The majority of electronic systems are too complex for convenient hand analysis and of the the most common tools for simplification is the small-signal approximation.
In general, electronic components are non-linear and very quickly become tedious to work with.
In fact, many common terms such as gain or output impedance rely on the small-signal approximation.
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Taylor’s Theorem
From calculus we have Taylor’s theorem that lets us approximate almost any function as a sum of terms, near a specific point.
f(x) near x= a can be approximated as:
$$f(x) = f(a) + \frac{df(a)}{dx}(x – a) + \frac{1}{2!}\frac{d^2f(a)}{dx^2}(x – a)^2 + \frac{1}{3!}\frac{d^3f(a)}{3! dx^3}(x – a)^3 + …$$
The individual terms are called first order, second order etc. based on the power of x.
This is a very powerful approximation that is used very frequently but rarely explicitly mentioned.
Operating Point
The first step to the approximation is to define the operating point. In a typical transistor circuit design, this would be the DC bias point.
If f(x) is a transfer function then ‘a’ is the input and f(a) is the output.
This is also frequently called the bias point, or steady state solution.
Gain
The gain, A, of a circuit is the linear multiplication term relating the output to input:
$$v_o(t) = Av_i(t)$$
In general gain varies with the operating point and intuitively it is equal to:
$$A(a) = \frac{df(a)}{dx}$$
This is of course the first-order term from Taylor’s theorem.
Distortion
The typical amplifier or receiver is designed to be linear, therefore the gain is the only desirable term.
The second and third order terms then represent unwanted non-linear distortion.
Typically the second and third order terms are the dominant distortion products and rarely are higher order terms discussed.
A large amount of effort is spent in analog design to minimize these distortion terms.
Small-Signal
Each term in the Taylor expansion has an 1/k!(x-a)^k term. When x is near a, each higher order term is less and less significant. For example:
$$0.9 > 0.9^2 > 0.9^3 …$$
When this assumption is true, we can say it is a small-signal and can calculate the transfer function using just the linear terms.
Typically, electronic circuits are analyzed using the small-signal assumption, with the higher-order effects added in afterwards if required.