5  Analysis of Electrical Circuits

The information gleaned from a measurand and carried by a measurement system can be instantiated/handled in a variety of means. There are many reasons favoring electronic measurement systems (as opposed to, say, pneumatic or hydraulic, which are steam-punk alternatives that could have emerged), though Putten (1988) provides a list of such reasons, including:

So it’s not surprising that we are using electronic measurement systems today virtually everywhere. This also means that the key to understanding modern instruments lies in understanding electrical circuit theory, the basic laws that dictate how dc and ac circuits behave, and how they affect the signals that are being transformed by them.

Today we’ll make sure you have a good understanding of how to analyze electrical circuits by going through Chapter 2 of Dally, Riley, and McConnell (1993), in detail. You can find a copy of this chapter on Canvas.

After the lecture, there were some questions that remained open and answering them would prove beneficial to our learning:

  1. Why is the following equation true?

\[w = C \int_{0}^{V} v dv = \frac{1}{2} C V^2\]

  1. Is the resistor the only element that produces a voltage drop in a dc circuit?
  2. Why is \(e^{j\omega t} = cos(\omega t) + j sin(\omega t)\)?
  3. In class, we analyzed a simple circuit with an ac voltage source \(v_s=v_i(t)=v_i e^{j\omega t}\) and two impedances in series, \(Z_1 = R\) and \(Z_2 = \frac{1}{j\omega C}\). We were interested in the voltage drop across \(Z_2\) and by applying KVL found that \(v_o(t) = \frac{1}{1+j\omega RC} v_i e^{j\omega t} = \frac{1-j\omega RC}{1 + (\omega RC)^2} v_i e^{j\omega t}\). Knowing that we can easily find the magnitude and phase of this complex periodic signal \(v_o(t)\) if we know its real \((Re)\) and imaginary \((Im)\) parts (i.e., \(\sqrt{(Re)^2 + (Im)^2}\) and \(\text{tan}^{-1}\frac{(Im)}{(Re)}\)) then what are they?
  1. Here’s the answer using the sympy library for Python:
import sympy
sympy.init_printing()
w, R, C, vi, vo = sympy.symbols(('\omega', 'R', 'C', 'vi', 'vo'),real=True)
vovi = 1/(1+sympy.I*w*R*C)
sympy.simplify(vovi)

\(\displaystyle \frac{1}{i C R \omega + 1}\)

The real part of this expression would be:

re_vovi = sympy.re(vovi)
sympy.simplify(re_vovi)

\(\displaystyle \frac{1}{C^{2} R^{2} \omega^{2} + 1}\)

And the imaginary part of the expression would be:

im_vovi = sympy.im(vovi)
sympy.simplify(im_vovi)

\(\displaystyle - \frac{C R \omega}{C^{2} R^{2} \omega^{2} + 1}\)

Using those we can now calculate the magnitude/amplitude of the phasor:

mag_vovi = sympy.sqrt(((re_vovi*re_vovi))+(im_vovi*im_vovi))
sympy.simplify(mag_vovi)

\(\displaystyle \frac{1}{\sqrt{C^{2} R^{2} \omega^{2} + 1}}\)

Similarly, we can compute its phase:

ph_vovi = sympy.atan(im_vovi/re_vovi)
sympy.simplify(ph_vovi)

\(\displaystyle - \operatorname{atan}{\left(C R \omega \right)}\)

  • What is the hydraulic component equivalent to an inductor?