Four canonical second-order responses built from elementary passive networks. For each topology we derive the transfer function $H(s)$ from Kirchhoff's laws, locate its poles and zeros in the $s$-plane, and examine the resulting steady-state response to a composite sinusoidal input. The same four objects (circuit, transfer function, pole-zero diagram, frequency response) describe the filter from four complementary viewpoints, and the purpose of this page is to make the correspondence between them explicit.
Consider a resistor $R$ in series with a capacitor $C$, driven by an input voltage $v_{\text{in}}(t)$, with the output $v_{\text{out}}(t)$ taken across the capacitor. The capacitor presents an impedance $Z_C(s) = 1/(sC)$ in the Laplace domain, and the network forms a voltage divider between $R$ and $Z_C$.
Defining the cutoff frequency $\omega_c = 1/(RC)$ gives the canonical first-order low-pass form
The transfer function has a single pole at $s = -\omega_c$ on the negative real axis and no finite zeros. At $\omega = 0$ the capacitor is open-circuit and $H(0) = 1$. At $\omega \to \infty$ the capacitor short-circuits and $|H(j\omega)| \to 0$. The magnitude passes through $1/\sqrt{2}$ (the $-3$ dB point) at $\omega = \omega_c$ and rolls off asymptotically at $20$ dB per decade.
The pole location $s = -1/(RC)$ is the reciprocal of the network time constant $\tau = RC$. The same quantity that governs the exponential charging of the capacitor in the time domain determines the corner frequency in the Laplace domain. The choice of component values $R$ and $C$ therefore fixes both the transient duration and the frequency response in one step.
Interchanging the positions of the resistor and capacitor produces the dual topology: the capacitor $C$ is placed in series with the input, and the output is taken across the resistor $R$ to ground. The voltage divider now contains $Z_C(s) = 1/(sC)$ in the upper arm and $R$ in the lower arm.
With $\omega_c = 1/(RC)$ this reduces to
The denominator is identical to the low-pass case, so the filter shares the same pole at $s = -\omega_c$. The numerator, however, introduces a zero at the origin. At $\omega = 0$ the capacitor blocks the direct-current component and $|H(0)| = 0$. At $\omega \to \infty$ the capacitor acts as a short and $|H(j\omega)| \to 1$. The magnitude reaches $1/\sqrt{2}$ at $\omega = \omega_c$ and approaches the asymptotic pass gain at $+20$ dB per decade.
The zero at the origin is a direct algebraic consequence of the series capacitor: evaluating $H(s)$ at $s = 0$ corresponds to the steady direct-current case, for which the capacitor carries no current and the output voltage across $R$ must vanish. The pole location is unchanged because the impedance sum $R + 1/(sC)$ forms the denominator of both topologies.
Adding an inductor to the high-pass network produces a series RLC circuit: $L$, $C$, and $R$ in series between the input terminal and ground, with the output taken across the resistor. The total impedance seen by the source is $Z(s) = sL + 1/(sC) + R$, and the transfer function is the fraction of this impedance contributed by $R$.
Introducing the resonant frequency $\omega_0 = 1/\sqrt{LC}$ and the quality factor $Q = \omega_0 L / R$ yields the second-order standard form
The transfer function possesses a zero at the origin and a conjugate pair of poles at $s = -\omega_0/(2Q) \pm j\omega_0\sqrt{1 - 1/(4Q^2)}$. The magnitude is zero at both direct-current and infinite frequency, and reaches its maximum value of unity at $\omega = \omega_0$, where the inductive and capacitive reactances cancel and the impedance reduces to $R$ alone. The $-3$ dB bandwidth is $B = \omega_0 / Q$, and increasing $Q$ narrows the passband while leaving the centre frequency unchanged.
The quadratic denominator arises because the circuit contains two independent energy-storage elements, $L$ and $C$. The resonance condition $\omega_0 = 1/\sqrt{LC}$ is the frequency at which the stored magnetic and electric energies exchange coherently; the damping $\omega_0/Q = R/L$ is set by the resistive loss that removes energy from this exchange. The zero at the origin again reflects the fact that the series capacitor blocks any steady component of the input.
Retaining the series RLC topology but relocating the output terminals so that $v_{\text{out}}$ is taken across the LC branch rather than across $R$ produces the complement of the band-pass filter. The output impedance is now $sL + 1/(sC)$, and
With the same definitions $\omega_0 = 1/\sqrt{LC}$ and $Q = \omega_0 L / R$ this becomes
The denominator is identical to that of the band-pass filter, so the pole pair is unchanged. The numerator, however, now carries a pair of conjugate zeros at $s = \pm j\omega_0$, located exactly on the imaginary axis. At $\omega = \omega_0$ the factor $(j\omega_0)^2 + \omega_0^2 = 0$ and the response vanishes identically, producing an ideal transmission null. Away from the notch, $|H(j\omega)| \to 1$ at both low and high frequency.
The zero on the imaginary axis corresponds to the series resonance of $L$ and $C$: at $\omega_0 = 1/\sqrt{LC}$ the inductive and capacitive reactances are equal in magnitude and opposite in sign, the combined impedance $j\omega L + 1/(j\omega C)$ is zero, and no voltage appears across the LC branch regardless of the current delivered by the source. The width of the null in the frequency domain is again governed by $Q$: small $R$ (large $Q$) produces a sharp, narrow notch; large $R$ (small $Q$) produces a broad, shallow attenuation.
Each of the four filters considered above is fully specified by the positions of its poles and zeros in the complex plane. The transfer function is the rational function whose numerator vanishes at the zeros and whose denominator vanishes at the poles; the frequency response is obtained by restricting this function to the imaginary axis. The summary below collects the defining structure of each filter.
The second-order pair (band-pass and notch) share the same denominator because they arise from the same physical network; the distinction between them lies entirely in the location at which the output is measured, which selects a different numerator polynomial. This observation generalises: for a passive network with fixed topology, relocating the output port changes only the zeros of the transfer function, leaving the poles invariant. The pole pattern is therefore a property of the network; the zero pattern is a property of the measurement.