Electronic – How to determine the peak power consumption of the office

The formula is correct, but it's normally given as:

period = 0.693×(R1 + 2R2)×C

Since frequency = 1/period and 1.44 = 1/0.693, your formula is equivalent to this.

R1 is the variable resistor in the schematic, R2 is the fixed resistor, and the capacitor C is the one attached to R2. Units are in ohms and farads.

ADDITIONAL NOTES:

In case you're wondering where the 0.693 comes from, it's the natural log of 2, and this derives from how the 555 timer operates, driving the capacitor voltage between 1/3 Vcc and 2/3 Vcc.

Be aware that in the example circuit, you cannot allow the adjustable resistor to go all the way to zero ohms, as this would then tie the DIS (discharge) pin of the 555 directly to the power supply. The first time the chip tries to discharge the capacitor, it will short out the power supply.

The input to this system is denoted by \$x(t)\$ and its output by \$y(t)\$.

• To prove that the system is homogeneous, what you need to show is that

  if input \$x(t)\$ produces output \$y(t)\$, then the output produced by input \$\alpha\cdot x(t)\$ is \$\alpha\cdot y(t)\$.

  Here, \$\alpha\$ is an arbitrarily chosen real number, that is, the highlighted statement must hold regardless of the value of \$\alpha\$. Similarly, the requirement must be satisfied regardless of the choice of input \$x(t)\$. In other words, finding one pair of input and output signals \$x(t)\$ and \$y(t)\$ and one real number \$\alpha\$ for which the highlighted statement is true is not sufficient; it has to hold for all such choices.

• To prove that the system is additive, what you need to show is that

  if inputs \$x_1(t)\$ and \$x_2(t)\$ produce outputs \$y_1(t)\$ and \$y_2(t)\$ respectively, then the output produced by input \$x_1(t)+x_2(t)\$ is \$y_1(t)+y_2(t)\$.

  Once again, cherry-picking is not allowed; the statement has to hold for all choices of input signals and corresponding output signals.

It is not too hard to verify that your system is both homogeneous and additive.