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A matrix is a type of shorthand notation for systems of linear equations.
This includes systems of linear differential equations as well.
Matrices lead to very powerful and concise matrix equations that greatly simplify finding solutions to linear systems.
In this post, matrices and their basic operations are introduced but the connection to linear systems is left out.
Efficiency is the ratio of output power to input power in a power converter or other device, with the difference being power loss.
The ratio can be expressed as a percentage, decimal or fraction.
Usually we know the output power and efficiency and need to figure out the total input power and loss.
For example, a circuit design may require 2W of regulated power and our options include a DC-DC converter which is 90% efficient.
Efficiency is always less than 1.00 or 100%.
A linear ODE with constant coefficients looks like this:
$$a_n\frac{\mathrm{d}^ny}{\mathrm{d}x^n} + a_{n-1}\frac{\mathrm{d}^{n-1}y}{\mathrm{d}x^{n-1}} + … + a_1\frac{\mathrm{d}y}{\mathrm{d}x} + a_0y = f(x)$$
For example:
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} – 6\frac{\mathrm{d}y}{\mathrm{d}x} + 13y = x^3$$
This is the most common form of differential equation in circuit theory.
Note that if the coefficients are not constant (i.e. a function of x) then most analytical methods do not work.
Continue reading “Linear Differential Equations with Constant Coefficients”
Partial fraction expansion is a nice trick to simplify certain kinds of problems.
The two main applications are integral problems and inverse Laplace transforms.
Differential equations are key to solving electromagnetic and electronic problems.
While most problems are too complex to be worth solving analytically, the concepts are worth knowing.
A differential equation is an equation which relates a function to its derivatives, where the solution is the function itself.
Decibels are the most common way to discuss RF voltage, current and power.
Continue reading “Decibels”