Network Analysis
A network function gives a complete s-domain description of how a circuit transforms its input into output. It is one of the most powerful ways to study poles, zeros, stability, time response, and frequency response in one framework.
A network function is the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions. It treats the whole circuit as one mathematical object instead of solving the circuit from scratch every time.
H(s) = Output(s) / Input(s)
Key idea: the entire network is represented by a single function that tells how signals are shaped.
Input and output are observed at the same port. Common examples are impedance and admittance.
Z(s) = V(s) / I(s)
Input and output are measured at different ports, such as voltage gain or current gain.
Vo(s) / Vi(s), Io(s) / Ii(s)
H(s) = (bm s^m + ... + b0) / (an s^n + ... + a0)
Poles are values of s that make the denominator zero. They dominate stability and response speed.
Zeros are values of s that make the numerator zero. They shape attenuation and transmission.
Animated explanation
A network function compresses the whole input-output behavior of a circuit into one s-domain expression. From that one function we can read poles, zeros, stability, time response, and frequency shaping.
Vo = Vi x ZC / (R + ZC)
Vo = Vi x 1 / (1 + sRC)
H(s) = 1 / (1 + sRC)
To obtain frequency response, replace s with j omega. This converts the network function into a frequency-dependent expression.
s = j omega
Magnitude = |H(j omega)|
Phase = angle H(j omega)
Initial value: f(0) = limit s to infinity of sF(s)
Final value: f(infinity) = limit s to 0 of sF(s)
These checks are very useful when validating whether a derived response starts and ends where physics says it should.
Impulse response: h(t) = L^-1 {H(s)}
Step response: Output(s) = H(s) x 1 / s
Once H(s) is known, impulse response and step response follow directly through inverse Laplace methods.
H(s) = 1 / (s + 2)
For a unit step, Y(s) = 1 / (s(s + 2))
y(t) = 1 - e^(-2t)
This result immediately shows a stable response that rises smoothly toward a final value.
A network function is the s-domain language of a circuit. It tells how input signals are transformed, how poles and zeros shape behavior, and how time response and frequency response emerge from one compact expression.
