Relation Cross-Scans

LC Product Frequency-Square Cross-Scan

Three exact LC rows show the reciprocal relation between storage product and mode square. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

LC product 1 sets omega-square

Row 1 uses inductance 1 and capacitance 1. Their product is 1, so the reciprocal frequency square follows from the same source.

LC=1,ω2=11L C=1,\quad \omega^{2}=\frac{1}{1}
LC row 1The row checks L, C, product, and reciprocal mode square.inductance=1 Hcapacitance=1 FlcProduct=1 s^2omegaSquared=1 1/s^2

LC product 4 sets omega-square

Row 2 uses inductance 2 and capacitance 2. Their product is 4, so the reciprocal frequency square follows from the same source.

LC=4,ω2=14L C=4,\quad \omega^{2}=\frac{1}{4}
LC row 2The row checks L, C, product, and reciprocal mode square.inductance=2 Hcapacitance=2 FlcProduct=4 s^2omegaSquared=1/4 1/s^2

LC product 16 sets omega-square

Row 3 uses inductance 4 and capacitance 4. Their product is 16, so the reciprocal frequency square follows from the same source.

LC=16,ω2=116L C=16,\quad \omega^{2}=\frac{1}{16}
LC row 3The row checks L, C, product, and reciprocal mode square.inductance=4 Hcapacitance=4 FlcProduct=16 s^2omegaSquared=1/16 1/s^2

Frequency-square falls as the LC product rises

The storage pair is doubled together across the scan. The product goes one, four, sixteen while the reciprocal goes one, one fourth, one sixteenth.

LCLCω⋅ω1111224144416116\begin{array}{c|c|c|c}L&C&L C&\omega\cdot\omega\\1&1&1&1\\2&2&4&\frac{1}{4}\\4&4&16&\frac{1}{16}\\\end{array}
LC reciprocal cross-scanOnly the middle row is displayed; the table carries all three exact rows.middle row: LC=4 gives omega-square=1/4