Positive-real function

Key takeaways

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

Reference excerpt

Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. They are complex functions, Z(s), of a complex variable, s. A rational function is defined to have the PR property if it has a positive real part and is analytic in the right half of the complex plane and takes on real values on the real axis. That is:

ℜ [ Z ( s ) ] > 0 if ℜ ( s ) > 0 ℑ [ Z ( s ) ] = 0 if ℑ ( s ) = 0 {\displaystyle {\begin{aligned}&\Re [Z(s)]>0\quad {\text{if}}\quad \Re (s)>0\\&\Im [Z(s)]=0\quad {\text{if}}\quad \Im (s)=0\end{aligned}}}

In electrical network analysis, Z(s) represents an impedance expression and s is the complex frequency variable, often expressed as its real and imaginary parts;

s = σ + i ω {\displaystyle s=\sigma +i\omega \,\!}

in which terms the PR condition can be stated;

ℜ [ Z ( s ) ] > 0 if σ > 0 ℑ [ Z ( s ) ] = 0 if ω = 0 {\displaystyle {\begin{aligned}&\Re [Z(s)]>0\quad {\text{if}}\quad \sigma >0\\&\Im [Z(s)]=0\quad {\text{if}}\quad \omega =0\end{aligned}}}

The importance to network analysis of the PR condition lies in the realisability condition. Z(s) is realisable as a one-port rational impedance if and only if it meets the PR condition. Realisable in this sense means that the impedance can be constructed from a finite (hence rational) number of discrete ideal passive linear elements (resistors, inductors and capacitors in electrical terminology).

Definition The term positive-real function was originally defined by Otto Brune to describe any function Z(s) which

is rational (the quotient of two polynomials), is real when s is real has positive real part when s has a positive real part Many authors strictly adhere to this definition by explicitly requiring rationality, or by restricting attention to rational functions, at least in the first instance. However, a similar more general condition, not restricted to rational functions had earlier been considered by Cauer, and some authors ascribe the term positive-real to this type of condition, while others consider it to be a generalization of the basic definition.

History The condition was first proposed by Wilhelm Cauer (1926) who determined that it was a necessary condition. Otto Brune (1931) coined the term positive-real for the condition and proved that it was both necessary and sufficient for realisability.

Properties The sum of two PR functions is PR. The composition of two PR functions is PR. In particular, if Z(s) is PR, then so are 1/Z(s) and Z(1/s). All the zeros and poles of a PR function are in the left half plane or on its boundary of the imaginary axis. Any poles and zeroes on the imaginary axis are simple (have a multiplicity of one). Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative. Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle). For a rational PR function, the number of poles and number of zeroes differ by at most one.

Generalizations A couple of generalizations are sometimes made, with intention of characterizing the immittance functions of a wider class of passive linear electrical networks.

Irrational functions The impedance Z(s) of a network consisting of an infinite number of components (such as a semi-infinite ladder), need not be a rational function of s, and in particular may have branch points in the left half s-plane. To accommodate such functions in the definition of PR, it is therefore necessary to relax the condition that the function be real for all real s, and only require this when s is positive. Thus, a possibly irrational function Z(s) is PR if and only if

Z(s) is analytic in the open right half s-plane (Re[s] > 0) Z(s) is real when s is positive and real Re[Z(s)] ≥ 0 when Re[s] ≥ 0 Some authors start from this more general definition, and then particularize it to the rational case. The second condition is sometimes stated as: Z ( s ∗ ) = ( Z ( s ) ) ∗ {\displaystyle Z(s^{*})=(Z(s))^{*}} in the open right half plane (which is equivalent given that the function is analytic there).

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Positive-real function

Start with the simplest possible case. Write down what Positive-real function 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 Positive-real function 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 Positive-real function 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 Positive-real function

In research: Positive-real function 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 Positive-real function 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: Positive-real function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Complex analysis, Electronic engineering, Types of functions, so understanding it makes those chapters shorter.
In everyday life: Look for Positive-real function 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 Positive-real function in 20 minutes

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

Frequently asked questions

What is Positive-real function in simple terms?

Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. They are complex functions, Z(s), of a complex variable, s.

Why does Positive-real function 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 Positive-real function?

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 Positive-real function.