Measurement and Instrumentation: cram pack

Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.

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1Formulas and methods

Wheatstone bridge & strain gauges

Divider voltages: V_a = V_sR₂/(R₁ + R₂), V_b = V_sR₄/(R₃ + R₄)
Output: V_ab = V_a − V_b
Balance: R₁/R₂ = R₃/R₄
Slide-wire bridge: R/R_x = l/(100 − l)
Thévenin resistance: R_th = R₁∥R₂ + R₃∥R₄
Gauge factor: ΔR/R = GF·ε
Bridge output: quarter V_s·GF·ε/4, half V_s·GF·ε/2, full V_s·GF·ε
Small change, one arm: V_ab ≈ V_s·ΔR/(4R)
Stress: σ = E·ε

Method

Treat the bridge as two voltage dividers across the same supply.
For balance, set R₁/R₂ = R₃/R₄ and solve for the unknown resistor.
For an unbalanced bridge, find V_a and V_b from the dividers and subtract; use the Thévenin equivalent when a meter loads the output.
For strain gauges, convert strain to ΔR/R = GF·ε and use V_s·GF·ε/4, /2 or ×1 for a quarter, half or full bridge.

Data acquisition & sampling

Nyquist criterion: f_s > 2·f_max, f_N = f_s/2
Alias frequency: f_a = |f − k·f_s|, k = round(f/f_s)
Step size: 1 LSB = V_FSR/2ᴺ
Quantisation error (rounding): ±½ LSB
Truncating ADC code: D = ⌊V_in/LSB⌋
Bits for a resolution r: 2ᴺ ≥ span/r
Ideal SQNR, full-scale sine: SQNR = 6.02·N + 1.76 dB
Data rate: channels × f_s × N bit/s
Multiplexed ADC: f_s = 1/(n·(t_c + t_set))
UART: T_b = 1/baud; 8N1 = 10 bits per byte; bytes/s = baud/bits per frame
Baud rate divider: baud = f_clk/(16·D)

Method

Write down what is given: sampling rate, signal frequencies, number of bits, input range, baud rate.
For sampling questions, compare each frequency with f_s/2; a higher frequency appears at f_a = |f − k·f_s| with the nearest multiple of f_s.
For resolution questions, find 1 LSB = V_FSR/2ᴺ, or the smallest N with 2ᴺ ≥ span/resolution.
For data rates and UART links, count everything that is sent: channels, samples per second, bits or whole bytes per sample, and the start, parity and stop bits of every frame.

Op-amp circuits

Inverting amplifier: V_out = −(R_f/R_in)·V_in
Non-inverting amplifier: V_out = (1 + R_f/R₁)·V_in
Summing amplifier: V_out = −R_f(V₁/R₁ + V₂/R₂)
Ideal op-amp: V₊ = V₋, I_in = 0

Series & parallel resistors

Resistors in series: R_eq = R₁ + R₂ + …
Resistors in parallel: 1/R_eq = 1/R₁ + 1/R₂ + …
Two in parallel: R_eq = R₁R₂ / (R₁ + R₂)
Ohm's law: V = I·R
Power: P = V·I = I²R = V²/R

Method

Start at the end of the network furthest from the terminals.
Combine resistors in series by adding them, and resistors in parallel with 1/R = 1/R₁ + 1/R₂ + …
Redraw the circuit after each step.
Repeat until one resistor is left between the terminals.

Thévenin & Norton

Thévenin voltage: V_th = V_{open circuit}
Thévenin resistance: R_th: all sources off, seen from a–b
Norton current: I_N = V_th/R_th
Load current: I_L = V_th/(R_th + R_L)
Maximum power transfer: R_L = R_th, P_max = V_th²/(4R_th)

Method

Remove the load and mark the two terminals a and b.
Find the open-circuit voltage between a and b: this is V_th.
Switch off all independent sources and find the resistance seen from a–b: this is R_th.
Put the load back on the equivalent circuit: V_th in series with R_th.
For the Norton equivalent, convert: I_N = V_th/R_th, with R_N = R_th in parallel.

2Practice problems

1.Wheatstone bridge & strain gaugesMedium

Find the Thévenin resistance R_th of the bridge shown, seen from the output terminals a and b.

V_s = 15 VR₁ = 150 ΩR₂ = 300 ΩR₃ = 1000 ΩR₄ = 800 Ω

R_th = Ω

2.Wheatstone bridge & strain gaugesHard

A strain gauge (GF = 2) on a member with E = 70 GPa forms a quarter bridge excited with 2.5 V. The output reads 0.938 mV. Find the axial stress σ in MPa.

V_s = 2.5 VGF = 2V_o = 0.938 mVE = 70 GPa

σ = MPa

3.Data acquisition & samplingMedium

An 8-bit ADC with an input range of 0 to 5 V truncates: its output code is ⌊V_in/LSB⌋ with 1 LSB = V_FSR/2ᴺ. Find the decimal output code D for an input of 2.946 V.

N = 8V_FSR = 5 VV_in = 2.946 V

D =

4.Data acquisition & samplingHard

A 1.6 kHz square wave is sampled at 8 kHz with no anti-aliasing filter. A square wave contains the odd harmonics 3f₁, 5f₁, 7f₁, … At which frequency f_a does its harmonic 3f₁ appear in the sampled data? Give it in Hz.

f₁ = 1.6 kHzf_s = 8 kHzharmonic: 3f₁

f_a = Hz

5.Op-amp circuitsMedium

The op-amp is ideal and runs from ±15 V supplies. Find the output voltage V_out.

V_in = −1.7 VR₁ = 10 kΩR_f = 22 kΩ

V_out = V

6.Op-amp circuitsHard

The op-amp is ideal and runs from ±15 V supplies. The resistor ratios are not matched. Find the output voltage V_out.

V₁ = 0.5 VV₂ = −1.1 VR₁ = 22 kΩR₂ = 22 kΩR₃ = 2 kΩR₄ = 22 kΩ

V_out = V

7.Series & parallel resistorsMedium

Find the equivalent resistance R_eq seen by the source V₁.

V₁ = 24 VR₁ = 20 ΩR₂ = 15 ΩR₃ = 2 ΩR₄ = 6 Ω

R_eq = Ω

8.Series & parallel resistorsHard

Find the equivalent resistance R_eq seen by the source V₁.

V₁ = 36 VR₁ = 2 ΩR₂ = 4 ΩR₃ = 2 ΩR₄ = 5 ΩR₅ = 2 ΩR₆ = 10 Ω

R_eq = Ω

9.Thévenin & NortonMedium

A load R_L = 4 Ω is connected across a–b. Replace the rest of the circuit with its Thévenin equivalent and find the load current I_L.

V₁ = 15 VR₁ = 5 ΩR₂ = 15 ΩR₃ = 2 ΩR_L = 4 Ω

I_L = A

10.Thévenin & NortonHard

Find the Norton current I_N: the current that flows from a to b if the terminals are shorted.

V₁ = 30 VI_s = 6 AR₁ = 12 ΩR₂ = 5 ΩR₃ = 6 Ω

I_N = A

3Solutions

1.Wheatstone bridge & strain gauges

Answer: R_th = 544.4 Ω

Wheatstone bridge: two voltage dividers share one supply V_s. The left divider R₁–R₂ gives V_a = V_s·R₂/(R₁ + R₂) and the right divider R₃–R₄ gives V_b = V_s·R₄/(R₃ + R₄); the output is V_ab = V_a − V_b. The bridge is balanced (V_ab = 0, no current in the meter) when R₁/R₂ = R₃/R₄, that is R₁·R₄ = R₂·R₃. Seen from the output, the bridge is a Thévenin source with R_th = R₁∥R₂ + R₃∥R₄.

Supply shorted

R₁∥R₂ = 150·300/450 = 100 Ω

R₃∥R₄ = 1000·800/1800 = 444.4 Ω

Thévenin resistance

R_th = 544.4 Ω

Answer

R_th = 544.4 Ω

2.Wheatstone bridge & strain gauges

Answer: σ = 52.53 MPa

Strain gauges: a gauge changes its resistance with strain: ΔR/R = GF·ε, where the gauge factor GF is about 2 for metal foil. In a bridge with all arms equal to R, one active gauge (quarter bridge) gives V_o ≈ V_s·GF·ε/4. Two active gauges with opposite strains in adjacent arms (half bridge) give V_s·GF·ε/2, and four (full bridge) give V_s·GF·ε. Stress follows from Hooke’s law, σ = E·ε. Load cells state their sensitivity in mV/V: the output at full load per volt of excitation.

Strain

ε = 4·0.000938/(2.5·2) = 0.0007504

Stress

σ = 70·10³ MPa·0.0007504 = 52.53 MPa

Answer

σ = 52.53 MPa

3.Data acquisition & sampling

Answer: D = 150

ADC resolution: an N-bit converter divides its full-scale range V_FSR into 2ᴺ codes, from 0 to 2ᴺ − 1. One step is 1 LSB = V_FSR/2ᴺ: the smallest change the converter can show. A truncating ADC gives the code ⌊V_in/LSB⌋; a rounding ADC is never more than ½ LSB away from the true value, its quantisation error. Every extra bit halves the LSB and adds about 6 dB to the ideal signal-to-quantisation-noise ratio SQNR = 6.02·N + 1.76 dB for a full-scale sine. To reach a resolution r over a span, choose the smallest N with 2ᴺ ≥ span/r.

Step size

1 LSB = 5/2⁸ = 19.53 mV

Code

V_in/LSB = 2946/19.53 = 150.8

code = ⌊150.8⌋ = 150

Answer

D = 150

In binary that is 10010110.

4.Data acquisition & sampling

Answer: f_a = 3200 Hz

Sampling: an ADC measures the signal f_s times per second. By the Nyquist–Shannon theorem a frequency f is captured correctly only while f < f_s/2, the Nyquist (folding) frequency, so the sampling rate must exceed twice the highest signal frequency. A higher frequency is not lost: it shows up as an alias at f_a = |f − k·f_s|, with the whole number k that brings the result between 0 and f_s/2. The samples cannot tell the two apart, so an analogue anti-aliasing filter must remove everything above f_s/2 before the ADC.

Harmonic frequency

3f₁ = 3·1.6 kHz = 4.8 kHz

Nearest multiple of the sampling rate

m = round(0.6) = 1

Alias frequency

f_a = |4800 − 1·8000| = 3200 Hz

Answer

f_a = 3200 Hz

The alias lands inside the signal band, where no digital filter can remove it. Only an analogue filter before the ADC helps.

5.Op-amp circuits

Answer: V_out = −5.44 V

Ideal op-amp rules: with negative feedback, the op-amp drives its output so that its two inputs sit at the same voltage (V₋ = V₊), and no current flows into either input. Writing KCL at the inverting input with these two facts gives the output.

Inverting input follows the input

V₋ = V₊ = −1.7 V

I = V₋/R₁ = −1.7/10 = −0.17 mA

Output voltage

The current through R₁ also flows through R_f.

V_out = V₋ + I·R_f = −1.7 + (−0.17)·22 = −5.44 V

Answer

V_out = −5.44 V

6.Op-amp circuits

Answer: V_out = −2.517 V

Voltage at the + input

V₊ = V₂·R₄/(R₃ + R₄) = −1.1·22/(2 + 22) = −1.008 V

Superpose the two inputs

With V₂ = 0, V₁ sees an inverting amplifier. With V₁ = 0, V₊ is amplified by the non-inverting gain 1 + R₂/R₁.

V_out = −(22/22)·0.5 + (1 + 22/22)·(−1.008) = −0.5 + (−2.017) = −2.517 V

Answer

V_out = −2.517 V

7.Series & parallel resistors

Answer: R_eq = 25.22 Ω

Series and parallel resistors: in series the same current flows through each resistor and R = R₁ + R₂ + …. In parallel each resistor has the same voltage and 1/R = 1/R₁ + 1/R₂ + …. A ladder network is reduced one pair at a time until a single equivalent resistance remains.

Combine resistors from the far end

Start at the end farthest from the source and work back towards it. Resistors carrying the same current are in series; resistors across the same two nodes are in parallel.

R₃ + R₄ = 2 + 6 = 8 Ω

R₂ ∥ 8 = 15·8/(15+8) = 5.217 Ω

R_eq = R₁ + 5.217 = 20 + 5.217 = 25.22 Ω

Answer

R_eq = 25.22 Ω

8.Series & parallel resistors

Answer: R_eq = 4.321 Ω

Combine resistors from the far end

Start at the end farthest from the source and work back towards it. Resistors carrying the same current are in series; resistors across the same two nodes are in parallel.

R₅ + R₆ = 2 + 10 = 12 Ω

R₄ ∥ 12 = 3.529 Ω

R₃ + 3.529 = 5.529 Ω

R₂ ∥ 5.529 = 2.321 Ω

R_eq = R₁ + 2.321 = 2 + 2.321 = 4.321 Ω

Answer

R_eq = 4.321 Ω

9.Thévenin & Norton

Answer: I_L = 1.154 A

Thévenin’s theorem: any linear two-terminal network behaves like a single voltage source V_th in series with a resistance R_th. Norton’s theorem gives the same network as a current source I_N = V_th/R_th in parallel with R_th.

Find R_th with every source switched off

Replace voltage sources with shorts and current sources with opens, then look into the circuit from a–b.

R₁ ∥ R₂ = 5·15/(5+15) = 3.75 Ω

R_th = (R₁ ∥ R₂) + R₃ = 3.75 + 2 = 5.75 Ω

Find V_th, the open-circuit voltage

With a–b open, no current flows through R₃, so there is no drop across it and V_b = V_A. R₁ and R₂ form a voltage divider.

V_th = V₁·R₂/(R₁+R₂) = 15·15/(5+15) = 11.25 V

Connect the load to the Thévenin equivalent

The Thévenin source and R_th in series behave exactly like the original circuit at a–b, so the load sees a simple series loop.

I_L = V_th/(R_th + R_L) = 11.25/(5.75 + 4) = 1.154 A

10.Thévenin & Norton

Answer: I_N = 6.926 A

Find R_th with every source switched off

Replace voltage sources with shorts and current sources with opens, then look into the circuit from a–b.

R₁ ∥ R₂ = 12·5/(12+5) = 3.529 Ω

R_th = (R₁ ∥ R₂) + R₃ = 3.529 + 6 = 9.529 Ω

Find V_th, the open-circuit voltage

With a–b open, the only path for Iₛ is through R₃ into node A. Find V_A by superposition of V₁ and Iₛ, then add the drop across R₃.

V_A = V₁·R₂/(R₁+R₂) + I_s·(R₁∥R₂) = 8.824 + 6·3.529 = 30 V

V_th = V_B = V_A + I_s·R₃ = 30 + 6·6 = 66 V

Norton equivalent

A Norton source is the same circuit seen as a current source in parallel with R_th. Its current is the short-circuit current at a–b.

I_N = V_th/R_th = 66/9.529 = 6.926 A

ExamProblems · examproblems.com/courses/measurement-and-instrumentation

Measurement and Instrumentation: cram pack

1Formulas and methods

Wheatstone bridge & strain gauges

Method

Data acquisition & sampling

Method

Op-amp circuits

Series & parallel resistors

Method

Thévenin & Norton

Method

2Practice problems

3Solutions

Supply shorted

Thévenin resistance

Answer

Strain

Stress

Answer

Step size

Code

Answer

Harmonic frequency

Nearest multiple of the sampling rate

Alias frequency

Answer

Inverting input follows the input

Output voltage

Answer

Voltage at the + input

Superpose the two inputs

Answer

Combine resistors from the far end

Answer

Combine resistors from the far end

Answer

Find Rth with every source switched off

Find Vth, the open-circuit voltage

Connect the load to the Thévenin equivalent

Find Rth with every source switched off

Find Vth, the open-circuit voltage

Norton equivalent

Find R_th with every source switched off

Find V_th, the open-circuit voltage

Find R_th with every source switched off

Find V_th, the open-circuit voltage