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1Formulas and methods
Wheatstone bridge & strain gauges
Divider voltages
Va = VsR2/(R1 + R2), Vb = VsR4/(R3 + R4)
Output
Vab = Va − Vb
Balance
R1/R2 = R3/R4
Slide-wire bridge
R/Rx = l/(100 − l)
Thévenin resistance
Rth = R1∥R2 + R3∥R4
Gauge factor
ΔR/R = GF·ε
Bridge output
quarter Vs·GF·ε/4, half Vs·GF·ε/2, full Vs·GF·ε
Small change, one arm
Vab ≈ Vs·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 Va and Vb 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 Vs·GF·ε/4, /2 or ×1 for a quarter, half or full bridge.
Data acquisition & sampling
Nyquist criterion
fs > 2·fmax, fN = fs/2
Alias frequency
fa = |f − k·fs|, k = round(f/fs)
Step size
1 LSB = VFSR/2ᴺ
Quantisation error (rounding)
±½ LSB
Truncating ADC code
D = ⌊Vin/LSB⌋
Bits for a resolution r
2ᴺ ≥ span/r
Ideal SQNR, full-scale sine
SQNR = 6.02·N + 1.76 dB
Data rate
channels × fs × N bit/s
Multiplexed ADC
fs = 1/(n·(tc + tset))
UART
Tb = 1/baud; 8N1 = 10 bits per byte; bytes/s = baud/bits per frame
Baud rate divider
baud = fclk/(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 fs/2; a higher frequency appears at fa = |f − k·fs| with the nearest multiple of fs.
For resolution questions, find 1 LSB = VFSR/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
Vout = −(Rf/Rin)·Vin
Non-inverting amplifier
Vout = (1 + Rf/R1)·Vin
Summing amplifier
Vout = −Rf(V1/R1 + V2/R2)
Ideal op-amp
V+ = V−, Iin = 0
Series & parallel resistors
Resistors in series
Req = R1 + R2 + …
Resistors in parallel
1/Req = 1/R1 + 1/R2 + …
Two in parallel
Req = 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
Vth = Vopen circuit
Thévenin resistance
Rth: all sources off, seen from a–b
Norton current
IN = Vth/Rth
Load current
IL = Vth/(Rth + RL)
Maximum power transfer
RL = Rth, Pmax = Vth²/(4Rth)
Method
Remove the load and mark the two terminals a and b.
Find the open-circuit voltage between a and b: this is Vth.
Switch off all independent sources and find the resistance seen from a–b: this is Rth.
Put the load back on the equivalent circuit: Vth in series with Rth.
For the Norton equivalent, convert: IN = Vth/Rth, with RN = Rth in parallel.
2Practice problems
1.Wheatstone bridge & strain gaugesMedium
Find the Thévenin resistance Rth of the bridge shown, seen from the output terminals a and b.
Vs = 2 VR1 = 80 ΩR2 = 120 ΩR3 = 60 ΩR4 = 800 Ω
Rth = Ω
2.Wheatstone bridge & strain gaugesHard
A bridge with 120 Ω arms is excited with 5 V and has two active gauges in adjacent arms, one in tension and one in compression (half bridge). The gauge factor is 2.1 and the strain is 500 µε. Find the bridge output Vo in mV.
Vs = 5 VGF = 2.1ε = 500 µε
Vo = mV
3.Data acquisition & samplingMedium
A UART sends bytes at 4800 baud in the format 8E1: 8 data bits, an even parity bit and 1 stop bit(s), plus the start bit. Find the largest number of data bytes per second it can transfer.
baud = 4800 Bd8E1
RB = B/s
4.Data acquisition & samplingHard
A 2.4 kHz square wave is sampled at 20 kHz with no anti-aliasing filter. A square wave contains the odd harmonics 3f₁, 5f₁, 7f₁, … At which frequency fa does its harmonic 5f₁ appear in the sampled data? Give it in kHz.
f₁ = 2.4 kHzfs = 20 kHzharmonic: 5f₁
fa = kHz
5.Op-amp circuitsMedium
Choose the feedback resistor Rf that gives this non-inverting amplifier a voltage gain of 3.
R1 = 22 kΩAv = 3
Rf = kΩ
6.Op-amp circuitsHard
The summing amplifier uses an ideal op-amp with ±15 V supplies. Find the output voltage Vout.
Wheatstone bridge: two voltage dividers share one supply Vs. The left divider R1–R2 gives Va = Vs·R2/(R1 + R2) and the right divider R3–R4 gives Vb = Vs·R4/(R3 + R4); the output is Vab = Va − Vb. The bridge is balanced (Vab = 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 Rth = R1∥R2 + R3∥R4.
Supply shorted
R₁∥R₂ = 80·120/200 = 48 Ω
R₃∥R₄ = 60·800/860 = 55.81 Ω
Thévenin resistance
Rth = 103.8 Ω
Answer
Rth = 103.8 Ω
2.Wheatstone bridge & strain gauges
Answer: Vo = 2.625 mV
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 Vo ≈ Vs·GF·ε/4. Two active gauges with opposite strains in adjacent arms (half bridge) give Vs·GF·ε/2, and four (full bridge) give Vs·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.
Relative resistance change
ΔR/R = 2.1·500·10⁻⁶ = 0.00105
Bridge output
Vo = Vs·GF·ε/2 = 5·0.00105/2 = 2.625 mV
Answer
Vo = 2.625 mV
3.Data acquisition & sampling
Answer: RB = 436.4 B/s
UART: an asynchronous serial link sends each byte as a frame: one start bit (low), the data bits, an optional parity bit and one or two stop bits (high). The baud rate is the number of bits per second, so one bit lasts Tb = 1/baud and a frame of n bits lasts n/baud. The common 8N1 format has 10 bits per byte. The baud generator divides the clock: baud = fclk/(16·D) with a whole number D, so the real baud rate differs slightly from the target; an error below about 2 % is usually acceptable.
Bits per frame
1 + 8 + 1 + 1 = 11
Byte rate
4800/11 = 436.4 B/s
Answer
RB = 436.4 B/s
4.Data acquisition & sampling
Answer: fa = 8 kHz
Sampling: an ADC measures the signal fs times per second. By the Nyquist–Shannon theorem a frequency f is captured correctly only while f < fs/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 fa = |f − k·fs|, with the whole number k that brings the result between 0 and fs/2. The samples cannot tell the two apart, so an analogue anti-aliasing filter must remove everything above fs/2 before the ADC.
Harmonic frequency
5f₁ = 5·2.4 kHz = 12 kHz
Nearest multiple of the sampling rate
m = round(0.6) = 1
Alias frequency
fa = |12 − 1·20| = 8 kHz
Answer
fa = 8 kHz
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: Rf = 44 kΩ
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.
Gain of a non-inverting amplifier
V₋ = V₊ = Vin, and the current through R₁ also flows through Rf, so Vout = Vin·(1 + Rf/R₁).
Av = 1 + Rf/R1
Rf = (3 − 1)·22 = 44 kΩ
Answer
Rf = 44 kΩ
6.Op-amp circuits
Answer: Vout = 0.2715 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.
Input currents
V₋ = 0, so each input resistor has its full input voltage across it.
I1 = V1/R1 = −1.4/10 = −0.14 mA
I2 = V2/R2 = 0.6/4.7 = 0.1277 mA
If = I1 + I2 = −0.01234 mA
Output voltage
Vout = −Rf·If = −22·(−0.01234) = 0.2715 V
Answer
Vout = 0.2715 V
7.Series & parallel resistors
Answer: IR1 = 6.17 A
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 = 12 + 15 = 27 Ω
R2 ∥ 27 = 2·27/(2+27) = 1.862 Ω
Req = R1 + 1.862 = 3 + 1.862 = 4.862 Ω
Current from the source
The source sees only Req, so Ohm’s law gives the current it delivers. That current flows through R₁.
IR1 = V1/Req = 30/4.862 = 6.17 A
Answer
IR1 = 6.17 A
8.Series & parallel resistors
Answer: IR1 = 0.4705 A
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.
R5 + R6 = 2 + 12 = 14 Ω
R4 ∥ 14 = 5.091 Ω
R3 + 5.091 = 25.09 Ω
R2 ∥ 25.09 = 11.13 Ω
Req = R1 + 11.13 = 8 + 11.13 = 19.13 Ω
Current from the source
The source sees only Req, so Ohm’s law gives the current it delivers. That current flows through R₁.
IR1 = V1/Req = 9/19.13 = 0.4705 A
Answer
IR1 = 0.4705 A
9.Thévenin & Norton
Answer: IL = 0.4235 A
Thévenin’s theorem: any linear two-terminal network behaves like a single voltage source Vth in series with a resistance Rth. Norton’s theorem gives the same network as a current source IN = Vth/Rth in parallel with Rth.
Find Rth 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 = 15·10/(15+10) = 6 Ω
Rth = (R1 ∥ R2) + R3 = 6 + 8 = 14 Ω
Find Vth, 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.
Vth = V1·R2/(R1+R2) = 18·10/(15+10) = 7.2 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.
IL = Vth/(Rth + RL) = 7.2/(14 + 3) = 0.4235 A
10.Thévenin & Norton
Answer: PL = 268.7 W
Thévenin’s theorem: any linear two-terminal network behaves like a single voltage source Vth in series with a resistance Rth. Norton’s theorem gives the same network as a current source IN = Vth/Rth in parallel with Rth.
Find Rth 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 = 20·15/(20+15) = 8.571 Ω
Rth = (R1 ∥ R2) + R3 = 8.571 + 20 = 28.57 Ω
Find Vth, 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.
VA = V1·R2/(R1+R2) + Is·(R1∥R2) = 3.857 + 6·8.571 = 55.29 V
Vth = VB = VA + Is·R3 = 55.29 + 6·20 = 175.3 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.
IL = Vth/(Rth + RL) = 175.3/(28.57 + 30) = 2.993 A