Second-order response and ringing on signal lines

A circuit with both inductance and capacitance can do something a first-order circuit never does: it can overshoot its final value and oscillate around it before settling. Energy sloshes back and forth between the capacitor's electric field and the inductor's magnetic field, losing a little in the resistance on each swing. On a digital board this behaviour appears as ringing on fast signal edges, as glitches that make a clock input see two edges instead of one, and as overshoot that can push a pin beyond its voltage rating. On long cables a related effect appears as reflections. This lesson introduces second-order response and then shows where the lumped-element assumption from lesson 1 starts to fail.

The NS-5 prototype has a new problem that the NS-4 never had. Its faster microcontroller drives the radar module's clock line through a 9 cm trace, and Jonah's oscilloscope shows the clock overshooting to 4.4 V and ringing below ground before settling. The radar module occasionally misses a configuration command. Ines suspects the ringing, and asks Jonah to find out why it happens and how to stop it.

The series RLC circuit

A logic output switching a trace that ends at an input is, to a first approximation, a series RLC circuit. The output resistance and any series resistor form R, the trace and the pin leads form L, and the input and trace capacitance form C. Applying KVL around the loop and differentiating gives a second-order differential equation. Its behaviour is set by two numbers: the natural frequency ω₀, the frequency at which the circuit would oscillate with no resistance, and the damping ratio ζ, which measures how quickly the resistance removes energy.

The damping ratio divides the step response into three cases. If ζ > 1 the circuit is overdamped: the voltage rises to its final value without overshoot, more slowly than necessary. If ζ = 1 it is critically damped, the fastest response without overshoot. If ζ < 1 it is underdamped: the voltage overshoots and rings at the damped frequency ω_d = ω₀√(1 − ζ²), with the oscillation decaying inside an exponential envelope. The smaller ζ is, the larger the overshoot and the longer the ringing.

Damping with a series resistor

The standard cure is a series resistor at the driver, typically 22 to 47 Ω, which raises R in the damping formula. Critical damping needs R = 2√(L/C), here 2√(60 nH/12 pF) = 141 Ω in total, but that slows the edge considerably. A modest damping ratio of about 0.5 to 0.7 gives a small overshoot and a fast edge. With the pin's 18 Ω, adding a 33 Ω resistor makes R = 51 Ω and ζ = 0.36, cutting the overshoot to about 30 percent; adding 56 Ω gives ζ = 0.52 and about 15 percent. The code below compares several choices. Designers place the resistor right at the driver pin, for reasons the transmission-line section explains.

# Second-order step response of a driver, trace and input: overshoot versus series resistance
import math

L, C, R_out = 60e-9, 12e-12, 18.0
w0 = 1 / math.sqrt(L * C)
print("natural frequency %.0f MHz" % (w0 / 2 / math.pi / 1e6))

def step_response(R, t):
    """Capacitor voltage for a unit step into a series RLC circuit."""
    z = R / 2 * math.sqrt(C / L)
    if z < 1:
        wd = w0 * math.sqrt(1 - z * z)
        return 1 - math.exp(-z * w0 * t) * (math.cos(wd * t) + z / math.sqrt(1 - z * z) * math.sin(wd * t))
    if z == 1:
        return 1 - math.exp(-w0 * t) * (1 + w0 * t)
    s1 = -w0 * (z - math.sqrt(z * z - 1)); s2 = -w0 * (z + math.sqrt(z * z - 1))
    return 1 + (s2 * math.exp(s1 * t) - s1 * math.exp(s2 * t)) / (s1 - s2)

for R_series in (0, 22, 33, 56, 100):
    R = R_out + R_series
    z = R / 2 * math.sqrt(C / L)
    ts = [k * 0.01e-9 for k in range(2000)]
    v = [step_response(R, t) for t in ts]
    peak = max(v)
    t10 = next(t for t, x in zip(ts, v) if x >= 0.1)
    t90 = next(t for t, x in zip(ts, v) if x >= 0.9)
    print("series %3d ohm: zeta %.2f, overshoot %4.0f%%, peak %.2f V, 10-90%% rise %.2f ns"
          % (R_series, z, (peak - 1) * 100, 3.3 * peak, (t90 - t10) * 1e9))

natural frequency 188 MHz
series   0 ohm: zeta 0.13, overshoot   67%, peak 5.51 V, 10-90% rise 0.95 ns
series  22 ohm: zeta 0.28, overshoot   40%, peak 4.61 V, 10-90% rise 1.11 ns
series  33 ohm: zeta 0.36, overshoot   30%, peak 4.28 V, 10-90% rise 1.19 ns
series  56 ohm: zeta 0.52, overshoot   15%, peak 3.78 V, 10-90% rise 1.43 ns
series 100 ohm: zeta 0.83, overshoot    1%, peak 3.33 V, 10-90% rise 2.21 ns

The table shows the trade. With no series resistor the edge is fastest but overshoots wildly. A 33 Ω resistor brings the overshoot down to about 30 percent while keeping the rise time near 1 ns, still far faster than the radar's 4 MHz configuration clock needs. Ines chooses 56 Ω for the NS-5, giving about 15 percent overshoot, and the missed commands stop.

When wires stop being wires

The lumped-element model assumes a signal arrives everywhere on a connection at the same instant. In reality signals travel at a finite speed, about 15 to 20 cm per nanosecond on a circuit board, roughly half to two thirds of the speed of light. When the time a signal takes to travel along a connection becomes comparable to its rise time, the connection must be treated as a transmission line. A common rule of thumb is that a trace needs transmission-line treatment when its length exceeds about one sixth of the distance the signal travels during the rise time.

For the radar clock, a 1 ns edge travels about 15 cm, so the critical length is about 2.5 cm and the 9 cm trace qualifies as a transmission line. The lumped RLC model above still gives the right picture of the ringing, but the more accurate view says that the edge travels down the trace, reflects from the high-impedance input and travels back, and the reflections add up to the observed ringing.

Characteristic impedance and reflections

A transmission line, such as a trace over a ground plane or a twisted pair, has inductance and capacitance spread along its length. A signal travelling along it sees not a capacitor or an inductor but a pure resistance called the characteristic impedance, Z₀ = √(L′/C′), where L′ and C′ are the inductance and capacitance per unit length. Typical board traces are designed for 50 Ω, and twisted-pair cables such as RS-485 and Ethernet are around 100 to 120 Ω.

When a travelling wave reaches a point where the impedance changes, part of it reflects. The reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀) gives the fraction of the voltage that comes back. An open end (Z_L very large) reflects the whole wave with the same sign, doubling the voltage there; a short (Z_L = 0) reflects it inverted; a matched load (Z_L = Z₀) absorbs it completely with no reflection. A CMOS input is nearly an open circuit, which is why an unterminated line overshoots.

Termination

Two termination methods are common. Series termination places a resistor at the driver so that the driver's output resistance plus the resistor equals Z₀. The wave launched down the line is then half the final voltage; it doubles on reflection at the open receiver, giving the full voltage there, and the returning wave is absorbed by the matched source. This is why the series resistor goes at the driver, and it uses no extra power. Parallel termination places a resistor equal to Z₀ across the far end, absorbing the wave on arrival; it gives the cleanest result but draws current continuously. It is the standard choice for multi-drop buses where several receivers share one line.

What this means for digital design

Ringing and reflections are set by edge speed, not clock frequency. A slow 1 MHz signal produced by a modern microcontroller with 1 ns edges rings just as badly as a fast one. Practical habits follow: use the slowest edge rate that meets timing, since many microcontrollers let firmware choose the drive strength; keep fast traces short and over a continuous ground plane; add a series resistor at the driver of any clock or fast signal longer than a few centimetres; and terminate long cables at their impedance. ELE-203 and later courses treat transmission lines and signal integrity in much more depth.