The Transfer Function and Generalized Impedance
One pole sets both the transient and the cutoff
On the left is an RC low-pass circuit; seen with generalized impedances, H(s) = (1/sC)/(R + 1/sC) = 1/(1+sRC). The pole, which makes the denominator 0, is at s = −1/(RC) = −1/τ, and the cross drawn on the negative real axis of the s-plane on the right marks it. Slide the time constant τ to put the cross on the target s = −2 inside the dashed circle. That pole is at once the speed of the transient e^(−t/τ) and the frequency cutoff ω_c = 1/τ.
Widen jω into s
In sinusoidal steady state, impedance was written with jω: jωL for an inductor and 1/jωC for a capacitor. Extend that jω to the complex frequency s = σ + jω and the inductor becomes sL and the capacitor 1/sC. Set s = jω (σ=0) and you are back in sinusoidal steady state; let s have a real part σ and transients that grow or die away fit inside the same impedance. A resistor does not depend on frequency, so it stays R, and this one extension lets steady state and transients be handled in the same language.
The circuit is a voltage divider in s
Seen with generalized impedances, two elements in series are still a voltage divider. Take the output of an RC low-pass across the capacitor and H(s) = Vout/Vin = Z_C/(Z_R + Z_C) = (1/sC)/(R + 1/sC). Multiply top and bottom by sC and it tidies into H(s) = 1/(1 + sRC), so one line of voltage division is the circuit's transfer function itself. Slot in an inductor as sL, or take a more complicated circuit, and the divisions and combinations written with impedances still build H(s).
One expression, three faces
The transfer function H(s) = 1/(1+sRC) reads three ways. First, at s = 0 (DC), H = 1, so the DC gain is 1 and slowly changing signals pass straight through. Second, setting s = jω gives H(jω) = 1/(1+jωRC), a low-pass frequency response whose magnitude bends to 1/√2 at ω = 1/RC. Third, the pole that makes the denominator 0, s = −1/RC, sets the transient: switch it on and the gap between the output and its final value shrinks as e^(−t/RC). The same single expression holds the DC gain, the frequency cutoff and the transient time constant, so one pole position compresses the whole behavior of the circuit. In the plot, the lightly shaded left half-plane is the stable region, where transients die away.
Back to the first screen
As you slid the time constant τ, the pole s = −1/τ moved along the negative real axis, and at τ = 0.5 it reached the target s = −2 in the dashed circle. That one pole set two things together: in time, the speed of the transient as the output settles by e^(−t/τ), and in frequency, the cutoff that bends at ω = 1/τ. Because jω was extended to s and the capacitor seen as 1/sC, the RC low-pass was written as one line of voltage division, H(s) = 1/(1+sτ), and that single expression held the DC gain, the frequency response and the transient all at once.