Poles, zeros & characteristic frequencies

Sinusoids use the special complex exponential e^jωt. A constant-coefficient linear differential operator is equally simple on the more general exponential

because

The real part σ controls exponential growth or decay; the imaginary part ω controls oscillation. Setting σ=0 returns the sinusoidal axis used in AC-01 and chapters 01–02 of AC-02.

Write the declared input/output relation as

with

Use an exponential trial input x(t)=Xe^st and seek a particular response y(t)=Ye^st. The common exponential cancels:

so, wherever the ratio is defined,

This is the generalized exponential input/output ratio. A later Laplace treatment gives the familiar transfer-function interpretation under its own assumptions, including treatment of initial conditions. We do not need that machinery to identify the algebraic pole-zero structure here.

With the external input set to zero, the homogeneous equation is

Try y_h=Ce^pt. Then

The roots p of the denominator polynomial are therefore the exponential natural-mode exponents. Those same roots are poles of the reduced transfer ratio H(s)=B(s)/A(s).

For ordinary continuous-time rational systems, a pole with negative real part corresponds to a decaying exponential mode; zero real part gives a non-decaying ideal mode; positive real part gives a growing mode.

A transfer zero z is a root of the reduced numerator:

For the corresponding exponential excitation, the ideal transfer ratio is zero. A zero on the imaginary axis, z=jω_z, means the selected output vanishes at that sinusoidal frequency in the ideal model.

Zeros are strongly tied to the chosen input and output. The same physical network can have different zeros when a different node voltage or branch current is declared as the output.

The sinusoidal frequency response is obtained by evaluating the generalized function at

when that evaluation corresponds to a meaningful bounded steady-state experiment. Geometrically, the real-frequency sweep walks along the imaginary axis of the s-plane.

This is why pole location influences magnitude and phase even when the pole itself is not on the jω axis: the distance and angle from each evaluation point jω to the pole change with ω.

The RC low-pass can be written

The denominator vanishes at

so there is one real negative pole. On the sinusoidal axis,

When ω≪ω₀, the imaginary term is small compared with ω₀. When ω≈ω₀, both are comparable. When ω≫ω₀, the jω term dominates. The characteristic frequency is therefore visible directly from the pole scale—before any Bode rule is memorized.

The complementary RC high-pass response is

It has the same pole

and one zero at

The zero at the origin forces

so DC is rejected. At high frequency the numerator and denominator have the same leading dependence on s, so the ratio tends to one.

Low-pass and high-pass therefore share one natural pole but differ by their transfer zeros and numerator structure.

A rational transfer function can often be written in factored form

or, for real first-order characteristic frequencies, in normalized factors such as

This form will make Bode reasoning systematic in chapter 04 because multiplication of factors becomes addition of logarithmic magnitudes and phase contributions. But chapter 04 will derive those rules; chapter 03 uses factorization only to expose structure.

Suppose an unreduced expression contains a common factor:

The external transfer ratio reduces algebraically to

For that exact reduced input/output ratio, s=−a is not a pole. But two cautions are essential.

First, a physical implementation may contain an internal mode associated with the canceled dynamics even though that mode is invisible in the chosen external transfer under ideal assumptions. Second, real component tolerances make exact pole-zero cancellation fragile: a near-cancellation can leave a residual feature.

Therefore “the algebra canceled” is not evidence that all internal energy storage or all physical sensitivity disappeared.

1. Show that D^ke^st=s^ke^st.
2. For τ dy/dt+y=Kx, derive H(s), identify its pole, and recover H(jω).
3. For the RC high-pass, identify its pole and zero and explain the DC limit from the zero.
4. A pole is at p=−2000 s⁻¹. What characteristic angular frequency and hertz frequency does it define?
5. What does a pole with positive real part imply about a natural exponential mode?
6. Why can two outputs of the same physical network have different zeros?
7. Reduce (s+10)/[(s+10)(s+100)] and explain what the reduction does—and does not—prove physically.
8. For H(s)=5(1+s/1000)/[(1+s/100)(1+s/10000)], list the finite poles, zero and low-frequency value without drawing a Bode plot.

Chapter 04 will derive Bode magnitude and phase rules from these factors. Slopes and corner-frequency contributions will appear as consequences of logarithms and asymptotic limits, not as a memorized chart.