Linear

From Syncleus Wiki

Linear electronic components are those which relate different quantities in circuits using linear relationships or differential equations.

For example, the ideal resistor which relates the voltage across it with with the current flowing through it via $V = I R$ is linear because the values in question (V and I) have exponents of 1. A resistor whose voltage drop was $V = I R 0 + I 3 R 1$ would not be linear, because of the $I 3$ term.

Linear behaviors and components are nice because any system of them is described by a corresponding system of coupled linear differential equations whose general solution is easily found. By modeling any driving signals as Fourier series, particular solutions are also relatively easy to find.

Linear Components

Resistor - $V = IR = R\frac{dQ}{dt}$

Capacitor - $V = \frac{Q}{C}$

Inductor - $V = L \frac{dI}{dt} = L \frac{d^2Q}{dt^2}$

Linear approximation

Few real-world behaviors are truly linear if you look close enough but very often the nonlinear terms are small. Assuming the behavior is represented by continuous & differentiable functions, which it almost certainly is in electronics, the nonlinear terms can be dropped to simplify analysis.

For example, if a nonlinear resistor drops voltage according to

$V = 500 I + .1 I 3$

it could be approximated as a 500Ω ideal resistor. As long as the current is less than about 4 amps the voltage drop would stay within 1% of the linear value - not bad at all. None the less, its actual resistance (= V/I) would continually increase with increasing current:

$R = \frac{V}{I} = \frac{500I + .1I^3}{I} = 500 + .1I^2$

It is also possible to imagine that the voltage drop depends on things other than I. Suppose it depends on temperature in some convoluted way: $V = I R * f (T)$ . In this case, one could linearize this function in temperature by keeping the zero and first order terms of a Taylor series:

$f(T_0 + T) \approx f(T_0) + (T - T_0)\frac{\partial f}{\partial T}$

The partial derivative would most likely be found at the workbench by measuring the resistance $R 0$ and $R 1$ at two temperatures $T 0$ and $T 1$ . Then

$f(T) \approx 1 + (T - T_0)\frac{R_1 - R_0}{T_1 - T_0} = 1 + (T - T_0)\frac{\Delta R}{\Delta T}$

In the real world, differentials are often replaced by finite differences. The new resistance will then be $R(T) \approx IR(1+(T-T_0)\Delta R/\Delta T)$ .

However, doing this clearly neglects all the higher-order terms of T. The range for which the approximation is usable depends on how nonlinear the part is (most thermistors are quite nonlinear - see the Wikipedia article for more) and the degree of accuracy required.