PHYS 3330
Physics Undergraduate Labs
A Review of Complex Numbers
This review covers complex number concepts used throughout PHYS 3330, particularly for analyzing AC circuits and transfer functions.
1 The Imaginary Number
The imaginary unit is \(\sqrt{-1}\). It is often referred to as \(i\), but in electronics, \(I\) is often used for a DC current (time independent) and \(i\) is used for an AC current (time dependent). For clarity, \(j\) is used in electronics to represent the imaginary unit.
\[j=\sqrt{-1}\]
2 Complex Numbers
Complex numbers are generically represented with the variable \(z\). In general, complex numbers have real and imaginary parts.
\[z=a+jb\]
where \(a\) is the real part and \(b\) is the imaginary part.
\[\text{Re}{[z]} = a\]
\[\text{Im}{[z]} = b\]
It is easy to add and subtract complex numbers in this form.
\[z_1 + z_2 = (a_1 + a_2) + j (b_1 + b_2)\]
Multiplying and dividing in this form is a bit trickier.
\[z_1\cdot z_2 = (a_1a_2-b_1b_2) + j(a_1b_2+a_2b_1)\]
\[\frac{z_1}{z_2} = \text{no thanks --- use polar form}\]
3 The Complex Plane
Another common representation of complex numbers is to write them in terms of their amplitude and phase.
\[z=|z|e^{j\phi}\]
where
\[|z| = \sqrt{a^2+b^2}\]
\[\tan\phi = \frac{b}{a}\]
This representation is better for multiplication and division.
\[z_1\cdot z_2 = |z_1|\cdot|z_2|\ e^{j(\phi_1 + \phi_2)}\]
\[\frac{z_1}{z_2} = \frac{|z_1|}{|z_2|}\ e^{j(\phi_1 - \phi_2)}\]
4 Complex Conjugates
A complex conjugate \(z^*\) can be determined by replacing all instances of \(j\) with \(-j\).
If
\[z = a+jb = |z|e^{j\phi}\]
then
\[z^* = a - jb = |z|e^{-j\phi}\]
The magnitude of a complex number \(|z|\) can be calculated by taking the square root of the product of the complex number with its complex conjugate.
\[|z| = \sqrt{zz^*}\]
5 Sinusoidal Functions
5.1 Taylor expansions of sin, cos, and exp
\[\begin{split} e^{x} &= \sum_{i=0}^\infty \frac{x^i}{i!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \frac{x^6}{720} + ...\\ \sin{x} &= \sum_{i=0}^\infty \frac{(-1)^i x^{2i+1}}{(2i+1)!} = x - \frac{x^3}{6} + \frac{x^5}{120} - ...\\ \cos{x} &= \sum_{i=0}^\infty \frac{(-1)^i x^{2i}}{(2i)!} = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + ... \end{split}\]5.2 Complex exponential
\[\begin{split} e^{jx} &= \sum_{i=0}^\infty \frac{(jx)^i}{i!} = 1 + jx - \frac{x^2}{2} - j\frac{x^3}{6} + \frac{x^4}{24} + j\frac{x^5}{120} - \frac{x^6}{720} + ...\\ &= \bigg(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + ...\bigg)+j\bigg(x - \frac{x^3}{6} + \frac{x^5}{120} - ...\bigg)\\ &= \cos{x} +j\sin{x} \end{split}\]This is the best form for keeping track of the phase of a wave.
6 Complex Transfer Functions
A transfer function is a time-independent relationship between an input and an output. It can be defined as
\[T = \frac{\text{output function}}{\text{input function}}\]
In this course, we will mostly use transfer functions to relate output voltages to input voltages, such that
\[V_\text{out}(t) = TV_\text{in}(t)\]
A transfer function can be complex. This means that applying it will not only scale the magnitude, but also shift the phase. A generic complex \(T\) is best put into the form
\[T = |T|e^{j\delta}\]
so that
\[V_\text{out}(t) = |T|V_\text{in}(t)e^{j\delta}\]
All waves can be represented in terms of sine or cosine wave components in a Fourier series.
\[\begin{split}V_\text{in}(t) =& \sum_i a_i \cos{(\omega_i t + \phi_i)}\\ V_\text{out}(t) =&\ T\sum_i a_i \cos{(\omega_i t + \phi_i)}\\ =&\ \sum_i a_i |T| \cos{(\omega_i t + \phi_i)}e^{j\delta} \end{split}\]
However \(\cos(\omega_i t+\phi_i)e^{j\delta}\) is not the easiest to work with. Instead, you can represent all cosines and sines as the real and imaginary parts of a single complex exponential.
\[\cos{(\omega t + \phi)} \rightarrow e^{j(\omega t + \phi)}\]
This will make the math much easier to work with and then at the end, you can just take the real part. The equations above become
\[\begin{split}\tilde{V}_\text{in}(t) =& \sum_i a_i e^{j(\omega_i t + \phi_i)}\\ \tilde{V}_\text{out}(t) =& \sum_i a_i |T| e^{j(\omega_i t + \phi_i)}e^{j\delta}\\ =& \sum_i a_i |T| e^{j(\omega_i t + \phi_i+\delta)} \end{split}\]
And then the real part can be taken to find \(V_\text{out}\).
\[\begin{split} V_\text{out}&=\text{Re}[\tilde{V}_\text{out}]\\ &= \sum_i a_i |T| \cos{(\omega_i t + \phi_i+\delta)} \end{split}\]
So each cosine component gets scaled by \(|T|\) and shifted by an amount of time of \(\delta/\omega_i\) (aka a phase shift).
7 More Identities
In the following, \(n\) is an integer.
\[\begin{split} \cos{x} &= \frac{e^{jx}+e^{-jx}}{2}\\ \sin{x} &= \frac{e^{jx}-e^{-jx}}{2j}\\ \cosh{x} &= \frac{e^x+e^{-x}}{2}\\ \sinh{x} &= \frac{e^x-e^{-x}}{2}\\ \cos{x} &= \cosh{jx}\\ \sin{x} &= -j\sinh{jx}\\ e^{j 2\pi n} &= 1\\ e^{j (\pi + 2\pi n)} &= -1\\ e^{j (\frac{\pi}{2} + 2\pi n)} &= j\\ e^{j (\frac{3\pi}{2} + 2\pi n)} &= -j \end{split}\]8 Practice Problems
Write the following complex numbers in terms of magnitude and phase.
\(z=\dfrac{A}{1+jx}\)
\(z=\dfrac{jx}{1+jx}\)
Let \(z_1=\sqrt{8}e^{j\frac{3\pi}{4}}\), \(z_2=2e^{j\frac{\pi}{6}}\).
Represent \(z_1\) and \(z_2\) in the complex plane and find their real and imaginary parts.
Evaluate \(z_1 + z_2\) and \(z_1^2z_2^3\).
By writing out \(\cos\theta\) in terms of exponentials and using the binomial expansion, express \((\cos\theta)^5\) in terms of \(\cos\theta\), \(\cos 3\theta\), and \(\cos 5\theta\).
Evaluate the sum
\[\sum_{n=-N}^N\cos(\theta+n\phi)\]
Suppose that frequencies \(\omega_1\) and \(\omega_2\) differ only slightly. Using the complex exponential, express the sum
\[A_0\cos\omega_1 t +A_0\cos\omega_2 t\]
where \(A_0\) is a constant, in the form
\[A(t)\cos\frac{\omega_1+\omega_2}{2}t\]
where \(A(t)\) is a slowly varying function of time.