Circuit Lab cheat sheet: the formulas, each with a worked example
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This is a one-page reference for Science Olympiad Circuit Lab, meant to be copied into your own notes sheet or binder. Each formula has one worked example, and each example is a circuit you can open and measure. The number in the example is what the simulator reads, so you can see the formula and the circuit agree.
Whether you may bring notes, how many pages and what calculator, is set by the current Circuit Lab rules at soinc.org, not by this page, and it changes from season to season. Read them before you build your sheet. This page is not affiliated with Science Olympiad.
Ohm's law: V = I x R
I = V / R and R = V / I. Example: 9 V across 470 ohms is 9 / 470 = 19.1 mA, and the meter in series reads 19.1 mA.
Power: P = V x I = I^2 x R = V^2 / R
Example: the same 470 ohm resistor has about 8.98 V across it and 19.1 mA through it, so P = 8.98 x 0.0191 = 0.17 W. That is more than a 1/8 W resistor is rated for.
Series: R = R1 + R2 + R3
The current is the same everywhere and the voltages add to the supply. Example: 1 k + 2.2 k + 3.3 k = 6.5 k, so 12 V drives 12 / 6.5 k = 1.85 mA, and the 2.2 k drops 1.85 mA x 2.2 k = 4.06 V. The meter across the 2.2 k reads 4.06 V.
Parallel: 1 / R = 1 / R1 + 1 / R2 + 1 / R3
The voltage is the same across every branch and the currents add. For two: R = R1 x R2 / (R1 + R2). The total is always less than the smallest branch. Example: 1 k, 2 k and 3 k in parallel are 545 ohms, so 6 V draws 6 / 545 = 11.0 mA, and the meter on the supply reads 11.0 mA.
Voltage divider: Vout = Vin x R2 / (R1 + R2)
R2 is the resistor you measure across, R1 everything above it. Example: three equal resistors from 5 V with the meter across the bottom one is Vout = 5 x 1 / 3 = 1.67 V, and the meter reads 1.67 V.
Current divider: I1 = I x R2 / (R1 + R2)
The current splits in inverse proportion to the resistances. Example: 10 V into 1 k and then two 2 k in parallel. The pair is 1 k, so 5.00 mA flows, and it splits 5.00 x 2 / 4 = 2.50 mA into each 2 k. The meter reads 5.00 V across the pair, which is 2.50 mA through each 2 k.
Kirchhoff's laws
Current law: the currents into a node add up to the currents out. Voltage law: the voltages around any closed loop add up to zero. Example: 12 V through 1 k and 6 V through 2 k into node A, with 3 k to ground. The meter reads 8.18 V at A. 3.82 mA comes in through the 1 k and leaves as 2.73 mA through the 3 k and 1.09 mA through the 2 k.
Thevenin and Norton: Vth, Rth and In = Vth / Rth
Vth is the voltage across the terminals with no load. Rth is the resistance looking back in with every voltage source shorted. Example: 12 V into 4.7 k over 10 k, then 2.2 k, is Vth = 8.16 V and Rth = 5.40 k, so In = 1.51 mA, and a 10 k load gets 8.16 x 10 / 15.4 = 5.30 V. Both meters read 5.30 V.
RC time constant: tau = R x C
Charging: V = Vs x (1 - e^(-t / tau)), 63 percent at one tau. Discharging: V = V0 x e^(-t / tau), 37 percent left at one tau. Example: 10 k x 100 uF = 1 s, and from 5 V the scope reads about 3.14 V one second after the switch.
RL time constant: tau = L / R
The current rises the same way a capacitor’s voltage does. Example: 100 mH / 100 ohms = 1 ms. One millisecond after the switch, the 100 ohms reads about 3.16 V of 5 V, so the current is 31.6 mA of 50 mA.
Capacitive reactance: Xc = 1 / (2 pi f C)
Example: 1 uF at 60 Hz is 1 / (2 pi x 60 x 0.000001) = 2.65 k. In the AC circuit the meters read 6.60 V across it and 2.50 mA through it, which is 2.64 k. With 1 k in series the impedance is Z = root(R^2 + Xc^2) = 2.83 k. RMS of a sine is the peak / root 2, so 10 V peak is 7.07 V RMS.
Inductive reactance: XL = 2 pi f L
Example: 160 mH at 1 kHz is 2 pi x 1000 x 0.16 = 1.01 k, about equal to the 1 k resistor in the RL filter. Where XL equals R the output is 1 / root 2 of the input, and the scope shows the output at 0.705 of the input at 1 kHz.
Resonance: f0 = 1 / (2 pi x root(L x C))
Series Q = (1 / R) x root(L / C). Example: 10 mH and 2.5 nF give f0 = 31.8 kHz, and with 250 ohms Q = 8.0. A sweep of the capacitor’s voltage peaks near 31.6 kHz at 17.9 dB, about 7.9 times the input.
LED resistor: R = (Vs - Vf) / I
Example: a red LED at 1.65 V and 10 mA from 5 V needs (5 - 1.65) / 0.010 = 335 ohms. The nearest common value, 330 ohms, gives 10.2 mA in the circuit, with 1.65 V across the LED. A silicon diode in the same circuit drops 0.74 V.
Capacitors in series and parallel
Parallel capacitors add: C = C1 + C2. In series, 1 / C = 1 / C1 + 1 / C2, the same shape as resistors in parallel. Example: 1 uF and 2.2 uF in series are 1 x 2.2 / 3.2 = 0.688 uF, and the meter reads 0.688 uF.
Resistor color code
Black 0, brown 1, red 2, orange 3, yellow 4, green 5, blue 6, violet 7, gray 8, white 9. On a four-band resistor the first two bands are digits, the third is how many zeros follow, and the fourth is the tolerance: gold 5 percent, silver 10.
Example: yellow, violet, red is 4 and 7 and two zeros, 4,700 ohms. The resistor in the part codes circuit is drawn with those bands, and the meter on ohms reads 4.70 k.
Capacitor codes
Three digits on a small capacitor are picofarads: two digits, then how many zeros. Example: 104 is 10 and four zeros, 100,000 pF, which is 100 nF or 0.1 uF. The meter on capacitance in the part codes circuit reads 0.100 uF across a 104. Set the capacitor to a 102 and it reads 1.00 nF; a 473 reads 47.0 nF.
A letter after the digits is the tolerance: J is 5 percent, K 10 and M 20.
SI prefixes
pico (p) 10^-12, nano (n) 10^-9, micro (u) 10^-6, milli (m) 10^-3, kilo (k) 10^3, mega (M) 10^6. Convert to volts, amps, ohms and farads before you multiply, then convert back.
Example: 10 k x 100 uF is 10,000 x 0.0001 = 1 s, the RC circuit’s time constant. Volts over kilohms gives milliamps: 9 V / 0.47 k = 19.1 mA, which is 0.0191 A.
Meter loading
A voltmeter is a resistor across what it measures, 10 megohms on most digital meters and on this one. Example: across the bottom of two 10 megohm resistors on 10 V it makes 5 megohms against 10, so Vout = 10 x 5 / 15 = 3.33 V, and the meter reads 3.33 V, not 5.00 V.
Making the sheet your own
Copy what you reach for, in the order you reach for it, and write the unit on every line. Add the mistakes you make in practice, next to the formula they belong to. Then estimate before you calculate: 9 V on 470 ohms has to be around 20 mA, so an answer of 2 mA or 200 mA is a slip.
For the topics behind these formulas, read the study guide and the Division B or Division C guide. To test yourself, take the practice test.
Questions
Can I bring this cheat sheet to a Circuit Lab tournament?
This page does not say what is allowed on test day. The current Circuit Lab rules at soinc.org say whether notes are allowed, in what form and how much. Use this page as material for notes you make yourself.
Where are Circuit Lab notes for each topic?
The study guide explains each topic with a circuit, the Division B and Division C guides go at each level’s pace, and the practice test has 30 questions you answer by measuring.
Why do some measured values differ slightly from the formula?
Because the instruments are modeled. The meter has a 10 megohm input on volts and a 1 ohm shunt on amps, and the scope a 1 megohm input, the same as real ones, so they load the circuit a little.
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