

Worked Example: Series RLC Circuit

Consider a series circuit with $R=4\Omega$, $L=0.5\text{ H}$, $C=0.125\text{ F}$, driven by a source $v_s(t)=20\cos (4t)$ V.

1. Find impedances: ${\mathbf{Z}}_R=4\Omega$, ${\mathbf{Z}}_L=j\omega L=j(4)(0.5)=j2\Omega$, ${\mathbf{Z}}_C=1/(j\omega C)=1/(j\ast 4\ast 0.125)=-j2\Omega$.
2. Find total impedance: ${\mathbf{Z}}_{total}={\mathbf{Z}}_R+{\mathbf{Z}}_L+{\mathbf{Z}}_C=4+j2-j2=4\Omega$.
3. Source phasor: ${\mathbf{V}}_s=20\angle 0^{\circ }$ V.
4. Find current phasor: $\mathbf{I}={\mathbf{V}}_s/{\mathbf{Z}}_{total}=(20\angle 0^{\circ })/(4)=5\angle 0^{\circ }$ A.
5. Convert to time domain: $i(t)=5\cos (4t)$ A.

Common Pitfalls

1. Forgetting the Common Frequency Assumption: Phasor analysis only applies to sinusoidal signals at a single, common frequency $\omega$. You cannot directly use phasors to analyze circuits with multiple source frequencies or non-sinusoidal waveforms (for those, you use superposition with separate phasor analyses for each frequency).
2. Mixing Time-Domain and Phasor-Domain Quantities: You cannot add a time-domain expression like $5\cos (2t)$ to a phasor like $3\angle 45^{\circ }$. All quantities in a phasor-domain equation must be phasors. Ensure all sources are converted to phasors and all elements to impedances before writing your circuit equations.
3. Incorrect Phase Reference for Sine Functions: The most frequent calculation error is misrepresenting a sine function. Remember: $\sin (\omega t)=\cos (\omega t-90^{\circ })$. Its phasor is $1\angle -90^{\circ }$, not $1\angle 0^{\circ }$. Always convert sines to cosines before creating the phasor.
4. Misinterpreting the Imaginary Part: The phasor itself is not a physical, measurable signal. The physical voltage or current is always the real part of the phasor multiplied by $e^{j\omega t}$. The "j" in inductive and capacitive impedance tracks the $90^{\circ }$ phase shift, not a mystical property.

Summary

• A phasor is a complex number representation ($A\angle \varphi$) of a sinusoid ($A\cos (\omega t+\varphi )$) that suppresses the common time-varying factor $e^{j\omega t}$.
• Phasor addition, subtraction, and multiplication follow standard complex arithmetic rules, simplifying the combination of sinusoidal signals.
• Kirchhoff's laws (KVL and KCL) are valid in the phasor domain, providing the foundation for writing circuit equations.
• Impedance $\mathbf{Z}$ generalizes resistance to inductors and capacitors in the frequency domain, allowing the use of Ohm's Law ($\mathbf{V}=\mathbf{ZI}$) and all DC circuit analysis methods for AC circuits.
• Successful application requires strict adherence to a common frequency and careful conversion of all sine functions to the cosine reference before generating phasors.