Analog Electronics I · 120 h · Topic 5 of 8

Fundamental electrical quantities

Voltage, current and resistance: the three quantities used to describe any circuit. Together with Ohm's law and Joule's law they form the basic language of the whole field, and they are the first things you need to learn to measure safely.

Circuits and fundamentals Measurement Ohm's law Power Multimeter

01The quantities and their units

Quantity	Symbol	Unit	What it represents
Electric charge	Q	coulomb (C)	Amount of electricity. One coulomb is 6.25 × 10¹⁸ electrons.
Current	I	ampere (A)	Charge passing through a conductor's cross-section each second.
Voltage or potential difference	V or U	volt (V)	Energy per unit charge. It is what “pushes” the electrons.
Resistance	R	ohm (Ω)	Opposition to the flow of current.
Power	P	watt (W)	Energy converted per second.
Energy	W or E	joule (J)	Power multiplied by time. On the electricity bill it is measured in kWh.

The water analogy

It helps you keep your bearings, as long as you remember that it is only an analogy:

Voltage = pressure / height of the tank. It exists even when nothing is flowing.
Current = flow rate. It exists only if the circuit is closed.
Resistance = a narrowing of the pipe.
Power = the work done by the falling water.

The practical consequence: voltage is measured between two points (there is always a “with respect to”), whereas current is measured at a point, by passing through the circuit.

Prefixes

In electronics you work with values ranging from picofarads to megaohms. Handling the prefixes fluently is essential.

Prefix	Symbol	Factor	Typical example
giga	G	10⁹	1 GΩ: insulation
mega	M	10⁶	1 MΩ: leakage resistance
kilo	k	10³	4.7 kΩ: base resistor
—	—	1	8 Ω: speaker
milli	m	10⁻³	20 mA: LED current
micro	µ	10⁻⁶	100 µF: filter capacitor
nano	n	10⁻⁹	100 nF: decoupling
pico	p	10⁻¹²	22 pF: crystal oscillator

🔤 SI prefix converter Converts between all the prefixes using exact decimal arithmetic, decodes the 3-digit capacitor code (104 → 100 nF) and includes a copier for Greek letters and electronics symbols. ›

02Ohm's law

This is the fundamental relationship. In a conductor at constant temperature, the current is directly proportional to the applied voltage and inversely proportional to the resistance.

I = \frac{V}{R} V = I \cdot R R = \frac{V}{I}

The three forms are the same equation solved for different variables. V in volts, I in amperes, R in ohms.

Watch out for mixed units

If V is in volts and I in milliamperes, then R comes out in kilohms. That combination (V, mA, kΩ) is the most widely used in electronics because it avoids long strings of zeros: 5 V / 2 mA = 2.5 kΩ. What you can never do is mix them: 5 V / 2 mA ≠ 2.5 Ω.

Direct-application exercises

A 220 Ω resistor with 9 V applied: I = 9/220 = 40.9 mA.
A resistor carries 15 mA with 3.3 V across it: R = 3.3/0.015 = 220 Ω.
A 1 kΩ resistor with 25 mA: V = 0.025 × 1000 = 25 V.

Which components do NOT obey Ohm's law

Ohm's law holds for conductors and resistors, not for everything. A diode has an exponential curve; an incandescent lamp changes its resistance as it heats up; a thermistor is made precisely so that its resistance depends on temperature. In these cases we speak of nonlinear components and work with their characteristic curve, as seen in semiconductors and diodes.

03Power and Joule's law

Electric power is the product of voltage and current. Combined with Ohm's law, it gives three equivalent expressions:

P = V \cdot I P = I^{2} \cdot R P = \frac{V^{2}}{R}

Use whichever fits the data you have. They all give the same result.

In a resistor, all of that power is converted into heat. This is Joule's law, and it is the reason resistors have a power rating: 1/4 W, 1/2 W, 1 W, 5 W.

Worked example · Choosing a resistor's power rating

A 100 Ω resistor is going to have 12 V applied. How many watts should it be rated for?

P = \frac{12^{2}}{100} = 1.44 W

You always choose at least double the calculated power, because the rated value assumes open air and 25 °C: you need a 3 W or 5 W part. A 1/4 W resistor would burn out in seconds, and a 2 W one would work at its limit, very hot.

Energy and the electricity bill

W = P \cdot t

With P in watts and t in seconds it gives joules. With P in kilowatts and t in hours it gives kWh, the unit on the bill. 1 kWh = 3,600,000 J.

Example · What it costs to leave something on

A charger drawing 5 W, plugged in 24 hours a day for a month:

W = 0.005 kW × 24 h × 30 days = 3.6 kWh per month. For reference, a 1500 W electric water heater used 2 hours a day consumes 90 kWh in the same period: 25 times more. Calculating consumption is the first step in any battery- or solar-powered project.

04Resistors

The most common component of all. It is characterized by three data: value in ohms, tolerance as a percentage and power rating in watts.

Figure 1. Color code, animated: the bands keep changing and the reference table is below. The first three give the value and the fourth, the tolerance.

🎨 Resistor color-code calculator 4 and 5 bands, with a live drawing of the resistor, the value, the tolerance, the actual range and a reference table. It also works in reverse. ›

Standard values: the E12 series

Resistors do not exist in every value. The E12 series (5 % tolerance) has 12 values per decade, chosen so that the tolerance ranges of neighboring values touch without leaving gaps:

10 · 12 · 15 · 18 · 22 · 27 · 33 · 39 · 47 · 56 · 68 · 82

And they repeat multiplied by 10, 100, 1000… That is why design calculations are always rounded to the nearest E12 value. The E24 series (1 %) adds the intermediate values.

Series and parallel

In series

Same current through all of them; the voltages add up.

R_{T} = R_{1} + R_{2} + R_{3}

The total is greater than the largest of them.

In parallel

Same voltage across all of them; the currents add up.

\frac{1}{R_{T}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}

The total is smaller than the smallest of them. For two: R_T = R1·R2/(R1+R2).

These two rules of thumb (“greater than the largest” / “smaller than the smallest”) let you spot calculation errors at a glance, and you should always apply them.

Voltage divider

V_{2} = V_{T} \cdot \frac{R_{2}}{R_{1} + R_{2}}

It is the most widely used formula in all of electronics. It appears in transistor biasing, in resistive sensors and at the input of every instrument.

The divider only holds without a load

The formula assumes nothing is connected to the output. As soon as a load R_L is connected, you must use R2 ∥ R_L instead of R2, and the voltage drops. Rule of thumb: the divider works well if R_L is at least 10 times larger than R2.

05DC measurement

Figure 2. How each instrument is connected, animated. The third case is the classic mistake: the ammeter in parallel is a short circuit, because its internal resistance is almost zero.

Instrument	How it is connected	Ideal internal R	Precautions
Voltmeter	In parallel with the element	Infinite (in practice ≥ 10 MΩ)	Observe polarity on analog meters. Start with the highest range.
Ammeter	In series, opening the circuit	Zero (in practice < 1 Ω)	Never in parallel with a source. Check the fuse and the 10 A jack.
Ohmmeter	Across the isolated component	—	The circuit must be unpowered and the component disconnected at least at one lead.

The three rules that prevent accidents

Never measure resistance with the circuit powered. The ohmmeter injects its own current; if there is also external voltage, the reading is false and the instrument can be damaged.
Never leave the leads in the current jack and measure voltage. This is the mistake that blows the internal fuse (or the whole multimeter). When you finish measuring current, put the lead back in the V/Ω jack.
Always start with the highest range and work down. On manual-range instruments, measuring 220 V on the 2 V range destroys the instrument.

Instrument loading effect

Every instrument modifies the circuit it measures. A voltmeter connected in parallel with a resistor puts its own internal resistance in parallel with it, and the measured voltage is lower than the real one.

Example · When the instrument lies

A 12 V divider with two 1 MΩ resistors. The real voltage at the midpoint is 6 V. It is measured with a digital multimeter with 10 MΩ input impedance:

R2 in parallel with the instrument: 1 M ∥ 10 M = 909 kΩ.
Measured voltage: 12 × 909/(1000+909) = 5.71 V.
Error: 4.8 %, without the instrument being faulty.

With 10 kΩ resistors instead of 1 MΩ the error would be 0.05 %, negligible. That is why the loading effect only matters in high-impedance circuits. An analog voltmeter with 20 kΩ/V on the 10 V range presents 200 kΩ and in that same divider would read 2 V: nonsense. This is the main reason the digital multimeter displaced the analog one.

Measurement errors

absolute error = | V_{m e d} - V_{r e a l} |

relative error = \frac{absolute error}{V_{r e a l}} \times 100 %

The accuracy of a digital multimeter is specified as “± (a % of reading + n digits).” For example, ± (0.5 % + 2 d) when measuring 10.00 V means ± (0.05 V + 0.02 V) = ± 0.07 V. The resolution (how many digits it displays) is not the same as accuracy: an instrument can show 4 digits and be wrong in the second.

06In the lab

Lab 1 · Verifying Ohm's law

Variable power supply, a 1 kΩ (1/2 W) resistor, a voltmeter and an ammeter. Raise the voltage from 1 V up to 10 V in 1 V steps, noting the current at each step. Plot I versus V: you should get a straight line through the origin, and its slope is 1/R. Repeat with 2.2 kΩ and check that the line is less steep.

Lab 2 · Series, parallel and divider

With three resistors of different values (for example 1 k, 2.2 k and 4.7 k):

Measure each one separately with the ohmmeter and compare with its color code. Verify that they fall within tolerance.
Connect them in series, calculate the total resistance and measure it.
Connect them in parallel, calculate and measure.
With the series combination powered at 10 V, measure the voltage across each resistor and verify that the sum is 10 V (Kirchhoff's second law) and that each is proportional to its value.

Lab 3 · Joule's law and heating

A 100 Ω / 1/4 W resistor powered at 5 V dissipates 0.25 W: it works right at its limit and gets clearly warm to the touch. Measure its resistance cold and again after two minutes powered: the value rises by a few ohms. This is the practical check that Ohm's law holds at constant temperature. Do not leave it on for more than two minutes, and do not touch it if it looks discolored.

Lab 4 · Loading effect

Build the 1 MΩ + 1 MΩ divider from the example, power it with 12 V and measure the midpoint. Compare the reading with the theoretical 6 V. Then redo it with 1 kΩ + 1 kΩ and check that now it does read 6.00 V. This is the lab that teaches you to distrust a measurement.

07Common mistakes

Symptom	Usual cause
The multimeter does not measure current and everything stopped working	Internal fuse blown by measuring voltage with the leads in the ampere jack.
The measured resistance does not match the code	It is soldered in the circuit and there is another path in parallel. You have to lift one lead.
The resistor burns out or discolors	Insufficient power rating. The value was calculated but not the watts.
The divider voltage is lower than calculated	The connected load is in parallel with R2. Lower the divider values or use an emitter follower.
Erratic readings, or readings that keep dropping	Multimeter battery worn out, or poor contact at the leads.
Calculation with an absurd result (megaamperes)	Mixed units: volts were divided by milliamperes expecting ohms.
The parallel result came out greater than the individual resistors	You forgot to invert the result of the sum of reciprocals.

08Self-assessment

Why is voltage measured “between two points” and current “at a point”?

Because voltage is a potential difference: it makes no sense to speak of the voltage of a single point; it is always with respect to another (usually ground). Current, on the other hand, is the flow of charge through a cross-section, so it is measured by intercepting it.

A 470 Ω resistor with 6 V applied: current and power.

I = 6/470 = 12.8 mA. P = 6 × 0.0128 = 76.6 mW. A 1/4 W part is more than enough.

What is the parallel combination of 1 kΩ with 1 kΩ? And of 1 kΩ with 10 Ω?

Two equal resistors in parallel give half: 500 Ω.
1 k ∥ 10 = (1000×10)/1010 = 9.9 Ω. When one is much smaller, the result is practically that one: the large one hardly takes part.

A 10 kΩ and 10 kΩ divider powered at 9 V. What is the voltage at the middle with and without a 10 kΩ load?

Without load: 4.5 V.
With load: R2 ∥ R_L = 5 kΩ, so V = 9 × 5/(10+5) = 3 V. It dropped 33 %: the load is not negligible compared with R2.

Why must the ammeter have a very low internal resistance?

Because it is connected in series: any resistance it adds reduces the current it is trying to measure, altering the circuit. Ideally it would be zero.

You want to measure the consumption of a 5 V circuit. Where do you connect the ammeter?

You open the supply wire (positive or ground, it makes no difference) and insert the ammeter in that gap. Never in parallel with the supply: that is a short circuit through the instrument.

A 2000 W appliance runs 3 hours a day. How many kWh does it consume per month?

2 kW × 3 h × 30 days = 180 kWh.

What is the difference between resolution and accuracy in a multimeter?

Resolution is the smallest change it can display (how many digits). Accuracy is how far it deviates from the true value, and it is specified as “± (x % of reading + n digits).” An instrument can show 4 digits and be wrong in the second.

Why can't you measure resistance with the circuit powered?

Because the ohmmeter injects its own current and calculates R from the drop it produces. If there is an external voltage, that measurement is meaningless, the reading is false and the instrument can be damaged.

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