AC power, frequency response and filters

The last lesson of the course puts impedance to work. It begins with power in AC circuits, where voltage and current may be out of phase and the simple product vi no longer gives the average power. It then looks at frequency response, the way a circuit's output changes as the input frequency is swept, and at the Bode plot that engineers use to show it. Filters, circuits designed to pass some frequencies and block others, are the payoff: the RC low-pass filter in front of an ADC, the high-pass coupling capacitor, and the LC filter that keeps a switching regulator's ripple away from a sensitive radio. The lesson ends by collecting the design decisions Jonah and Ines made for the NS-5 board.

Power in AC circuits

For sinusoidal steady state, the instantaneous power p(t) = v(t)i(t) oscillates at twice the signal frequency. Its average over a cycle is what heats a resistor or drains a battery. If the voltage and current phasors have RMS magnitudes V and I and the current lags the voltage by angle θ, the average or real power is V I cos θ. The factor cos θ is the power factor. A resistor has θ = 0 and a power factor of 1; an ideal capacitor or inductor has θ = ±90° and absorbs no average power at all, storing energy in one half-cycle and returning it in the next.

The quantity Q, measured in volt-amperes reactive (var), describes the energy that sloshes back and forth without being used; S, measured in volt-amperes (VA), is the apparent power, the product of the RMS voltage and current. Power factor matters for anything plugged into the mains. A supply with a poor power factor draws more current than its real power needs, which heats wiring and transformers, so power supplies above a certain size must include power factor correction. The roadside cabinet's 24 V supply, which powers the cabinet's own equipment, is rated in both watts and volt-amperes for exactly this reason.

For a single element the formulas reduce to familiar ones. A resistor absorbs I²R of real power, using RMS current, and an inductor or capacitor absorbs only reactive power, I²X. The statement that total power in a circuit sums to zero, from lesson 1, holds separately for real power and for reactive power, which gives two checks on any AC calculation.

Frequency response and transfer functions

A circuit's frequency response is its output phasor divided by its input phasor, as a function of frequency. This complex ratio is the transfer function H(jω). Its magnitude |H| says how much the circuit scales a sinusoid at that frequency, and its angle says how much it shifts the phase. For a voltage divider made of impedances, the transfer function is simply the divider ratio with ω left as a variable.

The RC low-pass filter takes its output across the capacitor. At low frequencies the capacitor is nearly open and the output follows the input; at high frequencies the capacitor is nearly a short and the output falls toward zero. The cutoff frequency f_c = 1/(2πRC) marks the boundary: there the magnitude is 1/√2, about 0.707, and the phase is −45°. Well above cutoff the magnitude falls in inverse proportion to frequency. Swapping the resistor and capacitor gives the RC high-pass filter, H = jωRC/(1 + jωRC), which blocks DC and passes high frequencies; a coupling capacitor in series with an input is a high-pass filter.

Decibels and Bode plots

Because filters change gain by factors of thousands across a wide range of frequencies, engineers plot gain on logarithmic scales. Gain in decibels is 20 log₁₀ |H| for voltage ratios. A factor of 10 is 20 dB, a factor of 2 is about 6 dB, 0.707 is −3 dB and 0.1 is −20 dB. A Bode plot draws the magnitude in decibels and the phase in degrees against frequency on a logarithmic axis. On a Bode plot the RC low-pass filter becomes two straight-line asymptotes: flat at 0 dB below cutoff, and falling at −20 dB per decade above it, meeting at the cutoff frequency, where the true curve is 3 dB below the corner.

Filtering before the ADC

Firmware samples the NS-5's analog inputs at fixed rates: the battery and thermistor once a second, but the loop-detector signal, which shows each vehicle as a change in a sensing coil's oscillator, at 1 kHz. Sampling theory, covered in ELE-250, says that any signal content above half the sampling rate, the Nyquist frequency, aliases: it shows up in the samples as a false lower frequency. An anti-aliasing filter before the ADC removes that content. A single RC stage is the simplest choice and, with the reservoir capacitor already on most ADC inputs, often costs only one resistor.

Second-order filters and resonance

An LC filter, an inductor in series with a capacitor across the output, is second order: above its resonant frequency its magnitude falls at −40 dB per decade, twice as fast as an RC stage, and it wastes almost no energy because inductors and capacitors are lossless in the ideal case. Near the resonant frequency f₀ = 1/(2π√(LC)) its response depends on the damping. With little resistance the filter peaks at resonance, amplifying signals there by the quality factor Q, which for a series RLC circuit is (1/R)√(L/C) = 1/(2ζ). A large peak can turn a filter into an amplifier for exactly the noise it was meant to remove, so LC filters are usually damped, for example with a capacitor that has some ESR.

# Bode data for the RC anti-aliasing filter and the LC supply filter
import math

def rc_lowpass(f, R, C):
    return 1 / complex(1, 2 * math.pi * f * R * C)

def lc_lowpass(f, L, C, R):
    w = 2 * math.pi * f
    return 1 / complex(1 - w * w * L * C, w * R * C)

def db(h):
    return 20 * math.log10(abs(h))

print("RC low-pass, 10 kohm and 100 nF (fc = %.0f Hz)" % (1 / (2 * math.pi * 10e3 * 100e-9)))
for f in (10, 50, 159, 500, 900, 5000):
    h = rc_lowpass(f, 10e3, 100e-9)
    print("  %5d Hz: %6.1f dB, phase %6.1f deg" % (f, db(h), math.degrees(math.atan2(h.imag, h.real))))

L, C = 10e-6, 10e-6
for R in (0.02, 0.3, 1.0):           # series resistance: inductor DCR plus capacitor ESR
    f0 = 1 / (2 * math.pi * math.sqrt(L * C))
    Q = math.sqrt(L / C) / R
    peak = max(db(lc_lowpass(f, L, C, R)) for f in [f0 * (0.8 + 0.001 * k) for k in range(400)])
    print("LC filter R = %.2f ohm: f0 = %.1f kHz, Q = %.1f, resonant peak %+.1f dB, at 400 kHz %.1f dB"
          % (R, f0 / 1e3, Q, peak, db(lc_lowpass(400e3, L, C, R))))

RC low-pass, 10 kohm and 100 nF (fc = 159 Hz)
     10 Hz:   -0.0 dB, phase   -3.6 deg
     50 Hz:   -0.4 dB, phase  -17.4 deg
    159 Hz:   -3.0 dB, phase  -45.0 deg
    500 Hz:  -10.4 dB, phase  -72.3 deg
    900 Hz:  -15.2 dB, phase  -80.0 deg
   5000 Hz:  -29.9 dB, phase  -88.2 deg
LC filter R = 0.02 ohm: f0 = 15.9 kHz, Q = 50.0, resonant peak +34.0 dB, at 400 kHz -56.0 dB
LC filter R = 0.30 ohm: f0 = 15.9 kHz, Q = 3.3, resonant peak +10.6 dB, at 400 kHz -56.0 dB
LC filter R = 1.00 ohm: f0 = 15.9 kHz, Q = 1.0, resonant peak +1.1 dB, at 400 kHz -56.0 dB

The LC results show the damping trade. With only 20 mΩ of resistance the filter's Q is 50 and it amplifies any noise near 16 kHz by about 34 dB, a real risk because the radar's pulses contain energy there. With 0.3 Ω the peak is a mild 10 dB, and with 1 Ω it almost disappears, all while the attenuation at 400 kHz stays above 50 dB. Ines chooses a capacitor with about 0.3 Ω of ESR in parallel with the low-ESR ceramic, a common way to damp an LC filter without wasting DC power.

Capstone: the NS-5 board decisions

Over the term, Jonah's analysis touched every part of the board, and the NS-5's design review lists the results. The energy budget from lesson 1 showed that the radar's listening duty cycle, not the radio, dominates battery life. The battery divider from lessons 2 and 6 gets a 100 nF reservoir capacitor, and the board gets a cleaning step after assembly to remove flux. The service button keeps its 47 kΩ pull-up and gains a 22 nF debounce capacitor with a Schmitt input (lessons 3 and 9). The I²C pull-ups stay at 2.2 kΩ, with a note that no module may add its own (lessons 3 and 9). The radar's 5 V output uses a divider with 1 percent resistors (lesson 3).

The analog section changes too. The thermistor network was solved with nodal analysis (lesson 4) and gains a buffer (lesson 7). The charger's current sense uses a gain of 33 (lesson 7), and the low-battery cutoff gets 0.8 V of hysteresis (lesson 7). Decoupling uses 100 nF and 1 nF capacitors beside each chip (lessons 8 and 11), and the relay keeps its flyback diode (lesson 8). The radar clock gets a 56 Ω series resistor (lesson 10), the RS-485 cable is terminated at both ends (lesson 10), the loop-detector input gets a 159 Hz anti-aliasing filter, and the radio's supply gets a damped LC filter (lesson 12). None of these decisions needed anything beyond the circuit laws in this course, applied carefully with real component values and worst-case thinking. That is what makes circuit analysis one of the most practical tools a computer engineer has.