Basic Electrical Quantities

Understanding basic electrical quantities is fundamental to mastering electronics. These quantities form the foundation of circuit analysis, component selection, and system design. Whether you are designing a simple LED circuit or a complex power distribution system, these measurements and their relationships are essential tools in any electronics toolkit.

This guide explores the fundamental electrical quantities, their units, measurement techniques, and practical applications. It examines how these quantities relate to each other through fundamental laws and how to apply them in real-world scenarios.

Voltage (Electric Potential Difference)

Voltage, measured in volts (V), represents the electric potential difference between two points in a circuit. It is the driving force that pushes electric charges through a conductor, analogous to water pressure in a plumbing system.

Understanding Voltage

Voltage is the energy per unit charge available to move charge through a circuit. One volt equals one joule of energy per coulomb of charge (1 V = 1 J/C). In practical terms, voltage determines how much work can be done by moving charges through a circuit.

Types of Voltage

DC Voltage: Constant polarity and magnitude over time, typical in batteries and regulated power supplies
AC Voltage: Alternating polarity with periodic variation, standard in power distribution systems
Peak Voltage (V_peak, also written V_p): Maximum instantaneous value of an AC waveform, measured from zero
Peak-to-Peak Voltage (V_pp): Total voltage swing from negative peak to positive peak, twice the peak value for a waveform symmetric about zero
RMS Voltage (V_rms): Effective value for power calculations in AC circuits, equal to V_peak/√2 for a sine wave

Practical Measurement

Voltage is always measured between two points using a voltmeter connected in parallel with the component or circuit section. Common reference points include ground (0 V), positive supply rails, and negative supply rails. Digital multimeters (DMMs) and oscilloscopes are standard tools for voltage measurement.

Common Voltage Levels

Logic circuits: 1.8 V, 3.3 V, and 5 V
Automotive systems: 12 V in cars, 24 V in many trucks and buses, and 48 V in mild-hybrid systems
Household AC: 120 V in North America, 230 V in Europe and many other countries, 220 V in China, and 100 V in Japan. These are nominal RMS values, so a 230 V supply peaks at about 325 V.
Industrial three-phase: 208 V, 480 V, and 600 V in North America (600 V chiefly in Canada), and 400 V in Europe

Current (Electric Charge Flow)

Current, measured in amperes (A), represents the rate of electric charge flow through a conductor. One ampere equals one coulomb of charge passing a point per second (1 A = 1 C/s).

Current Fundamentals

Electric current consists of moving charge carriers—electrons in metals, ions in electrolytes, or both electrons and holes in semiconductors. Conventional current is defined as the direction in which positive charge would move, which through an external circuit is from the source's positive terminal to its negative terminal. In a metal the electrons actually move the opposite way.

Types of Current

Direct Current (DC): Unidirectional flow with constant or varying magnitude
Alternating Current (AC): Bidirectional flow with periodic reversal
Pulsating Current: Unidirectional flow with periodic variation
Transient Current: Temporary current during circuit state changes

Current Measurement

Current measurement requires breaking the circuit and inserting an ammeter in series. A shunt resistor, also inserted in series, converts the current into a small voltage drop that is measured instead. Non-invasive methods include clamp meters and Hall-effect sensors: a clamp built on a current transformer measures AC only, while one built on a Hall-effect sensor measures DC as well.

Typical Current Ranges

Microelectronics: nanoamperes to milliamperes
Consumer electronics: milliamperes to amperes
Power electronics: amperes to kiloamperes
Industrial systems: tens to thousands of amperes

Resistance and Ohm's Law

Resistance, measured in ohms (Ω), quantifies a material's opposition to electric current flow. It determines the relationship between voltage and current in linear circuit elements.

Ohm's Law

The fundamental relationship between voltage (V), current (I), and resistance (R) is Ohm's law:

V = I × R

Rearranged, it gives the current through a known resistance, I = V/R, and the resistance that passes a given current at a given voltage, R = V/I. The ratio V/I defines the resistance of any two-terminal element at a particular operating point, but Ohm's law as a statement of proportionality holds only for ohmic elements, whose resistance does not change with the applied voltage. A metal-film resistor at constant temperature is ohmic; a diode is not, and neither is a filament lamp, whose resistance rises as it heats.

Factors Affecting Resistance

For a uniform conductor of length L and cross-sectional area A, made of a material with resistivity ρ:

R = ρL/A

Material properties: Resistivity, measured in ohm-meters, sets the inherent opposition to current. Copper is about 1.7 × 10⁻⁸ Ω·m at room temperature.
Temperature: The resistance of metals rises with temperature, that of copper by about 0.4 percent per degree Celsius near room temperature. Carbon, semiconductors, and NTC thermistors fall in resistance as they warm.
Geometry: Resistance rises in proportion to length and falls in inverse proportion to cross-sectional area
Frequency: Skin effect crowds alternating current toward the surface of a conductor and raises its effective resistance. In copper the skin depth is about 8.5 mm at 60 Hz and about 66 µm at 1 MHz.

Conductance

Conductance (G), measured in siemens (S), is the reciprocal of resistance (G = 1/R). It represents how easily current flows through a material, so a 50 Ω resistor has a conductance of 20 mS. High conductance indicates low resistance.

Practical Applications

Current limiting in LED circuits
Voltage division for signal conditioning
Pull-up and pull-down resistors in digital circuits
Impedance matching in transmission lines

Power and Energy

Power represents the rate of energy transfer or consumption, while energy is the total work done over time. Understanding these quantities is crucial for component selection, thermal management, and system efficiency.

Electrical Power

Power, measured in watts (W), equals one joule per second (1 W = 1 J/s). At any instant, and therefore at all times in a DC circuit, the power delivered to a circuit element is the product of the voltage across it and the current through it:

P = V × I

Substituting Ohm's law gives two further forms for a resistance, P = I²R and P = V²/R. The choice among them depends on which quantities are known. The I²R form shows why the loss in a conductor grows with the square of the current it carries.

AC Power Considerations

Real Power (P): Average power actually delivered to the load, measured in watts (W)
Reactive Power (Q): Power that flows back and forth between the source and the reactive elements without net delivery, measured in volt-amperes reactive (var)
Apparent Power (S): Product of the RMS voltage and the RMS current, measured in volt-amperes (VA)
Power Factor (PF): Ratio of real to apparent power (PF = P/S). When voltage and current are both sinusoidal, PF = cos φ, where φ is the phase angle between them, and S² = P² + Q².

Energy

Energy, measured in joules (J) or watt-hours (Wh), represents accumulated power over time:

E = P × t for a constant power P over a time t
1 kilowatt-hour (kWh) = 1,000 W × 3,600 s = 3.6 megajoules (MJ)
Battery capacity is often stated in ampere-hours (Ah) or watt-hours (Wh). The ampere-hour is a unit of charge, equal to 3,600 C; multiplying it by the nominal voltage gives the energy, so a 3 Ah cell at a nominal 3.7 V stores about 11.1 Wh.

Efficiency

System efficiency (η) compares output power to input power:

η = P_out/P_in × 100%

The difference P_in − P_out is lost, almost entirely as heat, which requires thermal management. A converter that delivers 90 W at 90 percent efficiency draws 100 W and dissipates 10 W.
Higher efficiency reduces operating costs and cooling requirements

Charge and Capacitance

Electric charge and capacitance are fundamental to energy storage, signal filtering, and timing circuits. These quantities govern the behavior of capacitors and charge storage systems.

Electric Charge

Charge, measured in coulombs (C), represents the quantity of electricity. Key relationships include:

Elementary charge: e ≈ 1.602 × 10⁻¹⁹ C, the magnitude of the charge on one proton or one electron. The electron carries −e. Since the 2019 revision of the SI, e is fixed at exactly 1.602176634 × 10⁻¹⁹ C, so one coulomb is about 6.24 × 10¹⁸ elementary charges.
Q = I × t for a constant current I flowing for a time t
Charge conservation: Total charge in an isolated system remains constant

Capacitance

Capacitance, measured in farads (F), represents the ability to store electric charge:

C = Q/V (capacitance equals charge divided by voltage)
One farad stores one coulomb at one volt
For two parallel plates of area A separated by a dielectric of thickness d and permittivity ε, C ≈ εA/d, neglecting fringing at the edges
Practical values range from picofarads to thousands of farads in supercapacitors

Capacitor Behavior

Charging: Through a resistance R from a source V_S, V(t) = V_S(1 − e^−t/RC), where τ = RC is the time constant. The capacitor reaches about 63.2 percent of V_S after one time constant and is within 1 percent of it after five.
Discharging: From an initial voltage V₀, V(t) = V₀e^−t/RC, falling to about 36.8 percent of V₀ after one time constant
Energy stored: E = ½CV² = ½QV = Q²/(2C)
Current-voltage relationship: I = C dV/dt, so the voltage across a capacitor cannot change instantaneously without an infinite current

Applications

Energy storage in power supplies and energy harvesting
Signal coupling and DC blocking
Timing circuits with RC networks
Filter circuits for noise reduction
Power factor correction in AC systems

Magnetic Flux and Inductance

Magnetic quantities are essential for understanding inductors, transformers, motors, and electromagnetic compatibility. These properties govern energy storage in magnetic fields and electromagnetic induction.

Magnetic Flux

Magnetic flux (Φ), measured in webers (Wb), represents the total magnetic field passing through a surface:

Φ = BA, where B is the magnetic flux density in teslas and A is the area, for a uniform field normal to the surface
One weber equals one volt-second (1 Wb = 1 V·s)
Flux linkage: Λ = NΦ for a coil of N turns, each linking the same flux

Inductance

Inductance, measured in henries (H), relates the flux linkage of a coil to the current that produces it, and so sets the voltage induced when that current changes:

L = Λ/I = NΦ/I (inductance equals flux linkage per unit current)
V = L dI/dt (induced voltage equals inductance times the rate of change of current)
One henry induces one volt when the current changes at one ampere per second

Inductor Behavior

Energy stored: E = ½LI²
Time constant: τ = L/R seconds, for an inductance in series with a resistance R
Reactance: X_L = 2πfL, the magnitude of the inductor's impedance jX_L
Quality factor: Q = X_L/R at a given frequency, where R is the inductor's series loss resistance. This Q is a dimensionless ratio, not to be confused with charge or reactive power.

Mutual Inductance

Mutual inductance (M) describes coupling between inductors:

Coupling coefficient: k = M/√(L₁L₂), ranges from 0 to 1
Transformer action depends on mutual inductance
EMI and crosstalk result from unwanted mutual inductance

Applications

Energy storage in switch-mode power supplies
Signal filtering and impedance matching
Transformers for voltage conversion and isolation
Chokes for EMI suppression
Sensors for position and current measurement

Frequency and Period

Frequency and period characterize repetitive signals and are fundamental to AC circuits, digital systems, and communications. These quantities determine circuit behavior, component selection, and system performance.

Frequency

Frequency, measured in hertz (Hz), represents cycles per second:

One hertz equals one cycle per second (1 Hz = 1/s)
Common prefixes: kHz (10³ Hz), MHz (10⁶ Hz), GHz (10⁹ Hz)
Audio range: 20 Hz to 20 kHz
Radio spectrum: kilohertz to hundreds of gigahertz

Period

Period (T) is the reciprocal of frequency:

T = 1/f (period equals one divided by frequency)
Measured in seconds, milliseconds, microseconds, and so on
One complete cycle occurs in one period, so a 50 Hz waveform repeats every 20 ms

Angular Frequency

Angular frequency (ω) relates to ordinary frequency:

ω = 2πf radians per second
Simplifies mathematical analysis of sinusoidal signals
Used in impedance calculations and transfer functions

Wavelength

For electromagnetic waves, wavelength (λ) relates to frequency:

λ = c/f, where c is the speed of light in free space, about 3 × 10⁸ m/s, so a 100 MHz signal has a wavelength of about 3 m
On a cable or a circuit board the wave travels more slowly, and the wavelength is shorter in proportion
Important for antenna design and transmission lines
Determines physical dimensions of RF components

Practical Considerations

Clock frequencies in digital systems
Switching frequencies in power converters
Carrier frequencies in communications
Resonant frequencies in filters and oscillators
Bandwidth requirements for signal processing

Impedance and Admittance

Impedance and admittance extend resistance and conductance concepts to AC circuits, accounting for both magnitude and phase relationships. These complex quantities are essential for AC circuit analysis and design.

Impedance

Impedance (Z), measured in ohms, represents total opposition to AC current:

Z = R + jX, where R is resistance and X is reactance
Magnitude: |Z| = √(R² + X²)
Phase angle: θ = arctan(X/R), positive for an inductive impedance and negative for a capacitive one
Polar form: Z = |Z|∠θ

Reactance Components

This article writes X_L and X_C as positive magnitudes and carries the sign in the impedance:

Inductive reactance: X_L = 2πfL. An inductor contributes +jX_L to the impedance, and its voltage leads its current by 90°.
Capacitive reactance: X_C = 1/(2πfC). A capacitor contributes −jX_C to the impedance, and its voltage lags its current by 90°.
Net reactance of an inductor and a capacitor in series: X = X_L − X_C. A positive X means the combination is inductive, a negative X that it is capacitive, and X = 0 marks series resonance.

Some texts instead treat capacitive reactance as a signed quantity, X_C = −1/(2πfC), and write X = X_L + X_C. The two conventions give the same result when either is applied consistently. At 1 kHz, for example, a 10 mH inductor has X_L ≈ 62.8 Ω and a 1 µF capacitor has X_C ≈ 159.2 Ω, so the pair in series has X ≈ −96.3 Ω and is net capacitive at that frequency.

Admittance

Admittance (Y), measured in siemens, is the reciprocal of impedance:

Y = 1/Z = G + jB
G = conductance (real part)
B = susceptance (imaginary part)
G equals 1/R only for a purely resistive impedance. For Z = R + jX, G = R/(R² + X²) and B = −X/(R² + X²).
Simplifies parallel circuit analysis

Series and Parallel Combinations

Series impedances: Z_total = Z₁ + Z₂ + Z₃ + ...
Parallel admittances: Y_total = Y₁ + Y₂ + Y₃ + ...
Parallel impedances: 1/Z_total = 1/Z₁ + 1/Z₂ + ..., which for two impedances gives Z_total = Z₁Z₂/(Z₁ + Z₂)

Applications

Filter design and frequency response analysis
Impedance matching for maximum power transfer
Transmission line calculations
Resonant circuit design
AC power system analysis

Decibel Notation

The decibel (dB) provides a logarithmic scale for expressing ratios of power, voltage, or current. This notation simplifies calculations involving large ranges and cascaded systems.

Power Ratios

For a ratio of two powers:

Ratio (dB) = 10 log₁₀(P₂/P₁)

+3 dB is very nearly a doubling of power (10 log₁₀2 ≈ 3.01 dB), and −3 dB a halving
10 dB represents a tenfold change in power
Negative dB indicates attenuation

Voltage and Current Ratios

For voltages or currents measured across equal impedances, power is proportional to the square of each, so the factor becomes 20:

Ratio (dB) = 20 log₁₀(V₂/V₁) = 20 log₁₀(I₂/I₁)

6 dB represents very nearly a doubling of voltage or current (20 log₁₀2 ≈ 6.02 dB)
20 dB represents a tenfold change in voltage or current

Absolute Units

dBm: Power relative to 1 milliwatt, so 0 dBm = 1 mW = −30 dBW
dBW: Power relative to 1 watt
dBV: RMS voltage relative to 1 volt
dBu: RMS voltage relative to 0.775 volts, referenced without assuming a load impedance (0.775 V is the value that dissipates 1 mW in 600 Ω; the "u" is commonly taken to mean "unloaded")
dBµV: Voltage relative to 1 microvolt

Cascaded Systems

Decibels simplify cascaded gain and loss calculations, because ratios that multiply become decibel values that add:

Net gain (dB) = (Gain₁ + Gain₂ + ...) − (Loss₁ + Loss₂ + ...), with each loss written as a positive number of decibels
Equivalently, a loss can be treated as a negative gain and every stage simply added
A 20 dB amplifier, a cable with 6 dB of loss, and a 10 dB amplifier in cascade give 20 − 6 + 10 = 24 dB, a power ratio of about 251

Common Applications

Audio system gain and attenuation
RF signal strength and antenna gain
Filter frequency response plots
Signal-to-noise ratio (SNR)
Dynamic range specifications

Root Mean Square (RMS) Values

RMS values provide the DC-equivalent value for AC quantities, essential for power calculations and comparing different waveforms. This measurement represents the effective value of a time-varying signal.

Mathematical Definition

RMS is the square root of the mean of the squared values, taken over one period T:

V_rms = √((1/T) ∫₀^T v²(t) dt)

Represents the DC value that would dissipate the same average power in a resistance
Never negative, whatever the polarity of the signal

Common Waveforms

Sine wave: V_rms = V_peak/√2 ≈ 0.707 × V_peak
Square wave (symmetric): V_rms = V_peak, because the magnitude is constant
Triangle or sawtooth wave (symmetric): V_rms = V_peak/√3 ≈ 0.577 × V_peak
Pulse train switching between 0 and V_peak with duty cycle D: V_rms = V_peak × √D

Crest Factor

Crest factor relates peak to RMS values:

CF = V_peak/V_rms
Sine wave: CF = √2 ≈ 1.414
Square wave: CF = 1
Triangle wave: CF = √3 ≈ 1.732
Indicates signal dynamics and meter requirements

True RMS vs. Average Responding

True RMS meters: Accurately measure any waveform by computing the RMS directly
Average responding meters: Sense the rectified average, then scale by the sine-wave form factor, so they read correctly for sine waves only
The form factor, the ratio of the RMS value to the rectified average, is π/(2√2) ≈ 1.11 for a sine wave, the scaling such meters assume. A symmetric square wave has a form factor of 1, so an average-responding meter reads it about 11 percent high.
Non-sinusoidal waveforms (for example, distorted mains or switching currents) require true RMS measurement

Practical Importance

AC power calculations: P = V_rms × I_rms × cos φ for sinusoidal voltage and current, where φ is the phase angle between them, equal to the impedance angle θ
A given RMS current heats a resistor equally at any frequency, skin effect aside
AC voltage and current ratings, including nominal mains voltages, are normally stated as RMS values
Noise and distortion measurements
Compliance with safety standards

Measurement Best Practices

Accurate measurement of electrical quantities requires proper technique, appropriate instruments, and understanding of potential sources of error.

Instrument Selection

Match instrument range to expected values
Consider accuracy, resolution, and input impedance
Account for frequency response limitations
Use calibrated instruments with valid certification

Common Measurement Errors

Loading effects: Instrument impedance affects circuit operation
Ground loops: Multiple ground paths cause measurement errors
Probe compensation: Improper adjustment distorts waveforms
Bandwidth limitations: Missing high-frequency components
Common-mode rejection: Inadequate CMR introduces noise

Safety Considerations

Verify meter ratings before measurement
Use appropriate CAT ratings for the application
Check probe insulation and test lead condition
Never exceed maximum input specifications
Use differential probes for floating measurements

Troubleshooting Tips

Start with DC measurements to establish baseline
Progress from low to high frequency measurements
Compare measurements at multiple points
Document measurement conditions and setup
Cross-check with calculations when possible

Practical Applications and Examples

Understanding how basic electrical quantities apply to real-world circuits helps bridge theory and practice. Here are common scenarios demonstrating these principles.

LED Current Limiting

To operate a red LED with a forward voltage V_F of about 2 V at a forward current I_F of 20 mA from a 5 V supply:

Voltage across the resistor: 5 V − 2 V = 3 V
Required resistance: R = V/I = 3 V / 0.02 A = 150 Ω
Resistor dissipation: P = I²R = (0.02 A)² × 150 Ω = 60 mW, while the LED itself dissipates about 2 V × 20 mA = 40 mW
A 150 Ω resistor, a standard E12 value, rated at 0.25 W gives a safety margin of about four times

Because the forward voltage varies from part to part and with temperature, the actual current varies somewhat too. A larger voltage across the resistor makes the current less sensitive to that variation.

RC Filter Design

For a first-order RC low-pass filter with a 1 kHz cutoff frequency:

Cutoff frequency: f_c = 1/(2πRC)
Choosing R = 10 kΩ: C = 1/(2π × 10³ Hz × 10⁴ Ω) ≈ 15.9 nF
The nearest standard value, 15 nF, gives f_c ≈ 1.06 kHz
At f_c the output is 3 dB below its passband level (the −3 dB point), and well above f_c the response falls at 20 dB per decade

The calculation assumes a source impedance much lower than R and a load impedance much higher.

Power Supply Filtering

To smooth the output of a full-wave rectifier running from 60 Hz AC, which produces ripple at 120 Hz:

Ripple frequency: f_ripple = 2 × 60 Hz = 120 Hz, because full-wave rectification doubles the line frequency
Peak-to-peak ripple voltage: V_ripple ≈ I_load/(f_ripple × C). The approximation assumes that the capacitor supplies the full load current for the whole ripple period, so it slightly overstates the ripple.
For a 1 A load and 1 V peak-to-peak ripple: C = I_load/(f_ripple × V_ripple) = 1 A / (120 Hz × 1 V) ≈ 8,333 µF
Use the 10,000 µF standard value for margin
The capacitor's voltage rating must exceed the peak rectified voltage, plus margin

Impedance Matching

To match a 50 Ω source to a 200 Ω load at 10 MHz, an L-network places its series element on the side of the lower resistance and its shunt element across the higher one. Here that means a series inductor next to the 50 Ω source and a shunt capacitor across the 200 Ω load:

Q = √(R_high/R_low − 1) = √(200/50 − 1) = √3 ≈ 1.732
Series reactance: X_L = Q × R_low = 1.732 × 50 Ω ≈ 86.6 Ω, so L = X_L/(2πf) ≈ 1.38 µH
Shunt reactance: X_C = R_high/Q = 200 Ω / 1.732 ≈ 115.5 Ω, so C = 1/(2πfX_C) ≈ 138 pF
Check: the capacitor in parallel with the 200 Ω load presents 50 − j86.6 Ω, and the inductor's +j86.6 Ω cancels the reactive part, leaving the 50 Ω the source requires

Because both reactances change with frequency, the match is exact only at 10 MHz.

Summary and Key Relationships

Basic electrical quantities form an interconnected framework essential for circuit analysis and design. Understanding their relationships enables effective problem-solving and system optimization.

Fundamental Relationships

Ohm's law: V = IR (foundation of DC analysis)
Power: P = VI = I²R = V²/R (energy transfer)
Charge: Q = It = CV (flow and storage)
Energy: E = Pt for constant power, with ½CV² stored in a capacitor and ½LI² in an inductor
Frequency: T = 1/f and ω = 2πf
Impedance: Z = V/I (AC extension of resistance)

Key Takeaways

Master the relationships between voltage, current, and resistance
Understand power calculations for component selection
Apply RMS values for AC power and heating calculations
Use impedance concepts for AC circuit analysis
Leverage decibel notation for system-level calculations
Consider frequency effects on circuit behavior
Practice proper measurement techniques for accuracy

Next Steps

With a solid understanding of basic electrical quantities, you are prepared to explore:

Circuit analysis techniques (Kirchhoff's laws, Thevenin and Norton equivalents)
AC circuit analysis with phasors and complex numbers
Transient analysis of reactive circuits
Frequency domain analysis and Bode plots
Digital signal measurements and timing analysis
Power electronics and energy conversion