02 · Intermediate

Magnitude, phase & normalized frequency

Decompose the complex response H(jω) into magnitude and phase without losing units or reference conventions. Derive exact first-order low-pass and high-pass magnitude/phase relations, introduce the dimensionless normalized frequency Ω=ω/ω₀, distinguish phase shift from true time delay, and use limiting values and complementary outputs as independent checks before logarithmic Bode scales appear.

135–175 minIntermediate

In this lesson

Write H(jω)=|H(jω)|e^{jφ(ω)} and interpret magnitude and phase as two views of the same complex response.
Track physical units correctly for dimensionless gains, transimpedance and transadmittance magnitudes.
Derive the exact magnitude and phase of the ideal RC low-pass HLP=1/(1+jΩ).
Derive the exact magnitude and phase of the complementary RC high-pass HHP=jΩ/(1+jΩ).
Define normalized frequency Ω=ω/ω₀ and explain why first-order responses with different component values collapse onto the same dimensionless curves.
Use low-, characteristic- and high-frequency limits to check magnitude and phase formulas.
Explain why a phase φ(ω) does not imply a constant time delay unless φ is linear in ω with the appropriate slope.
Handle principal phase and equivalent angles without confusing display convention with physical change.

Starting question

Chapter 01 produced a complex function H(jω). Engineers almost never inspect its real and imaginary parts first. They ask two direct physical questions: by what factor is the sinusoidal amplitude changed, and by how much is the output phase shifted relative to the declared input reference?

Those two answers are the magnitude and phase of the same complex response.

02.1One complex response, two coordinates

H(jω)=H_R(ω)+jH_I(ω)=|H(jω)|e^jφ(ω)

|H(jω)|=√(H_R²+H_I²)

φ(ω)=atan2(H_I,H_R)

For a dimensionless voltage or current gain, |H| is the amplitude ratio. If H is a transimpedance or transadmittance, its magnitude keeps the corresponding physical unit, Ω or S.

Figure 2.1 — What do magnitude and phase mean at one frequency?

The one question is: how does one complex number H(jω) encode both amplitude scaling and phase shift?

Magnitude is the distance from the origin; phase is the oriented angle from the real axis. They are two coordinates of the same complex number, not two independent responses.

02.2Magnitude scales amplitude; phase shifts the sinusoid

Let the input be

x(t)=X_mcos(ωt+θ_x)

with frequency response

H(jω)=M(ω)e^jφ(ω), M(ω)≥0

Then the output phasor is the product of the input phasor and H:

y̲=M(ω)X_m∠(θ_x+φ(ω))

and the real steady-state output is

y(t)=M(ω)X_mcos(ωt+θ_x+φ(ω))

Thus |H| controls amplitude scaling and φ controls relative phase. Neither statement requires a logarithmic scale.

02.3Normalize frequency before comparing first-order systems

For the ideal RC low-pass of chapter 01, define

ω₀=1/(RC)

Ω=ω/ω₀=ωRC

Ω is dimensionless. The response becomes

H_LP(jΩ)=1/(1+jΩ)

All ideal RC low-pass networks with the same topology have the same normalized curve versus Ω. Changing R or C moves the physical frequency scale through ω₀; it does not change the normalized first-order shape.

02.4Derive the exact low-pass magnitude and phase

H_LP(jΩ)=1/(1+jΩ)=(1−jΩ)/(1+Ω²)

Therefore

|H_LP|=1/√(1+Ω²)

φ_LP(Ω)=−atan(Ω), Ω≥0

Three exact checkpoints organize the whole curve:

RegionMagnitude / phase

Ω→0|H|→1, φ→0°

Ω=1|H|=1/√2, φ=−45°

Ω→∞|H|→0, φ→−90°

These are exact limits and one exact characteristic point; no straight-line Bode approximation has been used.

02.5The complementary high-pass output is a second independent check

Keep the same series R–C network and input reference, but now define the output across the resistor with a compatible polarity. Then

H_HP(jΩ)=jΩ/(1+jΩ)

Hence

|H_HP|=Ω/√(1+Ω²)

φ_HP(Ω)=90°−atan(Ω), Ω>0

The limiting behavior reverses:

Ω→0 : |H_HP|→0

Ω→∞ : |H_HP|→1

Because the two output voltages reconstruct the input under the chosen references,

H_LP(jΩ)+H_HP(jΩ)=1

At Ω=1 the two magnitudes are both 1/√2, but their phases are −45° and +45°. Their magnitudes do not add to one; their complex values do.

02.6Phase shift is not automatically a constant time delay

A pure delay τ has the frequency response

H_delay(jω)=e^−jωτ

|H_delay|=1

φ_delay(ω)=−ωτ

Its unwrapped phase is exactly linear in ω. By contrast, the RC low-pass phase is

φ_LP(ω)=−atan(ωRC)

which is not linear in ω over all frequencies, and its magnitude is not unity. Therefore the RC network is not simply a constant time delay.

02.7Principal phase is a display convention

φ ≡ φ+360°k, k∈ℤ

A complex number does not change when its displayed phase is shifted by an integer multiple of 360°. Calculators often return a principal angle such as (−180°,180°].

When frequency is swept, a continuous representation may require adding or subtracting 360° between adjacent samples. This is commonly called phase unwrapping. It changes the displayed branch of the angle, not the underlying complex response.

Do not unwrap a genuine reference reversal or sign error away. The physical reference definition must be correct before display continuity is considered.

02.8Numerical comparison at Ω=0.1, 1 and 10

ΩLow-pass

0.1|H|=0.9950, φ=−5.711°

1|H|=0.7071, φ=−45.000°

10|H|=0.09950, φ=−84.289°

The same table applies to every ideal first-order RC low-pass after converting physical frequency to Ω. For example, Ω=10 means f=10f₀ regardless of whether f₀ is 100 Hz or 100 MHz—provided the ideal lumped model is still valid.

02.9How to validate measured magnitude and phase

Use the same input/output references used to define H(jω).
Measure both amplitudes and their relative phase at the same frequency.
Form the linear magnitude ratio before any later logarithmic conversion.
Compare measured and predicted phase using a consistent angle convention.
Normalize frequency only after the characteristic frequency has been defined from the model or independently estimated from data.

If magnitude agrees but phase does not, possible causes include channel delay/skew, reference reversal, probe mismatch or an omitted reactive element. If phase agrees but magnitude does not, source/load scaling, attenuation settings or calibration may be the first suspects. Both are parts of the same complex response and should ultimately be checked together.

02.10Mastery exercises

Given H=3−j4, calculate |H| and principal phase.
For H(jω)=2/(1+jωτ), derive magnitude and phase.
For the same response, evaluate the exact values at ωτ=1.
Derive H_HP for the resistor output of a series RC network and verify H_LP+H_HP=1.
At Ω=1, show explicitly why the two RC output magnitudes do not add to one even though the complex outputs do.
Explain why Ω=ω/ω₀ is dimensionless and why that matters.
Compare φ(ω)=−atan(ωτ) with the pure-delay phase −ωT. Which one represents a constant delay?
A measured phase jumps from −179° to +179° between adjacent frequency points. Give two different possibilities that must be distinguished before “fixing” the plot.

Solutions

1. |H|=5 and φ=atan2(−4,3)=−53.13°.

2. |H|=2/√(1+(ωτ)²), φ=−atan(ωτ).

3. |H|=√2 and φ=−45°.

4. H_HP=jΩ/(1+jΩ); adding it to 1/(1+jΩ) gives one exactly.

5. Each magnitude is 1/√2, so magnitudes sum to √2; the complex values (1−j)/2 and (1+j)/2 sum to 1.

6. Both ω and ω₀ have units rad/s, so the ratio is dimensionless; normalized curves can therefore compare systems with different physical frequency scales.

7. −ωT is linear in ω and represents the pure constant delay T; −atan(ωτ) does not over the full frequency range.

8. It may be a harmless principal-angle wrap of a continuous phase, or a real reference/sign/measurement discontinuity. The complex data and setup must be checked before unwrapping.

Chapter 03 will explain how characteristic behavior is encoded structurally by poles and zeros. Bode slopes will still wait until those objects are derived rather than memorized.