Foundations

Voltage and Current

Current (I) is the rate of flow of charge through a certain point in a circuit, it is measured in amperes. Voltage (V) between two points in a circuit represents the driving force that causes current to flow between them, it is measured in volts.

Kirchhoff’s Current Law (KCL)

The algebraic sum of currents at a point in a circuit is zero. This is a consequence of the conservation of charge in a circuit.

Kirchhoff’s Voltage Law (KVL)

The sum of voltage drops going around a closed loop in a circuit is zero. This is a consequence of the conservative nature of the electrostatic field.

Signal Types, Amplitude and Frequency

A signal transports information, they come in two flavors: analog and digital.

In digital electronics we deal with pulses, which are signals that bounce between two voltages (e.g. +5 V and ground). Digital amplitudes are defined by these HIGH and LOW voltage levels.

In the analog world, sinewaves win the popularity contest. The amplitude of a sinusoidal signal can be given as (a) peak amplitude (or just “amplitude”), (b) root-mean-square (rms) amplitude or (c) peak-to-peak amplitude. If unstated, a sinewave amplitude is usually understood to be V(rms).

A periodic signal is characterized by its frequency f (in hertz) or, equivalently, period T (a time unit).

Ratios of signal amplitude or power

Ratios of signal amplitude or power are commonly expressed in decibels (dB), defined as
dB_{power} = 10 \times log_{10} (\frac{P_{2}}{P_{1}})
or
dB_{amplitude} = 20 \times log_{10} (\frac{A_{2}}{A_{1}})
An amplitude ratio of 10 (or power ratio of 100) is 20 dB; 3 dB is a doubling of power; 6 dB is a doubling of amplitude (or quadrupling of power).

Ratio Power (dB) Amplitude (dB)
1/10000 – 40 – 80
1/100 – 20 – 40
1/10 – 10 – 20
1 0 0
2 3 6
4 6 12
10 10 20
100 20 40
1000 30 60
1000000 60 120

Resistors, Capacitors and Inductors

The resistor is a linear device for which I=V/R (Ohm’s law).

For capacitors and inductors, there is a time-dependent relationship between voltage and current: I=CdV/dt and V=LdI/dt, respectively. Thinking instead in the frequency domain, these components are described by their impedances, the ratio of voltage to current as a function of frequency when driven by a sinewave. Impedances are complex, with the real part representing the amplitude of the response that is in-phase, and the imaginary part representing the amplitude of the response that is in quadrature (90° out of phase).

Sinewave current through a resistor is in phase with voltage, whereas for a capacitor it leads by 90°, and for an inductor it lags by 90°.

Series and Parallel

The impedance of components connected in series is the sum of their impedances.

R_{tot} = R_1 + R_2 + \ldots + R_n

L_{tot} = L_1 + L_2 + \ldots + L_n

C_{tot} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots + \frac{1}{C_n}

In a series circuit, a resistor much larger than the other dominates (given you are willing to ignore errors under 10%).

When connected in parallel, it’s the admittances (inverse of impedance) that add.

R_{tot} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}

L_{tot} = \frac{1}{L_1} + \frac{1}{L_2} + \ldots + \frac{1}{L_n}

C_{tot} = C_1 + C_2 + \ldots + C_n

For a pair of resistors in parallel this reduces to

R_{total} = \frac{R1 \times R2}{R1 + R2}

In a parallel circuit, a resistor much smaller than the others dominates (given you are willing to ignore errors under 10%).

Dynamic resistance

At any given moment in time, the resistance of an electronic device is defined by

R_{dyn} = \frac{\Delta V}{\Delta I}

This is the tangent to the slope of the device’s V-I curve at said moment in time.

Power

The power dissipated in a resistor R is

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

There is no dissipation in an ideal capacitor or inductor, because the voltage and current are 90° out of phase.

Basic circuits with R, L, and C

Voltage divider

Simple voltage divider

The unloaded voltage divider’s output voltage is

V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2}

V_{out} = I \times R_2

Since the current in the top and botton resistors are equal, the voltage drops are proportional to the resistances.

I = \frac{V_{in}}{R_1 + R2}

Filters

If one of the resistors in a voltage divider is replaced with a capacitor, you get a simple filter: lowpass if the lower leg is a capacitor, highpass if the upper leg is a capacitor.

Lowpass and highpass filters

In either case the -3 dB transition frequency is at

f = \frac{1}{2 \pi RC}

The ultimate rolloff rate of such a “single-pole” lowpass filter is -6 dB/octave, or -20 dB/decade. More complex filters can be created by combining inductors with capacitors.

A capacitor in parallel with an inductor forms a resonant circuit; its impedance goes to infinity at the resonant frequency

f = \frac{1}{2 \pi \sqrt{LC}}

The impedance of a series LC goes to zero at that same resonant frequency.

Bypass capacitors

A capacitor’s low impedance at signal frequencies suppresses unwanted singals, e.g. on a dc supply rail.

Blocking capacitors

A highpass filter blocks dc but passes all frequencies of interest.

Timing

An RC circuit generates a sloping waveform used to create an oscillation or a timing interval.

Energy storage

A capacitor’s stored charge Q=CV smooths out the ripples in a dc power supply.

Loading and Thévenin equivalent circuits

Connecting a load to the output of a circuit causes the unloaded output to drop; the amount of such loading depends on the load resistance and the signal source’s ability to drive it. The latter is usually expressed as the equivalent source impedance (or Thévenin impedance) of the signal. That is, the signal source is modeled as a perfect voltage source Vthev in series with a resistor Rthev. Any combination of voltage sources, current sources, and resistors can be modeled perfectly by a single voltage source in series with a single resistor (its “Thévenin equivalent circuit”), or by a single current source in parallel with a single resistor (its “Norton equivalent circuit”).

Vthev and Rthev can be found from:

Vthev: this is the open circuit voltage (ie. with nothing attached).

Rthev: this is the circuit’s output impedance. The resistance measured at the terminals with all voltage sources replaced by shorts and all current sources replaced by open circuits. Can also be formulated as Vthev / Ishort-circuit.

Experimental determination of the impedance at a point: apply a voltage DeltaV, measure DeltaI. The quotient is the impedance.

Because a load impedance forms a voltage divider with the signal’s source impedance, it’s usually desirable for the latter to be small compared with any anticipated load impedance.

A small rule of thumb for relating output impedance to input impedance: When circuit A drives circuit B, arrange things so that B loads A lightly enough to cause only insignificant attenuation of the signal. If the variation is 1:10, then the divider delivers 10/11 of the signal: attenuation is just under 10%.

Diodes

The diode is a non linear device. The ideal diode conducts in one direction only. The onset of conduction in real diodes is roughly at 0,5 V in the forward direction, and there is some small leakage current in the reverse direction. Diodes are commonly used to prevent polarity reversal, and their exponential current versus applied voltage can be used to fashion circuits with logarithmic response.

Useful diode circuits include power-supply and signal rectification, clamping and gating.

Signal rectifier applied to a differentiator, diode voltage clamp, diode OR gate

Diodes specify a maximum safe reverse voltage, beyond which avalanche breakdown occurs. This is an abrupt rise of current. This is useful in a zener diode for which a reverse breakdown voltage is specified. Zeners are usually used to establish a voltage within a circuit, or to limit a signal’s swing.

I hate loose sheets, and these pages are an attempt to bring order into my notes and thoughts. They are where I jot down circuit templates, formulas, and notes. Expect them to be full of errors, incomplete, and being updated frequently.