2.5 Nonlinear Circuits and Active Filters

2.5 Nonlinear Circuits and Active Filters

Introduction to Nonlinear Circuits and Active Filters

While op-amps are most commonly associated with linear amplification, their true power is unleashed in nonlinear and frequency-selective applications. Nonlinear circuits, such as logarithmic amplifiers and multipliers, perform essential functions like compression, multiplication, and frequency synthesis, enabling complex signal processing. Active filters, which incorporate op-amps alongside passive RC networks, provide superior performance—with gain, high input impedance, and low output impedance—compared to their passive counterparts. This unit explores how op-amps transcend linearity to enable precise mathematical operations, frequency control, and sophisticated signal shaping, forming the bridge between raw analog signals and measurable, processable data.

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1. Logarithmic and Exponential Amplifiers

These circuits exploit the nonlinear voltage-current relationship of a semiconductor junction (diode or transistor) to perform mathematical operations.

1.1 Logarithmic Amplifier

1. Principle: Utilizes the exponential relationship between the voltage across and current through a forward-biased p-n junction.

   • Diode equation: $I_D = I_S (e^{V_D / \eta V_T} - 1) \approx I_S e^{V_D / \eta V_T}$ for $V_D >> V_T$.

   • Where $I_S$ is reverse saturation current, $\eta$ is ideality factor (~1-2), and $V_T = kT/q \approx 26 \text{mV}$ at 300K.

2. Circuit Implementation: A diode or a bipolar transistor (with base-collector shorted, acting as a diode) is placed in the feedback path of an inverting op-amp configuration.

   • Input voltage $V_{in}$ is applied through resistor $R$.

   • Due to virtual ground, the input current is $I_{in} = V_{in}/R$.

   • This same current flows through the feedback diode, establishing a voltage $V_{out}$ across it.

3. Output Derivation: $I_{in} = \frac{V_{in}}{R} = I_D \approx I_S e^{V_{out} / \eta V_T}$ Taking natural log on both sides: $\ln\left(\frac{V_{in}}{R I_S}\right) = \frac{V_{out}}{\eta V_T}$ Therefore, $\boxed{V_{out} = -\eta V_T \ln\left(\frac{V_{in}}{R I_S}\right)}$

   • The negative sign appears because the output is taken across the diode in the inverting configuration.

   • The output is proportional to the natural logarithm of the input voltage.

4. Practical Considerations: $I_S$ and $\eta V_T$ are temperature-dependent. Precision log amps use matched transistor pairs and temperature compensation circuits.

5. Applications: Signal compression (wide dynamic range), analog computation (multiplication via addition of logs), decibel measurement (since dB is a logarithmic scale).

1.2 Exponential (Anti-Logarithmic) Amplifier

1. Principle: The inverse operation of the log amplifier. It generates an output proportional to the exponential (anti-log) of the input.

2. Circuit Implementation: The diode/transistor is placed in the input path of an inverting op-amp configuration.

   • Input voltage $V_{in}$ is applied across the diode.

   • The diode current becomes $I_D = I_S e^{V_{in} / \eta V_T}$.

   • This current flows through the feedback resistor $R_f$.

3. Output Derivation: $V_{out} = -I_f R_f = -I_D R_f$ $\boxed{V_{out} = -R_f I_S e^{V_{in} / \eta V_T}}$ The output is proportional to the exponential of the input voltage.

4. Applications: Used in analog multipliers, function generators, and as the decompression stage in signal processing chains.

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2. Logarithmic Multiplier

1. Principle: Multiplication is achieved by exploiting the logarithmic property: $\ln(a \times b) = \ln(a) + \ln(b)$. Therefore, $a \times b = \text{anti-log}(\ln(a) + \ln(b))$.

2. Block Diagram Implementation:

   • Step 1 (Log): Two logarithmic amplifiers convert the input voltages $V_x$ and $V_y$ into their logarithms: $V_1 = K \ln(V_x/V_{ref})$ and $V_2 = K \ln(V_y/V_{ref})$.

   • Step 2 (Sum): A summing amplifier adds these two logarithmic voltages: $V_3 = -(V_1 + V_2) = -K [\ln(V_x V_y / V_{ref}^2)]$.

   • Step 3 (Anti-Log): An exponential (anti-log) amplifier converts the sum back: $V_{out} = V_{ref}^2 \cdot e^{V_3 / K} = V_x \cdot V_y$.

   • The constants $K$ and $V_{ref}$ cancel out in the final product.

3. Significance: Enables analog multiplication and related operations (division, squaring, square-rooting) with high accuracy over a wide dynamic range.

4. Applications: Automatic gain control (AGC), modulation/demodulation, analog computation, RMS-to-DC conversion.

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3. Phase-Locked Loop (PLL) Basics

1. Definition: A Phase-Locked Loop is a negative feedback control system that generates an output signal whose phase is locked to (i.e., synchronized with) the phase of an input reference signal.

2. Fundamental Block Diagram and Function:

   1. Phase Detector (PD): Compares the phase of the input signal ($V_{in}$) with the phase of the Voltage-Controlled Oscillator (VCO) output. Its output voltage $V_{pd}$ is proportional to the phase difference.

   2. Low-Pass Filter (LPF): Filters the PD output to remove high-frequency noise and components, producing a smooth DC control voltage ($V_{ctrl}$).

   3. Voltage-Controlled Oscillator (VCO): Generates an output signal whose frequency is a linear function of the input control voltage $V_{ctrl}$.

      • Characterized by its free-running frequency ($f_o$) when $V_{ctrl}=0$ and its sensitivity ($K_{VCO}$ in Hz/V).

   4. Feedback Path: The VCO output is fed back to the phase detector, closing the loop.

3. Operation Modes:

   • Lock Range (Hold-in Range): The range of input frequencies over which the PLL can maintain lock once acquired.

   • Capture Range (Pull-in Range): The range of input frequencies over which the PLL can acquire lock from an initially unlocked state. Always smaller than the lock range.

   • Lock Time: Time taken to achieve phase lock.

4. Key Applications:

   • Frequency Synthesis: Generating a precise, stable output frequency that is a multiple of a lower-frequency reference (used in radios, clocks).

   • Frequency Modulation/Demodulation (FM): The VCO can act as an FM modulator. A PLL can extract the message from an FM signal (as an FM demodulator).

   • Clock Recovery: Extracting a clock signal from a data stream in digital communications.

   • Signal Conditioning: Cleaning noisy signals and tracking frequency variations.

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4. Voltage-to-Frequency (V-to-F) and Frequency-to-Voltage (F-to-V) Conversion

4.1 Voltage-to-Frequency Converter (VFC)

1. Function: Produces an output pulse or square wave whose frequency is directly proportional to an analog input voltage. $f_{out} = K_{VFC} \cdot V_{in}$

2. Common Implementation (Charge-Balance Type):

   • An integrator ramps down (or up) linearly in response to $V_{in}$.

   • A comparator detects when the integrator output crosses a threshold.

   • This triggers a precision one-shot pulse generator.

   • The pulse causes a fixed amount of charge to be dumped into the integrator, resetting it.

   • The pulse rate (frequency) is proportional to $V_{in}$.

3. Applications: Analog-to-digital conversion (especially for noisy environments), remote telemetry, precision frequency modulation, digital voltmeters.

4.2 Frequency-to-Voltage Converter (FVC)

1. Function: Produces a DC output voltage proportional to the frequency (or pulse rate) of an input signal. $V_{out} = K_{FVC} \cdot f_{in}$

2. Common Implementation:

   • Input pulses trigger a precision monostable multivibrator (one-shot) to produce pulses of constant width ($T_w$) and amplitude ($V_p$).

   • These constant-area pulses are then integrated (low-pass filtered).

   • The average DC value of this pulse train is $V_{out} = V_p \cdot T_w \cdot f_{in}$.

3. Applications: Tachometers (speed measurement), FM demodulation, frequency monitoring and control loops.

4. Duality: A VFC and an FVC are essentially inverse systems. Some integrated circuits (e.g., LM331) can be configured to perform both functions.

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5. Active Filter Characteristics and Advantages

Active filters use op-amps in conjunction with resistors (R) and capacitors (C) to achieve frequency selection. Inductors (L) are generally avoided.

1. Key Advantages over Passive (RLC) Filters:

   • Gain (Amplification): Can provide voltage gain ($A_v > 1$), not just attenuation.

   • No Loading Effect: High input impedance and low output impedance prevent interaction between filter stages, allowing easy cascading.

   • Elimination of Inductors: Inductors are bulky, non-ideal, and unsuitable for low-frequency IC fabrication. Active filters use only R and C.

   • Flexibility: Easy to tune by adjusting resistor values. Can realize complex filter characteristics (e.g., Butterworth, Chebyshev, Bessel).

2. Key Filter Parameters:

   • Passband: Range of frequencies passed with minimal attenuation.

   • Stopband: Range of frequencies significantly attenuated.

   • Cutoff Frequency ($f_c$ or $f_{3dB}$): Frequency at which the gain falls to $1/\sqrt{2}$ (≈ -3 dB) of its maximum passband value.

   • Roll-off (Slope): Rate of attenuation in the stopband, expressed in dB/decade or dB/octave. Determined by the filter order (n). A first-order filter has a roll-off of 20 dB/decade.

   • Q (Quality Factor): For band-pass/band-stop filters, defines the selectivity (sharpness of the peak/notch).

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6. First-Order High-Pass and Low-Pass Active Filters

6.1 First-Order Low-Pass Active Filter

1. Function: Passes low-frequency signals and attenuates high-frequency signals.

2. Circuit (Non-inverting Configuration):

   • A simple RC low-pass network ($R_1$, $C_1$) is placed at the non-inverting input of an op-amp configured as a non-inverting amplifier.

   • Alternatively, an inverting configuration uses an RC network in the feedback path.

3. Transfer Function (Non-inverting): $A_v(s) = \frac{V_{out}(s)}{V_{in}(s)} = \left(1 + \frac{R_f}{R_g}\right) \cdot \frac{1}{1 + s R_1 C_1}$ Where $s = j\omega$ is the complex frequency.

4. Cutoff Frequency: $\boxed{f_c = \frac{1}{2\pi R_1 C_1}}$

5. Frequency Response:

   • For $f << f_c$: Gain = Passband Gain = $1 + \frac{R_f}{R_g}$.

   • For $f >> f_c$: Gain rolls off at -20 dB/decade.

6. Applications: Removing high-frequency noise from signals, anti-aliasing in analog-to-digital converters.

6.2 First-Order High-Pass Active Filter

1. Function: Passes high-frequency signals and attenuates low-frequency signals.

2. Circuit (Non-inverting Configuration):

   • A simple RC high-pass network ($C_1$, $R_1$) is placed at the non-inverting input of an op-amp non-inverting amplifier.

3. Transfer Function: $A_v(s) = \left(1 + \frac{R_f}{R_g}\right) \cdot \frac{s R_1 C_1}{1 + s R_1 C_1}$

4. Cutoff Frequency: $\boxed{f_c = \frac{1}{2\pi R_1 C_1}}$ (Same formula as LPF, but the roles of R and C in the network are interchanged.)

5. Frequency Response:

   • For $f << f_c$: Gain rises at +20 dB/decade from zero.

   • At $f = f_c$: Gain = Passband Gain / $\sqrt{2}$.

   • For $f >> f_c$: Gain = Passband Gain = $1 + \frac{R_f}{R_g}$.

6. Applications: Removing DC offset or low-frequency drift from signals, coupling stages in AC amplifiers, emphasis in audio systems.

Conclusion: The transition from linear to nonlinear op-amp applications marks a significant expansion of analog signal processing capabilities. Logarithmic circuits enable dynamic range compression and analog computation. PLLs provide robust frequency control and synthesis. VFCs/FVCs bridge analog and frequency domains. Finally, active filters offer a practical and high-performance method for frequency selection, shaping signals with precision and flexibility. Mastery of these circuits is essential for designing advanced systems in communications, instrumentation, and control.