Lab

Draw any linear circuit (R, L, C, independent and all four dependent sources) and get every node voltage, current and power in DC or AC. Pick any element to get the Thevenin equivalent it sees and the maximum-power answer, and see the nodal equations the solver wrote.

Throw a switch and watch charge flow, the capacitor fill and the inductor's field build. Every 0⁻, 0⁺ and ∞ value is tabulated, and the RLC roots move in the s-plane as you drag R.

Pick a system and run each definition as an experiment: superpose two inputs, delay the input, change only the future, feed a bounded input for a long time. The two curves that must coincide are drawn on top of each other, so a failure is something you see, not a rule you recall.

Pick an input and an impulse response, then slide the flipped one across with your finger. The overlap is shaded, its area is the output at that instant, and the output curve grows point by point. A discrete mode shows the sum of products term by term.

Place two poles in the s-plane or z-plane and pick the ROC. The signal it stands for is drawn at once, right-sided, left-sided or two-sided, with the verdict on causality and stability beside it, so you see why a formula alone is never a full answer.

Add harmonics one at a time and watch a square, triangle, sawtooth or rectified sine take shape. The line spectrum lights up as you go, Parseval reports how much of the power you have, and the Gibbs ripple refuses to shrink.

Pick a pulse, then shift it, squeeze it, reverse it, modulate it or split it into even and odd parts. The spectrum changes beside it, magnitude and phase or real and imaginary, with the property that explains the change and the symmetry shortcut GATE uses to eliminate options.

Sample one or two tones and watch the spectrum copy itself to every multiple of fs. The alias frequency is worked out the GATE way, the low-pass output is listed line by line for any cut-off, and a band-limited message shows its copies overlapping below the Nyquist rate.

A six-chapter visual story, not a calculator. Why eˢᵗ is the one signal a system cannot reshape, how a sinusoid splits into two spinning arrows at ±jω, why Fourier is the Laplace map read along one line, why sin ωt for all time has no Laplace transform (its ROC is empty), and how switching it on adds a transient that dies to leave the frequency-response answer. A sixth chapter shows why “take FT both sides” turns L di/dt into jωL·I(jω): impedance.

Every pole of F(s) becomes one term of f(t). Watch partial fractions split F(s) one pole at a time, each pole's colour turn into its time-domain mode, and the modes add up to f(t). Drag the poles, apply shift, delay, scaling or integration, and see when the final value theorem lies.

Set the doping on each side and the bias. Watch the depletion region, charge, field, potential and band diagram respond together, with Vbi, W, xp, xn, Emax and Cj worked out live.

Set the bias and watch the Q-point move on the load line. Then feed in a sine: see the gain, the 180° inversion and the moment the output clips as the transistor leaves saturation, with small-signal and exact output side by side.

Click cells to set 1s, 0s and don't-cares. The minimal groups are drawn on the map, wrap-arounds included, and you get the minimal SOP or POS, every prime implicant, and which ones are essential.

Drag a pole in the s-plane and watch overshoot, peak time, rise time and settling time change. Formula values sit beside measured ones, and you can add a third pole to test dominance. A second tab shows type 0/1/2 loops tracking step, ramp and parabola inputs.

Watch s travel round the D-contour while its image L(s) is traced in step, the same colour for the same stretch in both planes. An arrow from −1 spins as the plot goes round; count its turns, apply Z = P − N, and check the answer against the closed-loop poles. Drag K and watch stability flip.

One message, four modulations. See the waveform and envelope, the spectrum line by line, and where the power goes. Push AM past μ = 1 to see overmodulation, and dial FM to β = 2.405 to watch the carrier vanish.

Watch a one-bit staircase chase a sine. Shrink the step or the sampling rate and it falls behind (slope overload); grow the step and it hunts on the flat tops (granular noise). Then give each sample n bits and compare DPCM's SNR with plain PCM's.