Gyrator Circuits and Simulated Inductors

In the design of electronic filters and oscillators, inductors are often essential. However, at audio and low frequencies, their required physical size becomes impractically large and expensive. Gyrator circuits solve this problem elegantly by using operational amplifiers, resistors, and a capacitor to mimic the behavior of an inductor. This technique, known as creating a simulated inductor, allows you to design compact, integrated circuits that perform as if they contained a massive coil, revolutionizing the implementation of low-frequency analog filters.

The Inductor Problem and the Gyrator Solution

A real inductor's impedance increases with frequency ($Z_L=j\omega L$), making it fundamental for frequency-selective circuits. The challenge is that inductance ($L$) is proportional to the square of the number of coil turns and the core's magnetic properties. To achieve a high inductance value—say, 1 Henry or more—for a low-frequency audio filter, the resulting component is physically large, heavy, prone to picking up electromagnetic interference, and difficult to integrate onto a silicon chip.

A gyrator is an active two-port network that can "gyrate" or transform one type of impedance into its dual. In its most common form for inductor simulation, it uses an operational amplifier configured with resistors and a single capacitor. The magic of the circuit is that it presents an impedance at its input terminals that is mathematically identical to that of an inductor, even though no physical magnetic component is present. This enables you to build high-performance, tunable filter networks entirely with small, cheap, and integratable components.

Core Gyrator Circuit Operation and Impedance Derivation

The most straightforward gyrator circuit for inductor simulation uses a single op-amp in a specific configuration. Consider a circuit where the non-inverting input is grounded. A resistor $R_1$ connects the input terminal to the inverting input. A capacitor $C$ is connected between the inverting input and the op-amp's output. Finally, a second resistor $R_2$ is connected from the output back to the input terminal.

To find the input impedance ($Z_{in}$), we analyze the circuit using standard op-amp assumptions (infinite input impedance, zero output impedance, and the virtual short condition where the voltage difference between the inputs is zero). Let the input voltage be $V_{in}$. Due to the virtual short, the inverting input is also at 0V (ground potential). The current through $R_1$ is therefore $I_1=V_{in}/R_1$. This same current must flow into the capacitor-impedance path, as no current enters the op-amp input. The voltage across the capacitor is $V_{out}-0=V_{out}$. The current through the capacitor is $I_C=V_{out}/(1/(j\omega C))=j\omega C{V}_{out}$.

Setting $I_1=I_C$ gives $V_{in}/R_1=j\omega C{V}_{out}$, so $V_{out}=V_{in}/(j\omega C{R}_1)$. Now, consider the current $I_2$ flowing through resistor $R_2$ from the output to the input. This current is $I_2=(V_{out}-V_{in})/R_2$. The total input current $I_{in}$ is the sum of $I_1$ and $I_2$: $I_{in}=I_1+I_2$.

Substituting the expressions for $I_1$, $V_{out}$, and $I_2$ and solving for the input impedance $Z_{in}=V_{in}/I_{in}$ yields a crucial result:

$$Z_{in}=\frac{V_{in}}{I_{in}}=j\omega (R_1{R}_{2}C)$$

This is precisely the impedance of an inductor, where $j\omega L$ defines its frequency-dependent opposition to current flow. Therefore, the simulated inductance value is given by the product of the two resistors and the capacitance:

$$L_{sim}=R_1{R}_{2}C$$

For example, if $R_1=R_2=10\text{ k}\Omega$ and $C=100\text{ nF}$, the simulated inductance is $L_{sim}=(10^4)(10^4)(10^{-7})=1\text{ H}$. You have created a 1 Henry inductor using only small, standard components.

Practical Behavior: Q Factor, Frequency, and Voltage Limits

While the ideal derivation is compelling, practical gyrators have limitations dictated by the non-ideal behavior of real op-amps. The first major constraint is the finite Q factor (Quality factor). For a real inductor, Q represents the ratio of its reactive energy storage to its resistive energy loss. In a gyrator, losses are introduced by the non-infinite open-loop gain and finite bandwidth of the op-amp, as well as the inherent resistance of the resistors and capacitor. At the target frequency of operation, the op-amp's gain begins to roll off, preventing it from perfectly enforcing the virtual short condition. This results in a simulated inductor with a series resistance, lowering its Q. A low-Q simulated inductor in a filter circuit leads to poor selectivity and increased passband attenuation.

Second, the circuit is subject to frequency range constraints. It only behaves as a high-quality simulated inductor within a specific frequency window. At very low frequencies, op-amp DC offsets and noise can dominate. At high frequencies, the op-amp's slew rate and gain-bandwidth product cause the phase shift to deviate from the ideal 90 degrees, making the impedance no longer purely inductive. The useful frequency range is typically bounded at the high end by a significant fraction of the op-amp's unity-gain frequency.

Finally, the voltage swing limits of the op-amp directly constrain the signal levels the simulated inductor can handle. The output of the op-amp in the gyrator circuit must swing to generate the required voltage across the capacitor. If the input signal is too large, the op-amp output will saturate at its supply rails, causing severe distortion and a breakdown of the inductive simulation.

Applications in Filter Design and Tuning

The primary application of gyrator circuits is in active filter design, particularly for low-frequency applications like audio equalizers, tone controls, and subsonic or rumble filters. For instance, a gyrator can replace the inductor in a classic RLC bandpass or notch filter topology, creating a fully integrated active filter. A major advantage is tunability: by making one of the resistors variable (e.g., a digital potentiometer), you can electronically adjust the simulated inductance value and, consequently, the center frequency of a filter. This is far more practical than trying to adjust a large, shielded inductor.

Another elegant application is in simulating grounded inductors for ladder filter synthesis. By using multiple gyrators, you can construct sophisticated filter responses that closely mimic the performance of passive LC ladders but with the benefits of small size, no magnetic shielding requirements, and easy gain integration.

Common Pitfalls

1. Ignoring Op-Amp Bandwidth: Selecting an op-amp with a gain-bandwidth product too close to your operating frequency is a common error. This drastically reduces the Q factor and usable frequency range. Correction: Always choose an op-amp whose unity-gain frequency is 50 to 100 times higher than your highest frequency of interest for reasonable performance.

1. Overdriving the Circuit: Applying an input signal that is too large causes the op-amp output to saturate, as the required $V_{out}$ may exceed the supply rails. The circuit stops behaving linearly. Correction: Perform a worst-case AC analysis to ensure the internal voltage swing at the op-amp output remains within its linear region for all input frequencies and amplitudes.

1. Neglecting DC Bias Paths: In the basic gyrator circuit, the non-inverting input is grounded, and the inverting input is a virtual ground. If the input source is capacitively coupled, the resistor $R_1$ provides a necessary DC path to ground for the op-amp's input bias current. Omitting this or getting the resistor values wrong can lead to DC offset saturation. Correction: Ensure all op-amp inputs have a defined DC path to a voltage within the common-mode range. In AC-coupled designs, use appropriately sized bias resistors.

1. Assuming Ideal Component Behavior: Using poor-quality, high-tolerance resistors or a capacitor with high Equivalent Series Resistance (ESR) directly adds unwanted series resistance to your simulated inductor, lowering its Q. Correction: Use low-tolerance (1%), low-noise metal-film resistors and a high-quality film or C0G ceramic capacitor for the critical capacitance $C$.

Summary

• Gyrator circuits use op-amps, resistors, and a capacitor to electronically synthesize the impedance of a large inductor, solving the problem of bulky magnetic coils in low-frequency designs.
• The value of the simulated inductor is calculated as $L_{sim}=R_1{R}_{2}C$, allowing for precise and tunable inductance from small, inexpensive components.
• Practical performance is limited by the finite Q factor, imposed primarily by the non-ideal gain and bandwidth of the real op-amp used in the circuit.
• The useful operating frequency is constrained at the high end by the op-amp's bandwidth and slew rate, and the maximum signal level is limited by the op-amp's voltage swing capabilities before saturation.
• Despite these limitations, gyrators are invaluable for designing compact, integrable, and electronically tunable active filters for audio and instrumentation applications.