#38 Fundamental Signals #38 — Step, impulse, ramp, and circuit waveforms

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Unit step and time shift

   

   Unit step u(t): the basic switching signal

   u(t)=0 for t<0

   u(t)=1 for t≥0

   u(t−a) turns on at t=a

   Physical meaning: a switch connects a DC source

   Amplitude scaling: K·u(t)

   Circuit inputs can be built from shifted steps

   Narration transcript

   The unit step function, written u of t, is the simplest switching signal in circuit analysis. It equals zero for all negative time, and one for t greater than or equal to zero. Physically, it represents a switch closing at t equals zero, connecting a DC source. A shifted step, u of t minus a, turns on at time t equals a instead of zero. The step function is so fundamental that almost every circuit input can be described using it.

2. 2. Step operations and rectangular pulse

   

   Shift, scale, and reverse

   u(t−1)−u(t−3)

   t<1: 0

   1≤t<3: 1

   t≥3: 0

   General pulse: u(t−a)−u(t−b)

   Pulse width is b−a

   Narration transcript

   We can scale and shift the step function to build any rectangular waveform. Three times u of t gives a step of height three. A rectangular pulse is simply the difference of two shifted steps: u of t minus one, minus u of t minus three, creates a pulse of height one from t equals one to t equals three. We can also compress time: u of 2t minus 1 turns on at t equals one-half instead of one. And time inversion: u of minus t is one for all negative time. This is how we mathematically describe real circuit inputs.

3. 3. Unit impulse and sifting

   

   Unit impulse δ(t): model of an instantaneous action

   Ideal limit: width→0, height→∞

   Total area: ∫δ(t)dt=1

   δ(t)=du(t)/dt

   ∫f(t)δ(t−t₀)dt=f(t₀)

   Sifting samples the signal at t₀

   δ(t) is not an ordinary finite point value

   Narration transcript

   The unit impulse, delta of t, is a mathematical idealization. Imagine a rectangular pulse of width delta and height one over delta, so its area is always one. As we make the pulse narrower and taller, in the limit we get the impulse: zero width, infinite height, unit area. The impulse is the derivative of the step function: d u d t equals delta of t. Its most powerful property is sifting: the integral of f of t times delta of t minus t-zero, d t, extracts the value f of t-zero. Multiplying any signal by a shifted impulse samples that signal at the impulse location.

4. 4. Ramp–step–impulse chain

   

   Unit ramp: r(t)=t·u(t)

   Zero for t<0

   Slope 1 for t≥0

   dr(t)/dt=u(t)

   du(t)/dt=δ(t)

   Integration travels in the reverse direction

   ∫δ→u and ∫u→r

   Narration transcript

   The unit ramp, r of t, equals t times u of t. It starts at zero and increases linearly with slope one. The ramp is the integral of the step function. This gives us a beautiful chain connected by differentiation: the derivative of the ramp is the step, and the derivative of the step is the impulse. Going the other direction through integration: integrating the impulse gives the step, and integrating the step gives the ramp. This chain — ramp, step, impulse — will appear throughout circuit analysis.

5. 5. Exponential circuit responses

   

   The basic first-order response shape

   Decay: A·e⁽−t/τ)u(t)

   At t=τ: A·e⁽−1)=0.368A

   Rise: A[1−e⁽−t/τ)]u(t)

   At t=τ: 0.632A

   At t=5τ: e⁽−5)=0.00674

   After 5τ the error is below one percent

   Narration transcript

   The decaying exponential, A times e to the minus t over tau, times u of t, is THE shape of every first-order natural response. Tau is the time constant. At t equals tau, the signal drops to 36.8 percent of its initial value. At two tau, it's down to 13.5 percent. The five-tau rule says: after five time constants, the signal is below one percent — practically zero. The rising complement, A times one minus e to the minus t over tau, starts at zero and approaches A. At t equals tau it reaches 63.2 percent. The decay is the RC discharge shape. The rise is the RC charging shape. Together they describe all first-order circuit responses.

6. 6. Sinusoid and damped oscillation

   

   General sinusoid: a sinωt+b cosωt

   Amplitude: √(a²+b²)

   Damped form: e⁽−αt)sin(ωt)u(t)

   Envelopes: ±e⁽−αt)

   For α>0, oscillation decays with time

   Frequency ω, decay rate α

   An underdamped RLC response rings before settling

   Narration transcript

   A general sinusoid can be written as a sine omega t plus b cosine omega t, which equals the square root of a squared plus b squared, times cosine of omega t minus theta. Here theta equals arc-tangent of a over b. The damped sinusoid, e to the minus t over tau times sine of omega t times u of t, combines exponential decay with oscillation. The envelope, plus and minus e to the minus alpha t, bounds the oscillation. As time increases, the oscillations die out within the decaying envelope. This is exactly the shape of an underdamped R-L-C circuit response — it rings before settling to its final value.

7. 7. Piecewise-linear decomposition

   

   Build each slope segment with shifted ramps

   Initial slope +1: r(t)

   At t=1 the slope changes from +1 to −1

   Slope change −2: −2r(t−1)

   At t=2 the slope changes from −1 to 0

   Slope change +1: +r(t−2)

   $v(t)=r(t)-2r(t-1)+r(t-2)$

   Narration transcript

   Any piecewise linear waveform can be expressed as a sum of shifted ramp functions. Here's the key technique: start with r of t for the first rising segment. At each corner or slope change, add a shifted ramp to adjust the slope. For example, a triangular wave rising from zero to one over one second, then falling back: that's r of t, minus 2 r of t minus 1, plus r of t minus 2. Each ramp adds its slope to the running total. This decomposition works for any piecewise linear signal — trapezoids, sawtooth waves, staircase functions. It's the most important practical skill for describing circuit inputs.

8. 8. Derivative chain of a piecewise waveform

   

   Differentiate the triangular waveform twice

   $v(t)=r(t)-2r(t-1)+r(t-2)$

   dv/dt=u(t)−2u(t−1)+u(t−2)

   The first derivative is the piecewise-constant slope

   d²v/dt²=δ(t)−2δ(t−1)+δ(t−2)

   The second derivative marks corners with signed impulses

   Each derivative moves one level up the chain

   Narration transcript

   Let's see the ramp-step-impulse chain in action. Take our triangular waveform: v of t equals r of t minus 2 r of t minus 1 plus r of t minus 2. Its first derivative replaces each ramp with a step: d v d t equals u of t minus 2 u of t minus 1 plus u of t minus 2. This gives us a rectangular pulse train. The second derivative replaces each step with an impulse: d squared v over d t squared equals delta of t minus 2 delta of t minus 1 plus delta of t minus 2. This impulse train shows exactly where and how the slope changes. Every differentiation moves us one level up the chain: ramps become steps, steps become impulses.

9. 9. Waveform toolbox

   

   Fundamental signals for circuit analysis

   u(t): switching

   r(t): linear change

   δ(t): instantaneous action and sifting

   e⁽−t/τ): first-order transient

   e⁽−αt)sinωt: underdamped second-order response

   Shift + scale systematically constructs complex inputs

   Narration transcript

   Let's review our waveform toolbox. The unit step u of t models switch actions. The unit ramp r of t models linear increases. The impulse delta of t is the derivative of the step and enables the sifting property. The exponential describes first-order circuit responses with a time constant tau. The damped sinusoidal describes underdamped second-order responses. And we learned to decompose any piecewise linear waveform into shifted ramps and steps. In the next lecture, we'll study series and parallel combinations of capacitors and inductors, and how to find initial conditions for circuit analysis.