Sallen–Key topology

Key takeaways

Sallen–Key topology belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sallen–Key topology to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sallen–Key topology from memory before moving on to harder problems.

Reference excerpt

The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology. It was introduced by R. P. Sallen and E. L. Key of MIT Lincoln Laboratory in 1955.

Explanation of operation A VCVS filter uses a voltage amplifier with practically infinite input impedance and zero output impedance to implement a 2-pole low-pass, high-pass, bandpass, bandstop, or allpass response. The VCVS filter allows high Q factor and passband gain without the use of inductors. A VCVS filter also has the advantage of independence: VCVS filters can be cascaded without the stages affecting each others tuning. A Sallen–Key filter is a variation on a VCVS filter that uses a unity gain amplifier (i.e., a buffer amplifier).

History and implementation In 1955, Sallen and Key used vacuum tube cathode follower amplifiers; the cathode follower is a reasonable approximation to an amplifier with unity voltage gain. Modern analog filter implementations may use operational amplifiers (also called op amps). Because of its high input impedance and easily selectable gain, an operational amplifier in a conventional non-inverting configuration is often used in VCVS implementations. Implementations of Sallen–Key filters often use an op amp configured as a voltage follower; however, emitter or source followers are other common choices for the buffer amplifier.

Sensitivity to component tolerances VCVS filters are relatively resilient to component tolerance, but obtaining high Q factor may require extreme component value spread or high amplifier gain. Higher-order filters can be obtained by cascading two or more stages.

Generic Sallen–Key topology

The generic unity-gain Sallen–Key filter topology implemented with a unity-gain operational amplifier is shown in Figure 1. The following analysis is based on the assumption that the operational amplifier is ideal. Because the op amp is in a negative-feedback configuration, its v + {\displaystyle v_{+}} and v − {\displaystyle v_{-}} inputs must match (i.e., v + = v − {\displaystyle v_{+}=v_{-}} ). However, the inverting input v − {\displaystyle v_{-}} is connected directly to the output v out {\displaystyle v_{\text{out}}} , and so

By Kirchhoff's current law (KCL) applied at the v x {\displaystyle v_{x}} node,

By combining equations (1) and (2),

v in − v x Z 1 = v x − v out Z 3 + v x − v out Z 2 . {\displaystyle {\frac {v_{\text{in}}-v_{x}}{Z_{1}}}={\frac {v_{x}-v_{\text{out}}}{Z_{3}}}+{\frac {v_{x}-v_{\text{out}}}{Z_{2}}}.}

Applying equation (1) and KCL at the op amp's non-inverting input v + {\displaystyle v_{+}} gives

v x − v out Z 2 = v out Z 4 , {\displaystyle {\frac {v_{x}-v_{\text{out}}}{Z_{2}}}={\frac {v_{\text{out}}}{Z_{4}}},}

which means that

Combining equations (2) and (3) gives

Rearranging equation (4) gives the transfer function

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Sallen–Key topology

Start with the simplest possible case. Write down what Sallen–Key topology claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sallen–Key topology before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sallen–Key topology ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sallen–Key topology

In research: Sallen–Key topology appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sallen–Key topology in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sallen–Key topology is common in secondary-school and first-year university syllabi. It links to neighbouring topics Electronic filter topology, Linear filters, so understanding it makes those chapters shorter.
In everyday life: Look for Sallen–Key topology outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Sallen–Key topology in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sallen–Key topology means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sallen–Key topology out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sallen–Key topology in simple terms?

The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology.

Why does Sallen–Key topology matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sallen–Key topology?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sallen–Key topology.