Is a usable Laplace function possible with a time shifted resistor, e.g. r = R + Re^-2s? For instance what is the Z(s) of a series RC circuit: 1/Cs + R + Re^-2s. (???)
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2\$\begingroup\$ The notion doesn't make sense since an ideal resistor has flat frequency response from DC to light, and zero phase. \$\endgroup\$– a concerned citizenCommented Sep 5, 2020 at 22:16
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\$\begingroup\$ Signals can be delayed, constant component values can't. \$\endgroup\$– ChuCommented Sep 6, 2020 at 1:21
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The idea of time shifted resistor is not impossible, but your attempt to use notations taken from analysis of time invariant circuits is full nonsense. Being possible to write by typing letters doesn't force the typed formula to have a meaning. As well I could claim there's 13 months in an year because I can write Gogbyary into a list where the 12 well known months already exist.
A proper meaning for time shifted resistor could be "it's assembled after the circuit was started". It should be handled like there's a switch in series or parallel with the resistor.
Formally you could define that "The time shifted resistor obeys such law that in every valid Laplace domain formula of ordinary resistor R the symbol R is replaced by R(exp(-sT)) where T is the delay of the resistor."
Ohm's law with it would be U=IR(exp(-ST)) . That seems to be a kind of linear amp with current input and voltage output in the same 2 wires. It registers its current, stores it to a delay buffer and sets the voltage between its poles to IR after time T. Not impossible to think at all.
But that declaration can contain a contradiction . The internal storage needs initial content which can lead impossible condition if both stability and non-trivial behaviour is expected. Checking that doesn't fit into the comfort zone of electricians. Sorry.
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\$\begingroup\$ Thanks. A switch was mentioned. I suspect that was just an analogy and there is no Laplace transform for a switch. Correct? \$\endgroup\$– RodBCommented Sep 7, 2020 at 13:22
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\$\begingroup\$ switches are handled as breakpoints where the circuit changes to another. One calculates every time period between the switch state changes with different circuit equations. The results at the end in one period are initial values for the next period. There's no Laplace transformation model for the switches, the circuit equations are changed when a switch changes its state. \$\endgroup\$– user136077Commented Sep 7, 2020 at 13:34