transient analysis with time varying Resistance

As we know in frequency domain V(s)= I(s)Z(s) then in time domain it should be V(t)= I(t) Convolution Z(t) instead of V(t)= I(t)Z(t) ?

• What are you asking for? It is unclear. – Unknown123 Feb 23 '19 at 9:12
• in simple RC circuit with R time varying, if we write KVL equation then voltage across resistance will be R(t)I(t) or R(t) convolution I(t) ? As per my knowledge in frequency domain we write R(s)I(s)= V(s), so It should be R(t) convolution I(t), But at many places I saw I(t) R(t) i.e. simple multiplication instead of convolution – Hariom Feb 23 '19 at 9:30
• What does the frequency domain have to do with this? – Edgar Brown Feb 23 '19 at 12:11

The expression $$\V(s) = I(s) Z(s)\$$ is only valid if the impedance $$\Z(s)\$$ is time-invariant (i.e. it is not $$\Z(s,t)\$$). For a time-varying resistor, the time-domain expression $$\v(t) = R(t) i(t)\$$ is correct.
Yes, $$\ V(s) = I(s) Z(s) \$$ is only valid for LTI (linear, time-invariant) circuits.