Time constants, poles & first-order filters
Unify the complete first-order picture. Starting from the state response, derive the meaning of one time constant and arbitrary settling thresholds, identify the pole s=−1/τ, derive ωc=1/τ for the normalized first-order pole, prove the exact 10–90% rise-time relation tr=τ ln9 and fc tr≈0.3497, map RC/RL output choices to low-pass/high-pass transfer functions, and show how transient and frequency-response measurements can cross-check the same model without treating −3 dB, five time constants or 0.35 as universal laws.
- Interpret τ quantitatively through the normalized residual e^(−t/τ), including arbitrary settling tolerances rather than a memorized five-τ rule.
- Derive the stable first-order pole p=−1/τ from the homogeneous differential equation.
- Derive H(s)=K/(1+sτ) and show that its characteristic angular frequency is ωc=1/τ.
- Derive the exact 10–90% rise time tr=τ ln9 for a monotonic first-order step response and therefore fc tr=ln9/(2π)≈0.3497.
- Explain why the common bandwidth–rise-time constant near 0.35 is model-dependent rather than universal.
- Show how series RC and RL circuits produce complementary low-pass/high-pass outputs while sharing the same first-order pole.
- Derive a settling-time formula ts=−τ ln ε for a declared fractional residual ε.
- Relate state-energy decay to e^(−2t/τ) and distinguish state time constant from energy decay.
- Estimate τ independently from a step response and from a first-order frequency response, then use disagreement as a model/loading diagnostic.
- Design simple RC/RL first-order networks from a target τ or characteristic frequency while checking source/load resistance and model limits.
RC and RL transients look different physically, but both reduce to one stable first-order mode. What does the time constant really measure, how is it related to the pole and cutoff frequency, and when can a transient measurement and a frequency-response measurement be expected to describe the same τ?
This chapter closes DY-01 by unifying the time-domain, pole and filter viewpoints.
05.1Normalize the residual first
For any stable first-order state x(t) with initial value x0 and final value x∞, chapters 02–04 established
Subtract the final value and normalize by the initial displacement from equilibrium:
This residual is independent of the absolute amplitudes. At one time constant,
so 36.8% of the initial displacement remains. For a monotonic rise from zero toward x∞, the achieved fraction is
Thus “one τ” is not the time at which the system is finished. It is the time over which the residual is multiplied by 1/e.
05.2Settling time comes from the tolerance, not from “five τ”
Suppose an application declares that the magnitude of the normalized residual must be no larger than ε:
For the first-order exponential,
Therefore the settling time is defined only after the tolerance is declared.
The familiar statement “wait five time constants” leaves a residual e−5≈0.674%, which is often useful but is not a definition of settling.
05.3The time constant is the inverse pole magnitude
Write the homogeneous first-order equation as
Try x=Xest. Then
For a nonzero mode,
A larger τ moves the pole toward the origin and produces slower decay. A smaller τ moves it farther left and produces faster decay.
Figure 5.1 — One pole, three views
The one question is: how does the same τ appear in time, the s-plane and frequency?
These are not three different phenomena. They are three representations of the same stable linear first-order system.
This is the direct bridge to AC-02: the natural exponential and the transfer-function pole are the same root viewed from time and complex-frequency descriptions.
05.4From the same differential equation to a first-order low-pass
For a driven first-order system
the exponential-response method gives
On the sinusoidal axis,
Its magnitude relative to K is
At ωτ=1,
so the characteristic angular frequency and hertz frequency are
For this normalized one-pole low-pass, the point is exactly −3.0103 dB relative to the low-frequency plateau. AC-02 already established why that should not be promoted to a universal bandwidth definition for arbitrary systems.
05.5Derive the exact 10–90% rise time
For a unit first-order step response starting from zero,
At 10%,
At 90%,
Therefore
Combine this with fc=1/(2πτ):
This is where the familiar value near 0.35 comes from for an ideal one-pole response using a 10–90% rise-time definition.
05.6RC and RL: the output choice decides low-pass or high-pass
For a series RC network driven by Vin, with τ=RC:
The capacitor output is low-pass; the resistor output is high-pass. They share the same pole p=−1/τ and satisfy
For a series RL network with τ=L/R:
Now the resistor output is low-pass and the inductor output is high-pass. “RC is low-pass” or “RL is high-pass” is therefore incomplete language; the transfer pair and output location must be specified.
05.7State energy decays twice as fast in the exponent
For a source-free first-order state, the voltage or current decays as
Stored energy is quadratic in the state:
Hence
If one chooses to describe this energy exponential by its own e-folding time, that energy time scale is τ/2. The circuit state time constant is still τ; the distinction matters because energy is the square of the state variable.
05.8Extract τ in two independent ways
Transient method. For a monotonic step, measure the initial and final levels and find the time when 63.212% of the total change has occurred. That time equals τ for an ideal first-order response. More robustly, use two points of the normalized residual:
Frequency method. For the normalized one-pole low-pass, estimate the pole frequency from the complex fit or, under the model assumptions, from the −3.0103 dB relative point:
If the two estimates disagree beyond uncertainty, do not average them blindly. Investigate source/load resistance, probe/input capacitance, extra poles/zeros, component tolerances, nonlinearities or an incorrect one-pole assumption.
05.9Design from τ or from characteristic frequency
For a simple RC first-order pole,
For an RL first-order pole,
The equivalent resistance is the resistance seen by the storage element in the actual network, not automatically the resistor printed nearest to it.
Pedagogical RC design: target fc=1 kHz and choose C=100 nF. Then
A real design must then include source resistance, load resistance, component tolerances and parasitics before claiming the target pole.
05.10Mastery synthesis
- A first-order residual must fall below 1%. Express the required settling time in τ.
- A system has τ=2 ms. Find its pole, fc and ideal 10–90% rise time.
- Derive the exact percentage of a rising step reached after 3τ.
- A source-free capacitor voltage has τ=5 ms. After how long has its stored energy fallen to 10% of its initial value?
- For a series RC circuit, state the transfer function when the output is taken across C and when it is taken across R.
- For a series RL circuit, identify which element voltage is low-pass and which is high-pass.
- A measured step gives τ≈100 μs, while a frequency sweep gives fc=1.2 kHz. Are those values consistent with an ideal one-pole model?
- Design an RC one-pole network with C=47 nF and fc=2 kHz. Find Req, τ and ideal 10–90% rise time.
Solutions
1. ts=−τln0.01=4.605τ.
2. p=−500 s⁻¹; fc=1/(2π·0.002)=79.58 Hz; tr=2.19722τ≈4.394 ms.
3. 1−e⁻³=95.021%.
4. Energy obeys w/w0=e^(−2t/τ). Set 0.1=e^(−2t/τ): t=(τ/2)ln10≈5.756 ms.
5. HC=1/(1+sRC); HR=sRC/(1+sRC).
6. VR/Vin=1/(1+sL/R) is low-pass; VL/Vin=(sL/R)/(1+sL/R) is high-pass.
7. τ=100 μs predicts fc≈1.592 kHz, not 1.2 kHz. The discrepancy is large enough to investigate loading, extra dynamics or extraction uncertainty.
8. Req=1/[2π(2000)(47 nF)]≈1.693 kΩ; τ≈79.58 μs; tr≈174.8 μs.
DY-01 is now conceptually complete. The next dynamics course, DY-02, adds a second independent storage state. The single real pole becomes a pair of natural roots, opening the door to damping, overshoot, resonance and quality factor.