Signals & Wireless (RF)
How an RC circuit shifts phase, and how to get 90°
An RC low-pass circuit, a resistor into a capacitor to ground, doesn't just shrink fast signals: it delays them. The output sine trails the input by a phase angle that depends on how the frequency compares with the circuit's corner. Far below the corner the lag is nearly nothing; at the corner it's exactly 45°; far above it the lag creeps towards 90° but never gets there. For a real quarter turn, use two legs.
The formula
The output's size falls at the same time, to 1 ÷ √(1 + (f ÷ fc)²) of the input:
| Frequency | Lag | Output size |
|---|---|---|
| A tenth of the corner | 5.7° | 99.5 % |
| At the corner | 45° | 70.7 % |
| Ten times the corner | 84.3° | 10 % |
Why it lags
The capacitor's voltage can only change as current flows into it, so it always arrives a little after the input moves. At low frequency it has plenty of time to keep up. At the corner, where the capacitor's reactance equals R, it falls behind by an eighth of a cycle. Far above, it hardly charges at all before the input turns round, and its small swing trails by almost a quarter cycle.
Worked example: 45° at 100 kHz
1.59 kΩ and 1 nF put the corner at 1 ÷ (2π × 1590 × 10⁻⁹) ≈ 100 kHz, so a 100 kHz sine comes out 45° late. Swap to 100 pF and the corner moves to 1 MHz: only 5.7° of lag. Swap to 10 nF and the corner drops to 10 kHz: 84° of lag, but the output is a tenth the size.
Live experiment
Make the output trail by 90°
A 100 kHz sine drives 1.59 kΩ into a capacitor. Pick C1: 100 pF lags 5.7°, 1 nF puts the corner on 100 kHz for exactly 45°, 10 nF lags 84° but shrinks the output. Then add a second leg, ten times lighter or identical, and compare the total lag and what's left of the signal.
“A quarter turn” is lesson 2 of 6 in Oscillator Outcrop, Signals & Wireless (RF) World 2. Free with an account. Lessons unlock in order, so you’ll do the 8 before it first.
Two legs make a quarter turn
Chain a second leg with the same corner and the lags add: 45° + 45° = 90° at that frequency. Make the second leg ten times lighter, 15.9 kΩ and 100 pF, so it barely loads the first. The output then lags about 90° and keeps about half the input's size.
Where phase shift matters
Radios split a signal into two copies a quarter cycle apart, called I and Q, to tell a station from its mirror image on the other side of the oscillator. The RC phase-shift oscillator chains three legs to reach 180° at one frequency. And in every feedback loop, the lag a filter adds eats into the loop's stability margin, which is why a Bode plot shows phase as well as gain.
Free lesson
Now build it yourself
Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.
“A quarter turn” is lesson 2 of 6 in Oscillator Outcrop, Signals & Wireless (RF) World 2. Free with an account. Lessons unlock in order, so you’ll do the 8 before it first.
Or start from the very first lesson, no account neededPractice questions
Three quick questions from the lessons. Pick an answer to check it.
Show the answer
Answer: 84°. Far above the corner the lag approaches 90°: at ten times, arctan(10) is 84°.
- Why not “0°”? Only far below the corner is the lag near zero. At ten times the corner the capacitor dominates.
- Why not “6°”? 6° is the lag at a tenth of the corner. Above the corner the lag grows towards 90°.
- Why not “45°”? 45° is the lag at the corner itself. Ten times above it, the lag is tan⁻¹(10), about 84°.
Show the answer
Answer: Still 90°, but only a third of the signal: the second leg loads the first. With both corners at 100 kHz the lag there is still 90°, but the second leg draws current from the first's node and the signal drops to a third. Ten times the impedance, 15.9 kΩ and 100 pF, keeps the same corner and barely loads: about half the signal is left.
- Why not “0°”? Each leg lags at the corner, so the pair can't be in step with the input.
- Why not “180°”? Two RC legs only approach 180° far above their corner. At the corner itself they give 90°.
- Why not “45°”? Each leg lags 45° at the shared corner, and the lags add, so the total is 90°, not 45°.
Show the answer
Answer: 1 MHz: it adds only 6°. 1 ÷ (2π × 15.9 kΩ × 10 pF) is 1 MHz. Ten times too high, and the second 45° never comes.
- Why not “DC”? An RC low-pass has its corner where R equals the capacitor's reactance, which is 1 MHz here, not zero.
- Why not “10 kHz”? A smaller capacitor moves the corner up, not down: 1 ÷ (2π × 15.9 kΩ × 10 pF) is about 1 MHz.
- Why not “100 kHz”? 100 kHz needs 15.9 kΩ with 100 pF. With 10 pF, R × C is ten times smaller and the corner ten times higher.
“A quarter turn” is lesson 2 of 6 in Oscillator Outcrop, Signals & Wireless (RF) World 2. Free with an account. Lessons unlock in order, so you’ll do the 8 before it first.
Questions people ask
What is the phase shift of an RC low-pass filter at its cutoff frequency?
The output lags the input by exactly 45°, an eighth of a cycle, and its amplitude is 70.7 % of the input. That's the point where the capacitor's reactance equals the resistance.
Can one RC stage give a 90° phase shift?
Only in the limit. The lag approaches 90° far above the corner, but by then the output has shrunk towards nothing: at ten times the corner it lags 84° and is a tenth the size. For a usable 90°, cascade two legs.
Does an RC high-pass filter shift the phase too?
Yes, the other way: its output leads the input. It leads by nearly 90° far below the corner, by 45° at the corner, and by almost nothing far above it.
Related topics
- Low-pass filter explainedA resistor and a capacitor that pass the bass and fade the treble, and where the cutoff sits.
- Capacitive and inductive reactance explainedWhy a capacitor passes treble and a coil passes bass: X_C, X_L and Ohm's law for AC.
- How to read a Bode plotGain in decibels against frequency on a log scale: corners, −3 dB and 20 dB per decade.
- RC time constant explainedτ = R × C: how long a capacitor takes to charge through a resistor, and the 63% rule.