Richards' theorem

Key takeaways

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

Reference excerpt

In mathematics, Richards' theorem is a result due to Paul I. Richards in 1947. The theorem states that for

R ( s ) = k Z ( s ) − s Z ( k ) k Z ( k ) − s Z ( s ) , {\displaystyle R(s)={\frac {kZ(s)-sZ(k)}{kZ(k)-sZ(s)}},}

if Z ( s ) {\displaystyle Z(s)} is a positive-real function (PRF) then R ( s ) {\displaystyle R(s)} is a PRF for all real, positive values of k {\displaystyle k} . The theorem has applications in electrical network synthesis. The PRF property of an impedance function determines whether or not a passive network can be realised having that impedance. Richards' theorem led to a new method of realising such networks in the 1940s.

Proof

R ( s ) = k Z ( s ) − s Z ( k ) k Z ( k ) − s Z ( s ) {\displaystyle R(s)={\frac {kZ(s)-sZ(k)}{kZ(k)-sZ(s)}}}

where Z ( s ) {\displaystyle Z(s)} is a PRF, k {\displaystyle k} is a positive real constant, and s = σ + i ω {\displaystyle s=\sigma +i\omega } is the complex frequency variable, can be written as,

R ( s ) = 1 − W ( s ) 1 + W ( s ) {\displaystyle R(s)={\dfrac {1-W(s)}{1+W(s)}}}

where,

W ( s ) = 1 − Z ( s ) Z ( k ) 1 + Z ( s ) Z ( k ) ( k + s k − s ) {\displaystyle W(s)={1-{\dfrac {Z(s)}{Z(k)}} \over 1+{\dfrac {Z(s)}{Z(k)}}}\left({\frac {k+s}{k-s}}\right)}

Since Z ( s ) {\displaystyle Z(s)} is PRF then

1 + Z ( s ) Z ( k ) {\displaystyle 1+{\dfrac {Z(s)}{Z(k)}}}

is also PRF. The zeroes of this function are the poles of W ( s ) {\displaystyle W(s)} . Since a PRF can have no zeroes in the right-half s-plane, then W ( s ) {\displaystyle W(s)} can have no poles in the right-half s-plane and hence is analytic in the right-half s-plane. Let

Z ( i ω ) = r ( ω ) + i x ( ω ) {\displaystyle Z(i\omega )=r(\omega )+ix(\omega )}

Then the magnitude of W ( i ω ) {\displaystyle W(i\omega )} is given by,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Richards' theorem

Start with the simplest possible case. Write down what Richards' theorem 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 Richards' theorem 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 Richards' theorem 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 Richards' theorem

In research: Richards' theorem 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 Richards' theorem 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: Richards' theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Circuit theorems, Electronic engineering, Network synthesis, so understanding it makes those chapters shorter.
In everyday life: Look for Richards' theorem 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 Richards' theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Richards' theorem 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 Richards' theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Richards' theorem in simple terms?

In mathematics, Richards' theorem is a result due to Paul I. Richards in 1947.

Why does Richards' theorem 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 Richards' theorem?

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 Richards' theorem.