Complex Zero
Understanding Complex Zero
In RF and microwave engineering, a circuit or filter is described by a transfer function H(s), the ratio of two polynomials in the complex frequency variable s = σ + jω. The roots of the numerator polynomial are the zeros of the network, and the roots of the denominator are the poles. A zero is a value of s at which H(s) goes to zero, meaning the network ideally transmits nothing at that complex frequency. When a zero has a nonzero imaginary part it is called a complex zero, in contrast to a real zero that lies directly on the σ axis. The position of a zero in the s-plane carries a clear physical meaning for how the network treats signals near that frequency.
Conjugate Pairs and the s-Plane
The coefficients of H(s) for any network made from real components are themselves real numbers. A fundamental property of real-coefficient polynomials is that their complex roots must occur in conjugate pairs. Consequently, a complex zero located at s = σ + jω is always accompanied by a matching zero at s = σ − jω. Plotted on the s-plane, the two zeros are mirror images across the real axis. Their horizontal position (σ) sets how heavily damped they are, while their vertical position (ω) sets the frequency at which their effect on the response is strongest. A real zero, by contrast, sits alone on the real axis because it equals its own conjugate.
How a Complex Zero Creates a Notch
The magnitude response of a network is found by evaluating H(s) along the imaginary axis, that is at s = jω. When the operating frequency approaches the imaginary part of a complex zero pair, the distance from the jω evaluation point to the nearby zero becomes very small, and the transfer-function magnitude drops sharply toward zero. This produces a transmission null, commonly called a notch. The closer the zero sits to the imaginary axis (the smaller σ is), the deeper and narrower the resulting notch. A zero placed exactly on the imaginary axis (σ = 0) gives, in theory, an infinitely deep null, while finite component losses keep real notches to a practical depth, often in the range of 40 to 80 dB.
Effect on Phase and Group Delay
Complex zeros do more than carve notches in the magnitude response. As frequency sweeps past the imaginary part of a left-half-plane zero pair, the phase contributed by that pair advances rapidly, and the group delay, which is the negative derivative of phase with respect to frequency, develops a sharp feature near the notch. This is why filters that use transmission zeros to sharpen selectivity, such as elliptic designs, trade flatter group delay for steeper skirts. If a zero instead lies in the right half-plane (positive σ), the network becomes non-minimum-phase, adding extra phase lag without a matching magnitude change, which matters in equalizer and all-pass design.
Where Complex Zeros Appear in RF Hardware
Complex zeros show up wherever a circuit deliberately suppresses energy at a specific frequency or sharpens a band edge. Cross-coupled cavity and dielectric resonator filters introduce them by adding signal paths that cancel at chosen frequencies, placing finite transmission zeros just outside the passband. Notch filters, band-stop networks, diplexer and multiplexer junctions, and stub-loaded lines all rely on complex or imaginary-axis zeros to reject interferers. The practical engineering challenge is that component loss pulls a zero away from the imaginary axis, so a designer who wants a deep, narrow notch must use high-Q resonators and accept tighter tuning tolerances.
Key Factors That Set Zero Behavior
- Real part (σ): controls damping and how close the zero sits to the imaginary axis, which sets notch depth and width.
- Imaginary part (ω): sets the approximate notch frequency where the magnitude null appears.
- Half-plane location: left half-plane keeps the response minimum-phase; right half-plane makes it non-minimum-phase.
- Resonator Q: finite loss moves a zero off the imaginary axis, limiting achievable null depth.
- Number of zero pairs: more finite transmission zeros allow steeper band edges and multiple notches.
Key Formula
H(s) = K · (s − z)(s − z*) / D(s)
= K · (s² − 2σs + σ² + ωz²) / D(s)
Notch (null) frequency:
fnotch ≈ ωz / (2π) Hz
Where s = σ + jω is the complex frequency variable (rad/s); z = σ + jωz and z* = σ − jωz are the complex zero and its conjugate; σ is the real part (damping, distance from the imaginary axis); ωz is the imaginary part (approximate notch frequency, rad/s); K is a real gain constant; and D(s) is the denominator polynomial whose roots are the poles. The quadratic factor keeps the numerator coefficients real, which is exactly why the zero and its conjugate appear together.
Typical Characteristics
| Parameter | Typical Range / Value | Notes |
|---|---|---|
| Zero location (σ) | 0 to small negative | Left-half-plane for stable, minimum-phase response |
| Notch frequency | ωz / 2π (Hz) | Set by the imaginary part of the zero |
| Achievable notch depth | 40 to 80 dB | Limited by resonator Q and component loss |
| Notch bandwidth | Narrows as σ → 0 | Closer to the jω axis means a sharper null |
| Conjugate symmetry | Always present | Real coefficients force pairs at σ ± jωz |
| Common use | Elliptic, notch, cross-coupled filters | Finite transmission zeros sharpen band edges |
Frequently Asked Questions
What is a complex zero?
A complex zero is a root of a transfer function's numerator that has a nonzero imaginary part, so it lies off the real axis in the s-plane. For networks with real component values it always appears as a conjugate pair, and it forces the magnitude response toward a sharp transmission null at its imaginary frequency.
Why do complex zeros occur in conjugate pairs?
Physical RF networks use real-valued inductors, capacitors, and resistors, so the polynomial coefficients of the transfer function are real. A polynomial with real coefficients can only have complex roots that come in conjugate pairs, so every complex zero at sigma plus j omega is matched by one at sigma minus j omega, mirrored across the real axis.
How does a complex zero affect a filter response?
A conjugate pair of complex zeros near the j omega axis creates a deep transmission null, or notch, at a frequency close to their imaginary part. This sharpens band-edge selectivity in elliptic and cross-coupled filters, but it also adds rapid phase change and group-delay distortion around the notch frequency.