The complex plane and its regions
Complex numbers are not a trick: they are the natural tool for everything that oscillates. A phasor, an impedance and a pole are points of this plane, and the regions drawn in it tell you whether a circuit is stable.
01Why an electronics engineer needs complex numbers
The word “imaginary” is a historical misfortune: it suggests that they do not exist. In electronics, the complex plane is the place where the most concrete things of the whole degree live.
| Object in the complex plane | What it is in electronics |
|---|---|
| A point in polar form | A phasor: amplitude and phase of a sinusoidal signal |
| A number \( R + jX \) | An impedance: resistance and reactance |
| Multiplying by \( e^{j\theta} \) | Phase-shifting a signal by \( \theta \) radians |
| A point in the left half-plane | A stable pole: the response dies out |
| The unit circle | The edge of stability of a discrete system |
| The circle \( |\Gamma| = \text{const} \) | A circle of the Smith chart |
This whole course is built on that plane: poles and zeros, the Laplace transform, the Z-transform and the frequency response are readings of points and regions inside it.
02The three ways of writing the same thing
| For | Use | Because |
|---|---|---|
| Adding and subtracting | Rectangular | The real and imaginary parts are added separately |
| Multiplying and dividing | Polar or exponential | The moduli are multiplied and the angles are added |
| Raising to a power | Exponential | \( z^n = r^n e^{jn\theta} \), De Moivre’s formula |
| Differentiating and integrating | Exponential | The exponential is its own derivative, up to a factor |
It is the bridge between what oscillates and what grows: it turns sines and cosines into exponentials, which are far easier to differentiate, integrate and multiply. The whole phasor method used in alternating current comes from here.
Click or tap the plane to move the point, or use the sliders. The operations are drawn on the same plane.
03What each operation does in the plane
Every operation with complex numbers is a geometric motion. Seeing it that way avoids memorizing formulas:
| Operation | In the plane | Where it appears |
|---|---|---|
| Adding | Placing the vectors head to tail | Adding two out-of-phase voltages |
| Multiplying by \( r\,e^{j\theta} \) | Scaling by \( r \) and rotating by \( \theta \) | Passing through a network that attenuates and shifts phase |
| Multiplying by \( j \) | Rotating 90° without changing the size | Differentiating a sinusoid; inductive reactance |
| Conjugating | Reflecting across the real axis | Impedance matching: \( Z_L = Z_S^* \) |
| Inverting \( 1/z \) | Inverting the modulus and changing the sign of the angle | Going from impedance to admittance |
| Squaring | Doubling the angle and squaring the modulus | A mixer, which generates twice the frequency |
A complex number has n nth roots, spread over the vertices of a regular polygon inscribed in a circle of radius \( \sqrt[n]{r} \):
Those roots are the ones that appear when you solve the characteristic equation of a system, and their position in the plane decides whether the response oscillates, dies out or grows.
04Regions: where an inequality draws a figure
An inequality between complex numbers does not define an interval, as with the reals, but a region of the plane. Recognizing them by heart saves a lot of work:
| Condition | Region | What it means in the degree |
|---|---|---|
| \( |z - z_0| = r \) | Circle centered at \( z_0 \) | A locus of constant modulus |
| \( |z - z_0| \lt r \) | Open disk | Neighborhood of a point: the basis of limits |
| \( r_1 \lt |z| \lt r_2 \) | Annulus | Region of convergence of the Z-transform |
| \( \operatorname{Re}(z) \lt 0 \) | Left half-plane | Stable zone of Laplace |
| \( |z| \lt 1 \) | Unit disk | Stable zone of the Z-transform |
| \( |\arg z| \lt \alpha \) | Angular sector | Minimum damping of a system |
The two highlighted rows are the reason this course starts with complex variable: the question “is this system stable?” is answered by looking at where its poles fall in the plane.
05The phasor, which is all of this together
A sinusoidal signal \( v(t) = V_p \cos(\omega t + \varphi) \) can be written as the real part of a rotating complex exponential:
With that, a circuit with derivatives and integrals becomes an algebraic one: differentiating is multiplying by \( j\omega \) and integrating is dividing by \( j\omega \). From there come \( Z_L = j\omega L \) and \( Z_C = 1/(j\omega C) \), and Ohm’s law works again, now with complex numbers.
It holds when everything oscillates at the same frequency and the transient has already died out. For switch-on, for a pulse or for an arbitrary signal, you need the more general tool studied later: Laplace, which is this same idea with an exponential that also grows or dies out, \( e^{st} \) with \( s = \sigma + j\omega \).
06In the lab
With a generator and a two-channel oscilloscope, measure the amplitude and phase shift in an RC circuit. Draw the input phasor and the output phasor on graph paper and check the angle with \( \varphi = -\arctan(\omega RC) \).
Compute by hand the five fifth roots of unity and plot them. Check that they form a regular pentagon and that their sum is zero.
For a series RLC at three frequencies, compute \( Z \) in rectangular and in polar form, and its inverse \( Y = 1/Z \). Verify that inverting reflects the point across the unit circle and changes the sign of the angle, as in the lab on this page.
07Common mistakes
- Confusing \( j \) with \( i \). In electronics \( i \) is current; the imaginary unit is \( j \).
- Adding in polar form. To add you have to convert to rectangular; only the product and the quotient are convenient in polar.
- Losing the quadrant when computing the angle with \( \arctan(b/a) \): you have to look at the signs of \( a \) and \( b \), or use the two-argument function.
- Mixing degrees and radians in the same calculation.
- Forgetting that the nth root has n results and keeping only one.
- Writing the phasor with the peak value when you had been working in RMS, or the other way around.
08Self-assessment
Write \( z = 3 + j4 \) in polar form.
\( r = \sqrt{9+16} = 5 \) and \( \theta = \arctan(4/3) \approx 53.13^\circ \), that is, \( 5\,e^{j53.13^\circ} \).
What does multiplying a phasor by \( j \) do to it?
It rotates it 90° counterclockwise without changing its modulus. It is what differentiating a sinusoid does.
What region is \( 1 \lt |z| \lt 2 \)?
An annulus centered at the origin, between radii 1 and 2. It is the typical shape of a region of convergence in the Z-transform.
Where do the poles of a continuous-time system have to be for it to be stable?
In the left half-plane, with negative real part: that way the exponential \( e^{st} \) dies out instead of growing.
How many cube roots does 8 have and where are they?
Three, of modulus 2, spaced 120° apart: \( 2 \), \( 2e^{j120^\circ} \) and \( 2e^{j240^\circ} \). Only the first is real.
Why does the conjugate appear in impedance matching?
Because maximum power transfer occurs with \( Z_L = Z_S^* \): the same resistance and the opposite reactance, so that they cancel.
09Further reading
- Murray R. Spiegel. Complex Variables. Schaum’s Outline Series (Spanish edition, “Variable compleja”), McGraw-Hill, 2011. Concise theory and hundreds of solved problems: the classic companion for this part of the course.
- Erwin Kreyszig. Advanced Engineering Mathematics. Wiley (Spanish edition, “Matemáticas avanzadas para ingeniería, volumen 2”, 3rd ed.), Limusa Wiley, 2003. Complex variable from the engineer’s point of view, with good application examples.
- Alan V. Oppenheim and Alan S. Willsky. Signals and Systems. 2nd ed. (Spanish edition, “Señales y sistemas”), Pearson, 1997. The book for the whole course; worth having from day one.
- Charles K. Alexander and Matthew N. O. Sadiku. Fundamentals of Electric Circuits. 5th ed. (Spanish edition, “Fundamentos de circuitos eléctricos”), McGraw-Hill, 2013. The chapter on phasors, to see these complex numbers at work measuring real circuits.