Phase, Reactance, Impedance and Power Factor

Ohm's law told you that a resistor opposes current by a fixed number of ohms. Feed a capacitor or an inductor with alternating current and something stranger happens: they oppose it too, in ohms, but the amount depends on the frequency, and the current is no longer in step with the voltage. That is the whole of this lesson, and it is what makes coupling capacitors, RF chokes, filters, tuned circuits and antenna matching possible. Without it a radio would be a box of resistors.

Phase

Two alternating quantities of the same frequency are in phase if they reach their peaks and their zeros at the same instants. If one gets there first, the two are out of phase, and the amount by which they are out of step is the phase angle, measured in degrees of the cycle — because one whole cycle is 360°, a quarter of a cycle is 90°.

The quantity that arrives first leads; the one that arrives later lags. In a radio circuit the two quantities being compared are almost always the voltage across a component and the current through it.

The three components and what each does to phase

A resistor: in phase

A resistor has no memory. The instant the voltage across it rises, the current through it rises in proportion, and both cross zero together. Voltage and current in a resistor are in phase, phase angle 0°.

A capacitor: the current leads by 90°

A voltage cannot appear across a capacitor until charge has been carried onto its plates — and carrying charge is current. The current therefore has to happen first: it is largest at the moment the plates are empty and the voltage is zero, and it has fallen to nothing by the time the voltage reaches its peak and the capacitor is full. In a pure capacitor the current leads the voltage by 90°.

An inductor: the current lags by 90°

An inductor resists any change in the current through it by generating a back-EMF. Apply a voltage and the current does not jump; it builds. The current is therefore always late, reaching its peak a quarter of a cycle after the voltage did. In a pure inductor the current lags the voltage by 90°.

ELI the ICE man

Write the two facts as words and they collapse into one name. Using E for voltage (electromotive force) and I for current:

• ELI — in an inductor (L), E comes before I: voltage leads, current lags.
• ICE — in a capacitor (C), I comes before E: current leads, voltage lags.

The other common form is CIVIL: in a Capacitor I leads V; V leads I in an L. Both encode the same pair. Learn whichever you can recall under pressure, but be able to say what it means rather than only chant it — the examiner sometimes asks for the relationship in words instead of as a mnemonic.

Capacitive reactance

The opposition a capacitor offers to alternating current is its capacitive reactance, symbol Xc, measured in ohms:

Both f and C sit underneath, so both work the same way: make either bigger and the reactance falls.

• As frequency rises, Xc falls. At high frequencies a capacitor is close to a short circuit — which is exactly what a decoupling capacitor is for.
• As frequency falls, Xc rises.
• At DC, where f is zero, Xc is infinite. A capacitor is an open circuit to DC. That is why a coupling capacitor passes the signal from one stage to the next while blocking the DC bias of each from the other.

Inductive reactance

Here f and L multiply, so the behaviour is exactly reversed.

• As frequency rises, XL rises. At radio frequencies a small coil presents thousands of ohms — that is an RF choke, a component that lets the DC supply through to a stage while keeping RF out of the power rail.
• As frequency falls, XL falls.
• At DC, XL is zero: an inductor is nothing but a piece of wire with a few ohms of resistance in it.

Set the chart below to 10 µH and 100 pF and drag the frequency slider from one end to the other. Watch the Xc line slide down while the XL line climbs, and note the one frequency where the two cross — that crossing is resonance, and it is the whole subject of the next lesson.

Impedance

Real circuits contain resistance and reactance together. Their combined opposition is called impedance, symbol Z, also in ohms. You cannot simply add them: the voltage across a resistance and the voltage across a reactance are 90° apart, so they combine like the two perpendicular sides of a right-angled triangle.

The same triangle gives the phase angle of the whole circuit:

Net reactance

When a circuit contains both an inductor and a capacitor, their reactances act in opposite directions — one makes the current lag, the other makes it lead — so they subtract before they enter the triangle:

If XL is the larger the circuit is net inductive and the current lags; if Xc is larger it is net capacitive and the current leads. If they are equal the reactance vanishes altogether and only R is left — that is resonance.

True power, apparent power and reactive power

Multiply the voltmeter reading by the ammeter reading in an AC circuit and you do not necessarily get the power. You get the apparent power, and only part of it is doing work.

This is why a transformer, a motor or any other reactive load is rated in volt-amperes and not in watts. Its copper has to carry the whole current whether that current is doing useful work or merely charging and discharging a reactance, and the heating of the copper depends on the current, not on how much of it is in phase.

Power factor

• A purely resistive circuit has φ = 0°, and cos 0° = 1. Power factor is unity: every volt-ampere drawn becomes a watt of heat.
• A pure reactance has φ = 90°, and cos 90° = 0. Power factor is zero: the true power is nil however many amperes flow. That current is real enough to warm the wiring, which is why it is called wattless current.
• Everything real lies between, and is described as lagging when the load is inductive and leading when it is capacitive.

Worked example 1 — true and apparent power

A load draws 5 A from a 100 V supply at a power factor of 0.6.

• Apparent power = 100 × 5 = 500 VA
• True power = 100 × 5 × 0.6 = 300 W
• Reactive power = √(500² − 300²) = √160 000 = 400 VAR

The supply and the wiring must be sized for 500 VA. Only 300 W of it is doing anything.

Worked example 2 — a series RLC circuit

A series circuit of R = 100 Ω, L = 20 mH and C = 2 µF is fed at 1 kHz. Find Xc, XL, Z, the phase angle and the power factor.

1. XL = 2πfL = 6.283 × 1000 × 0.02 = 125.7 Ω
2. Xc = 1 ÷ (2πfC) = 1 ÷ (6.283 × 1000 × 0.000002) = 1 ÷ 0.01257 = 79.6 Ω
3. X = XL − Xc = 125.7 − 79.6 = 46.1 Ω, inductive because XL is the larger
4. Z = √(100² + 46.1²) = √(10 000 + 2125) = √12 125 = 110 Ω
5. tan φ = 46.1 ÷ 100 = 0.461, so φ = 24.7°, current lagging
6. power factor = cos 24.7° = 0.91 lagging

Raise the frequency and XL grows while Xc shrinks, so the circuit becomes still more inductive. Lower it and the two converge; at about 796 Hz they are equal, Z falls to the bare 100 Ω and the power factor becomes unity.