Analytic functions and conformal mapping
Near a point, every analytic function reduces to multiplication by a complex number: scale and rotate. That, and nothing else, is what makes angles preserved and what lets the Smith chart be drawn with a compass.
01Analytic, and what comes with it
To recap: \( f \) is analytic on a domain if it is differentiable in the complex sense at all of its points, which by Cauchy-Riemann is equivalent to \( u \) and \( v \) being tied to each other. That requirement, which at first seems technical, brings very concrete geometric consequences, and from it comes the tool most used at high frequency.
The key is to look at the first-order approximation around a point:
And multiplying by a complex number \( f'(z_0) = \rho\,e^{j\alpha} \) is one single thing: scaling by \( \rho \) and rotating by \( \alpha \). It does not stretch more in one direction than another, does not shear, does not reflect. That is why a small figure, once transformed, comes out the same as itself apart from size and orientation.
02Preserving angles
The central result follows from the previous approximation. If two curves cross at \( z_0 \) forming an angle \( \varphi \), their images cross forming the same angle \( \varphi \), and in the same sense of rotation. A mapping with that property is called conformal.
Both tangents rotate by the same amount, so the difference between them —the angle— does not change. And since all lengths are multiplied by the same \( \rho \), small figures keep their shape. Large ones do not: the approximation holds only on a neighborhood, and at the scale of the whole plane an analytic function deforms as much as it likes.
Two rays leave the chosen point at the angles set below. On the right, what they turn into. Compare the angle between them on one side and on the other, and shrink the size to watch the small square become a square.
\( f(z) = \bar{z} \) keeps the absolute value of the angle between curves, but it reverses its direction: it is a reflection. This is called anticonformal, and it is not the same thing. The definition of conformal includes orientation, and that is why it is equivalent to being analytic with a nonzero derivative.
03Where it stops working: critical points
The whole argument rested on \( f'(z_0) \ne 0 \). If the derivative vanishes, the first-order approximation says nothing and you have to go to the first term that survives:
With \( w = z^2 \) at the origin, \( m = 2 \): two curves that crossed at 45° come out crossing at 90°. You can see this in the lab by moving the point to zero. The points where \( f' \) vanishes are called critical, and they are exactly the places where the mapping stops being conformal.
The function \( f(z) = z^3 - 3z \) in the lab has two: \( f'(z) = 3z^2 - 3 \) vanishes at \( z = \pm 1 \). Moving the point there shows the fold.
04What it is used for
| Application | What is transformed | What is gained |
|---|---|---|
| Smith chart | \( \Gamma = \dfrac{z-1}{z+1} \) takes the right half-plane to the unit disk | The whole impedance plane fits on one sheet, and the circles of \( r \) and of \( x \) remain circles |
| Nyquist criterion | The Nyquist contour, mapped by \( 1 + L(s) \) | Counting encirclements of −1 tells how many unstable poles the closed loop has |
| Fields and capacitances | A complicated geometry, taken to a rectangle or a half-plane | Laplace is solved in the simple geometry and the solution is brought back |
| Microstrip | Schwarz-Christoffel, which maps the half-plane onto the interior of a polygon | Gives the closed-form characteristic impedance formulas used by design software |
| Filter stability | \( z = e^{sT} \), from the \( s \)-plane to the \( z \)-plane | The left half-plane becomes the interior of the unit circle |
The last row will be seen in detail when we reach the Z-transform, but it is worth noting now: the stability criterion for a digital filter —poles inside the circle— is the usual criterion —poles on the left— seen through a mapping.
If you traverse a closed curve in the \( s \)-plane and watch how its image winds around the origin, the number of turns is \( Z - P \): the zeros minus the poles that ended up inside. Nyquist is that count applied to \( 1 + L(s) \), and the calculus of residues is the same idea carried over to integrals. Everything left in the first half of the course comes from here.
05In the lab
For \( f(z) = z^2 \) at \( z_0 = 1 + j \), compute \( f'(z_0) = 2 + j2 \), its modulus \( 2\sqrt{2} \approx 2.83 \) and its argument 45°. Transform a square of side 0.1 centered there and verify with a ruler and protractor that the result measures 0.283 on a side and is rotated 45°.
With \( f(z) = z^3 - 3z \), move the lab point to \( z = 1 \) and check that two rays 60° apart come out 120° apart. Verify analytically that \( f'(1) = 0 \) and that the first nonzero derivative is the second.
For \( L(s) = \dfrac{K}{s(s+1)(s+2)} \), evaluate \( L(j\omega) \) at about ten frequencies, plot the result in the complex plane and find the value of \( K \) that makes the curve pass through −1. Compare it with the one given by the Routh criterion.
06Common mistakes
- Believing that conformal means “does not deform.” It preserves angles and the shape of small figures; at large scale it deforms as much as it needs to.
- Forgetting the condition \( f'(z_0) \ne 0 \). At critical points the angles are multiplied.
- Taking the conjugate to be conformal. It preserves the angle but reverses its direction: it is anticonformal.
- Confusing the scale factor with the circuit gain. \( |f'(z_0)| \) is a local stretching of the plane, not an amplitude.
- Using the first-order approximation far from the point. It holds on a neighborhood, and the lab shows how quickly it breaks down as the size is increased.
07Self-assessment
What two things does \( f'(z_0) \) do locally?
Its modulus scales and its argument rotates. Near \( z_0 \), the function is that multiplication and nothing more.
Why are angles preserved?
Because all the tangents rotate by the same \( \arg f'(z_0) \); the difference between any two of them does not change.
What happens at a point where \( f' = 0 \)?
Angles are multiplied by \( m \), the order of the first nonzero derivative. With \( z^2 \) at the origin, they are doubled.
Where does \( f(z) = z^3 - 3z \) have critical points?
At \( z = \pm 1 \), where \( f'(z) = 3z^2 - 3 \) vanishes.
What transformation is behind the Smith chart?
The bilinear transformation \( \Gamma = (z-1)/(z+1) \), which takes the right half-plane to the unit disk while preserving circles and angles.
What does the argument principle count?
The turns that the image of a contour makes around the origin: they equal \( Z - P \), zeros minus enclosed poles. It is the basis of Nyquist and of residues.
08Further reading
- Ruel V. Churchill and James W. Brown. Complex Variables and Applications. 7th ed. (Spanish edition, “Variable compleja y aplicaciones”), McGraw-Hill, 2004. The chapters on conformal mapping and its applications to Laplace’s equation, with the classic examples worked out.
- David M. Pozar. Microwave Engineering. 4th ed., Wiley, 2011. The Smith chart derived as a bilinear transformation, and the use of mapping in planar lines.
- Katsuhiko Ogata. Modern Control Engineering. 5th ed. (Spanish edition, “Ingeniería de control moderna”), Pearson, 2010. The Nyquist criterion presented as a contour mapping, which is exactly what it is.