Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
6. AC Analysis
7. Magnetically Coupled Circuits
8. Polyphase Systems
9. Operational Amplifiers
10. Laplace Transforms
11. Time-Dependent Circuits
12. Two-Port Networks
Appendix

How to Type Equations

CircuitsU has two equations editors that you can use to write and submit formulas or equations:

  • The 2-D Equation Editor (default)
  • The 1-D Equation Editor
You can choose which equation editor you want to use by going to the your account settings (click on your name in the top right corner of the page and, then, Settings).

2-D Equation Editor

The 2-D Equation Editor allows you to type equations in a nice (printable) format. The editor is relatively easy to use and you can introduce subscripts, superscipts (i.e. powers), fractions, integrals and derivatives. You can activate the virtual keyboard to see the list of recommended symbols and variables.

Here are a few examples that you can try:

\frac{v_1-v_2}{R_1} + \frac{v_1-v_3}{R_2} + I_0=0

R_1\cdot[i_1(t)-i_2(t)] + L_1\cdot\frac{d}{dt}[i_1(t)-i_3(t)] + \frac{1}{C_2}\cdot\int_0^t{i_1(t)dt} + V_1(t)=0

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

1-D Equation Editor

The 1-D Equation Editor allows you to type equations as a one-dimensional string. The tables below shows what you need to type to submit various formulas or equations. If you want to submit multiple equations in the same textbox you needs to write each equation on a separate line. To submit a single line answer (such as the final answer in a problem) you can type the final numerical value with or without any units or the analytical formula. Notice that

Table 1. How to submit numerical answers.
To submit... type...
$1.234\times10^{-3}$ 0.001234
1.234e-3
1.234m (use postfixes for powers of 10: n for $10^{-9}$, u for $10^{-6}$, m for $10^{-3}$, k for $10^3$, M for $10^6$, and G for $10^9$)
$I_0=2\:{\textcolor{gray}A}$ I0=2 A
I0=2 (units are optional)
2 A (you don't need to specify the sought variable if you submit a single-line answer)
2
$V_0=2\:{\textcolor{gray}{kV}}$ V0=2000 V
V0=2000 (units are optional)
2000
2e3
2k
$45°$ 45 deg
45deg
45°
$45 \: \textcolor{gray}{rad}$ 45 rad
45rad
Table 2. How to submit mathematical formulas using the 1-D Equation Editor.
To submit... type...
$R_1+R_2+R_3$ R1+R2+R2
$\frac{R_1 R_2}{R_1+R_2}$ R1*R2/(R1+R2) (the multiplication sign between R1 and R2 is mandatory in this case)
$\frac{1}{\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}}$ 1/(1/C1+1/C2+1/C3) or (1/C1+1/C2+1/C3)^(-1)
$\frac{1}{\textcolor{blue}{j}\omega C}+j\omega L$ 1/(jwC)+jwL
$P_d= R_1 (i_2-i_3)^2$ Pd=R1*(i2-i3)^2
Pd=R1(i2-i3)^2 (the multiplication sign is optional when multiplying a number with a symbolic quantity)
R1(i2-i3)^2 (it is optional to specify the sought variable if you submit a single-line answer)
$2\times 10^6\times [3-(a+b)]$ 2M*(3-(a+b)) (replace square brackets with paranthesis)
2M(3-(a+b))
$2 i_1+ R_1 I_x -7 = 0$ 2*i1+R1/Ix-7=0 (if you submit an equation make sure you don't forget the equal sign)
2i1+R1/Ix-7=0
$i_1=\frac{v_1-v_2}{R1}-\frac{v_3}{6.5}$ (v1-v2)/R1-v3/6.5=0
$\frac{i_1}{2 \textcolor{blue}{j}}+3j(i_1-i_2)$ 0=i1/(2*j)+3*j*(i1-i2) ($\textcolor{blue}{j}$ denotes imaginary number $\textcolor{blue}{j}=\sqrt{-1}$)
0=i1/(2j)+3j(i1-i2)
Table 3. How to submit differential and integral equations using the 1-D Equation Editor.
To submit... type...
$\dfrac{dv_{1}(t)}{dt}$ derivative(v1(t),t)
$\dfrac{dv_{1}}{dt}$ (incorrect) It is incorrect to use derivative(v1,t) (see the above line for the correct expression). You need to show the time dependance explicitly when referring to time-dependent quantities in CircuitsU. Therefore, you need v1(t) instead of v1.
$\int_{0}^{t}i_{1}(t){dt}$ integral(i1(t),t)
$\int_{0}^{t}i_{1}{dt}$ (incorrect) It is incorrect to use integral(v1,t) (see the above line for the correct expression). You need to show the time dependance explicitly when referring to time-dependent quantities in CircuitsU. Therefore, you need i1(t) instead of i1.
$I(t)=C\dfrac{d[v_{1}(t)-v_{2}(t)]}{dt}$ I(t)=C*derivative(v1(t)-v2(t),t)
$I(t)=\frac{1}{L}\int_{0}^{t}[v_{1}(t)-v_{2}(t)]{dt}$ I(t)=1/L*integral(v1(t)-v2(t),t)

Note that the constants and functions shown in see Table 4 might not always be available when you submit answers. Their availability depends on what you are asked to submit at that particular step.

Table 4. Predefined constants and functions.
Constant/Function Code
$\textcolor{blue}{j}=\sqrt{-1}$ j
$\omega$ omega or w
$\pi$ pi or 3.14
$e^x$ exp(x)
e^x
$a^b$ a^b
$\sqrt{x}$ sqrt(x)
x^0.5
$\sin(x)$ sin(x)
$\cos(x)$ cos(x)
E.g.: cos(45 deg) or cos(0.785 rad)
See also