Nonlinear circuits and active Filters

Not every circuit is meant to be linear. Log and antilog amplifiers compress or expand signals and allow analogue multiplication; phase-locked loops lock onto a signal's frequency; voltage-to-frequency converters turn analogue values into pulse trains. Active filters use op-amps with resistors and capacitors to shape frequency response without bulky inductors. This topic covers these non-linear circuits and first-order active filters — common sources of definition-type NEC MCQs.

Key concepts

Logarithmic amplifier

• Uses the exponential I-V relationship of a diode or a BJT (transdiode connection) in the feedback path of an inverting op-amp.
• Since $I = I_s e^{V/V_T}$, the output is $V_o = -V_T\ln\left(\frac{V_i}{RI_s}\right)$.
• Works only for one input polarity; output depends on temperature (through $V_T$ and $I_s$), so practical log amps use matched transistors and temperature compensation.
• Applications: compressing wide dynamic range signals, dB meters, multiplication.

Exponential (antilog) amplifier

• Diode/transistor at the input and resistor in feedback: $V_o = -RI_se^{V_i/V_T}$. Performs the inverse of a log amp.

Logarithmic multiplier

• Uses $\ln(xy) = \ln x + \ln y$: take logs of two inputs with log amps, add them with a summing amplifier, then take the antilog → output ∝ $V_1V_2$.
• Division uses a subtractor instead: $\ln(x/y) = \ln x - \ln y$.
• Squaring and square-root circuits are built the same way (multiply log by 2 or 1/2).

Phase-locked loop (PLL)

• A feedback system that makes an oscillator's frequency and phase track an input signal.
• Blocks: phase detector (compares input and VCO phases), low-pass filter (removes high-frequency components), error amplifier, voltage-controlled oscillator (VCO).
• Free-running frequency: VCO frequency with no input.
• Lock range (tracking range): range over which the PLL stays locked once locked. Capture range: range over which it can acquire lock — capture range ≤ lock range.
• Applications: FM demodulation, frequency synthesis/multiplication, clock recovery, FSK decoding, motor speed control. IC example: 565.

Voltage-to-frequency (V/F) converter

• Produces an output frequency proportional to input voltage: $f_o = KV_i$. Typically an integrator charged by the input plus a comparator that resets it (or charge-balancing). VCO is a type of V/F converter.
• Uses: telemetry and isolation (pulses easily sent over long lines or optocouplers), integrating ADCs.

Frequency-to-voltage (F/V) converter

• Produces DC output proportional to input frequency, using a monostable that gives a fixed-width pulse per cycle followed by averaging (low-pass) filter. Used in tachometers and FM demodulation.

Active filters

• Characteristics: passband, stopband, cut-off frequency $f_c$ (−3 dB point), roll-off rate (−20 dB/decade per order), gain in passband.
• Advantages over passive filters: provide gain, no inductors (small, cheap, suitable for ICs), high input and low output impedance so stages do not load each other, easy tuning.
• Disadvantages: need power supply, limited by op-amp bandwidth and slew rate, limited voltage swing.

First-order low-pass: RC network at the non-inverting input of an op-amp (non-inverting gain stage), or an inverting amplifier with C across $R_f$. Passes frequencies below $f_c$; above $f_c$, gain falls at −20 dB/decade (−6 dB/octave).

First-order high-pass: swap R and C positions (C in series at input, R to ground). Passes frequencies above $f_c$; below it, gain rises at +20 dB/decade.

Higher orders (e.g. second-order Sallen-Key Butterworth) give −40 dB/decade; band-pass = HPF + LPF cascade; band-stop (notch) = parallel combination.

High-yield points

• Log amp: diode/BJT in feedback, $V_o \propto -\ln V_i$; antilog amp: diode/BJT at the input.
• Log multiplier: log of each input, add, then antilog.
• PLL: phase detector, low-pass filter, amplifier, VCO; capture range ≤ lock range.
• First-order filter: $f_c = 1/(2\pi RC)$, gain 0.707 (−3 dB) and phase ±45° at $f_c$, roll-off 20 dB/decade.
• Active filters give gain without inductors and do not load preceding stages.

Formulas

$f_c = \frac{1}{2\pi RC}\ \text{(Hz)}$

First-order low-pass gain: $\frac{V_o}{V_i} = \frac{A_F}{1 + j(f/f_c)},\quad A_F = 1 + \frac{R_f}{R_1}$ First-order high-pass: $\frac{V_o}{V_i} = \frac{A_F\,j(f/f_c)}{1 + j(f/f_c)}$

Phase at $f_c$: −45° (LPF), +45° (HPF), relative to passband.

Worked example

Design a first-order low-pass filter with $f_c = 1$ kHz and passband gain 2, using C = 0.01 μF.

1. $R = \frac{1}{2\pi f_cC} = \frac{1}{2\pi\times1000\times10^{-8}} = \frac{1}{6.283\times10^{-5}} = 15.9$ kΩ.
2. Gain $1 + R_f/R_1 = 2$ → $R_f = R_1$, e.g. both 10 kΩ.
3. At 10 kHz (one decade above), gain ≈ $2/\sqrt{1+100} \approx 0.2$, i.e. 20 dB lower. ✓

Exam tips

• Log amp: diode/BJT in feedback; antilog amp: diode/BJT at input.
• PLL blocks: phase detector → LPF → amplifier → VCO (in a loop). Capture range ≤ lock range.
• First-order filter roll-off = 20 dB/decade = 6 dB/octave.
• Active filters avoid inductors — the main reason they are used at audio frequencies.
• Gain at cut-off is 0.707 of passband (−3 dB).

Exam-style questions

From Calmon's question bank, which has 25 questions on this topic.

1. The output of a voltage-to-frequency (V/F) converter is a pulse train whose frequency is:

• A. Inversely proportional to the input voltage
• B. Proportional to the square of the input voltage
• C. Independent of the input voltage
• D. Directly proportional to the input voltage

Answer: D. A V/F converter produces $f_o = K V_{in}$. Typical uses are telemetry and integrating A/D conversion, and an F/V converter does the reverse.

2. What is the function of the loop filter in a phase-locked loop?

• A. To remove high-frequency components from the phase detector output and supply a slowly varying control voltage to the VCO
• B. To multiply the input frequency
• C. To amplify the VCO output to drive the load
• D. To generate the reference frequency

Answer: A. The phase detector output contains sum-frequency and ripple components; the low-pass loop filter passes the dc/slow error voltage that steers the VCO.

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