Phase Angle and Phasors
In the previous lessons you learned that capacitors and inductors shift the phase of current relative to voltage — a capacitor makes current lead by 90°, an inductor makes voltage lead by 90°. When resistance and reactance both exist in a circuit, the actual phase shift is somewhere between 0° and 90°, depending on the relative magnitudes. Phasors are the graphical tool that makes these phase relationships visible, intuitive, and easy to calculate. A phasor is a rotating arrow whose length represents the amplitude of an AC quantity and whose angle represents its phase.
Phasor diagram for a series RLC circuit. VR is in phase with current I (pointing right). VL leads current by 90° (pointing up). VC lags current by 90° (pointing down). The total voltage VS is the vector sum, at angle θ from the current reference.
View LargerWhat Is a Phasor?
A phasor is a mathematical representation of a sinusoidal quantity — a voltage or current — as a rotating vector in a two-dimensional plane. It captures two essential properties of any AC quantity:
- Length (magnitude): proportional to the amplitude (usually RMS) of the sinusoidal quantity
- Angle: representing the phase of the quantity relative to some chosen reference
The word "phasor" is a portmanteau of "phase" and "vector." Unlike regular vectors, phasors rotate counterclockwise at the angular frequency ω = 2πf of the signal. But because all quantities in a linear circuit rotate at the same frequency, we can freeze the picture at any convenient instant and compare their relative angles — those relative angles (the phase differences) are what matter for circuit analysis.
Phasors are not the same as the physical sinusoidal waveforms. The phasor is a shorthand notation that captures the two pieces of information you need (amplitude and phase) in a compact graphical form. The actual time-domain waveform v(t) = Vpeak × sin(2πft + φ) can always be reconstructed from the phasor's length and angle.
Drawing Phasor Diagrams for Series RLC
For a series RLC circuit, the same current I flows through every component. This makes current the natural reference phasor — draw it along the positive horizontal axis (0° reference). Then:
- VR = I × R: The voltage across a resistor is always in phase with the current. Draw VR along the same horizontal axis as I (0°).
- VL = I × XL: The voltage across an inductor leads the current by 90°. Draw VL vertically upward (90°).
- VC = I × XC: The voltage across a capacitor lags the current by 90°. Draw VC vertically downward (−90°).
The total supply voltage VS is the phasor sum of VR, VL, and VC. Because VL and VC are opposite (180° apart), they partially cancel. The net reactive voltage is |VL − VC|, directed either upward (if VL > VC, inductive) or downward (if VC > VL, capacitive). Combined with VR at a right angle:
VS = √(VR² + (VL − VC)²)
Phase angle: θ = arctan((VL − VC) / VR)
Note that this is entirely consistent with the impedance triangle from the Impedance lesson: since VR = IR, VL = IXL, VC = IXC, dividing all phasor lengths by I gives the impedance triangle. The voltage phasor diagram and the impedance triangle are the same shape — just scaled by the current.
A series circuit has R = 30 Ω, XL = 80 Ω, XC = 20 Ω. The current is 100 mA RMS. Draw the phasor diagram and find the supply voltage.
VR = I × R = 0.1 × 30 = 3.0 V (horizontal, at 0°)
VL = I × XL = 0.1 × 80 = 8.0 V (vertical up, at 90°)
VC = I × XC = 0.1 × 20 = 2.0 V (vertical down, at −90°)
Net vertical: VL − VC = 8.0 − 2.0 = 6.0 V (upward, inductive)
VS = √(3.0² + 6.0²) = √(9 + 36) = √45 = 6.71 V
θ = arctan(6.0 / 3.0) = arctan(2) = 63.4° (inductive, voltage leads current)
Adding Phasors Geometrically
Phasor addition is vector addition. To add two phasors geometrically:
- Draw the first phasor starting from the origin.
- Draw the second phasor starting from the tip of the first.
- The resultant phasor is the arrow from the origin to the tip of the second phasor.
For two perpendicular phasors (like VR and (VL − VC)), this becomes the Pythagorean theorem. For phasors at other angles, you need the general vector addition formula, but in most basic AC circuit analysis the perpendicular case covers all the essential situations.
Phasor subtraction works the same way — just reverse the direction of the phasor being subtracted before adding. This is how VL and VC combine: since they point in opposite directions, their net effect is their arithmetic difference.
What the Phase Angle Means Physically
The phase angle θ tells you how many degrees the total voltage leads the current (positive θ) or how many degrees the current leads the total voltage (negative θ).
| Circuit type | Phase angle θ | Who leads? | What it means |
|---|---|---|---|
| Purely resistive | 0° | In phase | No energy storage; all power dissipated as heat |
| Purely inductive | +90° | Voltage leads current | All energy stored in magnetic field; none dissipated |
| Purely capacitive | −90° | Current leads voltage | All energy stored in electric field; none dissipated |
| Inductive > Capacitive | 0° to +90° | Voltage leads current | Circuit appears inductive; some real power dissipated |
| Capacitive > Inductive | −90° to 0° | Current leads voltage | Circuit appears capacitive; some real power dissipated |
Understanding the sign of the phase angle is important when reading impedance specifications. A load described as having "impedance 50 − j30 Ω" is capacitive (negative imaginary part). A load with "impedance 50 + j20 Ω" is inductive. A matching network must add the opposite reactance (in this case +j30 or −j20 respectively) to cancel the reactive component and achieve a purely resistive impedance at the source's output.
Power Factor and Reactive Power
The phase angle between voltage and current has a direct impact on how efficiently power is delivered to a load. The power factor (PF) is defined as:
Power Factor = cos(θ)
Real Power P = Vrms × Irms × cos(θ) = S × cos(θ) (watts)
Apparent Power S = Vrms × Irms (volt-amperes, VA)
Reactive Power Q = Vrms × Irms × sin(θ) (volt-amperes reactive, VAR)
When θ = 0° (purely resistive load), cos(0°) = 1 and all the apparent power is real power — nothing wasted. When θ = 90° (purely reactive load), cos(90°) = 0 and no real power is delivered — all the current produces only magnetic or electric field energy that sloshes back and forth without being consumed.
A small motor presents an impedance of 50 + j40 Ω at 60 Hz. It draws an apparent power of 10 VA. How much real power does it use?
Z = √(50² + 40²) = √4100 = 64.03 Ω
θ = arctan(40/50) = 38.7°
Power factor = cos(38.7°) = 0.781
Real power = PF × apparent power = 0.781 × 10 = 7.81 W. The remaining reactive power, about 6.2 VAR, is energy sloshing back and forth between the supply and the motor's magnetic field every cycle.
Correcting it: A capacitor connected across the motor supplies the reactive current locally, so less current flows in the supply wiring. Choosing a capacitor whose reactance cancels the motor's +j40 Ω inductive component brings the power factor close to 1. This is power factor correction, used in fluorescent and LED lighting, motors and large power supplies.
Phase Angle in Practice
Power factor correction. Utilities charge large users for reactive power, and many countries require equipment above a certain power to have a power factor close to 1. Switching power supplies above about 75 W include an active power factor correction (PFC) stage that shapes the input current to follow the mains voltage waveform, bringing the phase angle close to zero.
Impedance analyzers and LCR meters. An LCR meter or a vector network analyzer measures both the magnitude and the phase of an impedance. The display typically shows R (resistance) and X (reactance) separately, or equivalently |Z| (magnitude) and θ (phase angle). A reading close to +90° means an almost pure inductor; close to −90°, an almost pure capacitor; close to 0°, a resistor. The phase reading reveals, for example, whether a capacitor has become lossy with age.
Filters and crossovers. Every filter shifts the phase of the signals passing through it, and the amount depends on frequency. In a loudspeaker crossover, the woofer and tweeter outputs must arrive in phase around the crossover frequency or they partly cancel, leaving a dip in the sound. Crossover designers use phasor analysis to choose components that keep the two outputs aligned.
Frequently Asked Questions
Are phasors the same as vectors?
Phasors are a type of vector, but with one important distinction: phasors represent sinusoidal quantities at a specific frequency and are assumed to rotate at that frequency. In steady-state AC analysis, we "freeze" the rotation and only care about the relative angles between phasors. Physical vectors represent static or slowly changing quantities with direction in physical space. Mathematically, phasors use complex number arithmetic (multiplication and division rotate phasors as well as scale them), which is why complex impedance Z = R + jX is so useful.
Can the voltage across a capacitor or inductor be larger than the supply voltage?
Yes — this seems paradoxical but is entirely correct. In a series RLC circuit near resonance, the voltages across the inductor and capacitor can greatly exceed the supply voltage. They are large but nearly equal in magnitude and opposite in phase, so they almost cancel in the phasor sum. The supply voltage only needs to equal the voltage across R (which provides the resistive loss). This phenomenon — where reactive component voltages exceed the supply voltage — is called voltage magnification and is one of the reasons resonant circuits can build up large oscillating voltages. It is the principle behind Tesla coil operation and the step-up action of series resonant circuits.
Why does power factor matter?
Power factor affects how efficiently power is delivered from the supply to the load. A power factor less than 1 means some of the current flowing in the circuit is not doing useful work — it is just charging and discharging reactive components every cycle. That extra current heats the wiring, the transformer and the supply, and it must be allowed for when sizing cables, fuses, generators and UPS units. Power factor correction cancels the reactive component of the load so that the supply only has to deliver the real power the load uses.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.