= Switch on-resistance flatness
<Switch on-resistance flatness> describes how little an <analog switch on-resistance> changes across a stated analog signal range, at specified supply and <temperature>. A common measure is the peak-to-peak difference
$$
\Delta R_{\rm on}=\max_{v\in\mathcal V}R_{\rm on}(v)-\min_{v\in\mathcal V}R_{\rm on}(v).
$$
The signal interval $\mathcal V$ is part of the definition. A small minimum resistance does not establish good <switch on-resistance flatness>. Analog Devices illustrates how complementary transistor paths flatten the resistance curve and explains the distinction between fixed attenuation and signal-dependent distortion.<MT-088: Analog Switches and Multiplexers Basics>[Printed pp2–5, Figures1–4 and6]{ref}
For an original quasistatic model, an ideal source drives a load $R_L$ through a switch with
$$
R_{\rm on}(v)=r_0+\beta v^2,\qquad
v_o=\frac{R_Lv}{R_L+r_0+\beta v^2}.
$$
Here the stipulated resistance law uses the source signal $v$; it is not a transistor model or a device specification. For $|\beta v^2|\ll R_L+r_0$, expansion gives
$$
v_o\simeq Kv-Dv^3,\qquad
K=\frac{R_L}{R_L+r_0},\quad D=\frac{R_L\beta}{(R_L+r_0)^2}.
$$
With $v=A\sin\omega t$, the identity $\sin^3\theta=(3\sin\theta-\sin3\theta)/4$ yields
$$
v_o\simeq\left(KA-\frac{3DA^3}{4}\right)\sin\omega t
+\frac{DA^3}{4}\sin3\omega t.
$$
A constant gain correction cannot remove the generated third harmonic.
Taking $R_L=1000\,\Omega$, $r_0=10\,\Omega$, $\beta=2\,\Omega/\mathrm V^2$ and $A=1\,\mathrm V$ gives a resistance range10–12ohms and an approximate third-harmonic amplitude0.490millivolts. Its ratio to the fundamental is about0.050percent. This selected low-frequency calculation isolates a static <nonlinearity>; it omits switch capacitances and amplifier dynamics.
A practical test holds the switch on, applies a small known <electric current>, and sweeps the terminal signal level while measuring the conducting-path drop. Measuring several <temperatures> and supply conditions separates flatness over signal from changes between operating conditions. A second test with a sinusoidal source and <Fourier analysis> checks whether the complete circuit produces harmonics; source and readout distortion must be measured independently.
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