Experimentation Exam 1

What is u₀?

Half the resolution of the instrument — the smallest amount the display can show, divided by 2.

What is u_c?

The accuracy of the instrument. Comes from the spec sheet, not from your data.

What is u_D?

The combined design-stage uncertainty. You get it by combining u₀ and u_c using RSS.

Where does u_D come from vs where does s_x come from?

u_D comes from the instrument spec sheet before you take any data. s_x comes from your actual measurements.

Why does a fancier display with more digits rarely improve u_D?

Because accuracy u_c dominates. Once u_c is much bigger than u₀, shrinking u₀ further makes almost no difference.

What is the difference between accuracy and precision?

Accuracy is how close you are to the truth. Precision is how repeatable your readings are. You can be precise but wrong.

Which type of error does averaging fix?

Random error. Systematic (bias) error is the same every time — averaging does nothing to remove it.

What does s_x tell you?

How spread out your individual readings are from the mean.

What does s_x/√N tell you?

How uncertain your mean x̄ is — how much it would shift if you repeated the experiment.

More data — what gets better and what stays the same?

More data shrinks s_x/√N, so the mean gets more reliable. s_x stays about the same. u_D never changes.

When do you use t instead of z?

When you have fewer than about 60 readings and σ is unknown — estimated from your data as s_x.

What does the z-table actually give you?

The area under the normal curve from the center out to z₁. It's always between 0 and 0.5.

Symmetric band ±z₁ — how do you get the probability?

Double the table value. Both sides are equal by symmetry.

Upper tail — how do you get the probability above z₁?

Subtract the table value from 0.5.

Both bounds on the same side of the mean — what do you do?

Subtract the two table values.

Bounds on opposite sides of the mean — what do you do?

Add the two table values.

One reading vs true mean — how do you know which interval to use?

"Where would another reading fall?" → use s_x. "Where is the true mean?" → use s_x/√N.

When an accuracy spec is given, what changes about the interval?

You add u_D under a square root alongside the random term. More data can never shrink the result below ±u_D.

What does the propagation formula do?

It tells you how uncertainty in your inputs flows through a formula to create uncertainty in the output.

Which input matters most in a propagation calculation?

The one whose term is largest inside the square root — it dominates because of squaring.

What is the shortcut for power-law formulas like R = x²y?

Use relative uncertainties: u_R/R = √((2·u_x/x)² + (u_y/y)²). The exponent multiplies the relative uncertainty.

What do you have to do when a formula has addition or subtraction?

No shortcut — you have to take partial derivatives. The power-law shortcut only works for pure multiplication and division.

What must every uncertainty answer include?

A value, a ± number, units, and a confidence level like 95%.

In y(t) = D + C cos(ωt − φ), what is D?

The DC offset — a constant shift up or down. It equals the mean of the signal over a full period.

In y(t) = D + C cos(ωt − φ), what is C?

The amplitude — how tall the wave is from center to peak. Always a positive number.

In y(t) = D + C cos(ωt − φ), what is ω?

The angular frequency in rad/s. Related to regular frequency by ω = 2πf.

In y(t) = D + C cos(ωt − φ), what is φ?

The phase angle in radians. It shifts the wave left or right in time.

What is the relationship between ω, f, and T?

ω = 2πf = 2π/T. These three are all ways of describing how fast the signal repeats.

What is the fundamental period T* for a signal made of multiple sinusoids?

The smallest period that all components fit into evenly — found using the coprime integer rule or GCD of the frequencies.

GCD shortcut for fundamental frequency?

Convert each ω to f in Hz, then f* = GCD of those frequencies.

What makes two signals NOT have a fundamental period?

If the ratio of their periods is irrational (like π). Then no common repeat time exists.

Does a constant term in a signal affect the period?

No. Only the sinusoidal terms determine the period.

How do you find the amplitude C when a signal is written as A cosωt + B sinωt?

C = √(A² + B²). This is not on the formula sheet — memorise it.

What is standard cosine form?

C cos(ωt − φ) where C is positive and φ is between −π and π.

If a sinusoid has a negative amplitude, what do you do?

Make C positive and add or subtract π from the phase.

What is the mean of D + C cos(ωt − φ) over a full period?

Just D. The cosine part averages to zero over complete cycles.

What is the RMS shortcut for D + C cos(ωt − φ) over full periods?

√(D² + C²/2). Only works over whole periods — partial periods need integration.

What is the amplitude spectrum?

A plot showing the amplitude of each sinusoidal component vs frequency in Hz.

What is the phase spectrum?

A plot showing the phase of each component vs frequency in Hz.

Where does a constant like D = 5 appear on a spectrum?

As a spike of height 5 at f = 0.

Why can amplitudes never be negative on a spectrum?

Amplitude is always positive. Any minus sign belongs in the phase.

What does time lag tell you?

How much later one signal reaches its peak compared to a reference signal.

Time lag comes out negative — what does that mean and what do you do?

The signal arrives early (it leads). Add one full period T to convert it to a positive lag.

What is the difference between an oscilloscope and a function generator?

The function generator creates the signal. The oscilloscope shows you what the signal looks like.

What setting do you use on a DMM to measure resistance?

The Ω setting. The circuit must be completely powered off first.

What is Pk-Pk on an oscilloscope?

The distance from the lowest point to the highest point of the wave — peak to peak.

What is Cyc RMS?

The RMS value computed over exactly one full cycle of the signal.

What is the RC cutoff frequency?

The frequency where the circuit starts significantly blocking the signal — output drops to about 70.7% of input.

What does −3 dB mean physically?

The output amplitude is 1/√2 ≈ 0.707 times the input. It marks the cutoff frequency.

In Lab 1, what was the most important finding about u_c vs u₀?

The accuracy spec dominated everything. The resolution was so small compared to accuracy that extra display digits made no difference.

What is the Excel formula for standard deviation of raw data?

=STDEV.S(A2:A9)

True or false — an oscilloscope generates signals.

False. It measures and displays them. The function generator generates them.

T*=

1/f* and 2pi/w

delta t =

phi/w

when sketching amplitude and phase spectrum of the signal, if you’re converting from sin→cos (5sin(12pit+.8), what do you need to add?

pi/2 to the phase shift

If sketching the amplitude and phase spectrum of the signal, if amplitude is negative and it’s cosine, what do you add to the phase shift?

pi