Sources and References

Primary texts — Nilsson, J. W. and Riedel, S. A. Electric Circuits, 11th ed. Pearson, 2019. Hambley, A. R. Electrical Engineering: Principles and Applications, 7th ed. Pearson, 2017.

Supplementary texts — Hayt, W. H., Kemmerly, J. E., and Durbin, S. M. Engineering Circuit Analysis, 9th ed. McGraw-Hill, 2018. Horowitz, P. and Hill, W. The Art of Electronics, 3rd ed. Cambridge University Press, 2015. Bentley, J. P. Principles of Measurement Systems, 4th ed. Pearson, 2005.

Online resources — MIT OpenCourseWare 6.002 Circuits and Electronics; MIT OCW 2.671 Measurement and Instrumentation; NIST reference data on uncertainty and traceability; IEEE Xplore open-access tutorial articles on instrumentation amplifiers; National Instruments white papers on data acquisition fundamentals.

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Chapter 1: Charge, Current, Voltage, and Power

1.1 The Fundamental Quantities

Electric charge \(q\), measured in coulombs, is the conserved quantity whose motion constitutes electricity. The elementary charge is \(e = 1.602 \times 10^{-19}\) C. The electric current \(i(t)\) through a surface is the time derivative of charge transported through it,

\[ i(t) = \frac{dq}{dt}, \]

measured in amperes (one coulomb per second). By convention, current is positive in the direction of conventional (positive-charge) flow, though in metallic conductors the actual carriers are electrons moving in the opposite direction.

Voltage \(v\), measured in volts, is the work per unit charge required to move charge between two points in an electric field,

\[ v = \frac{dw}{dq}. \]

The instantaneous power absorbed by an element is the product of the voltage across it and the current through it, with the passive sign convention (current enters the positive voltage terminal):

\[ p(t) = v(t)\,i(t). \]

1.2 Sources

An ideal independent voltage source maintains a prescribed voltage \(v_s\) between its terminals regardless of the current drawn. An ideal independent current source maintains a prescribed current \(i_s\) through itself regardless of the voltage across it. Dependent sources have an output controlled by another circuit quantity and model transistors, amplifiers, and transducers.

Real sources have internal resistance. A real battery modelled as a Thevenin equivalent is an ideal voltage source \(V_T\) in series with a resistance \(R_T\); the terminal voltage under load \(R_L\) is

\[ v_L = V_T\frac{R_L}{R_T + R_L}. \]

Chapter 2: Passive Elements

2.1 Resistors and Ohm’s Law

For a linear resistor the voltage-current relation is Ohm’s law,

\[ v = Ri, \]

where \(R\) is the resistance in ohms. The inverse relation uses conductance \(G = 1/R\) in siemens, \(i = Gv\). The power dissipated is

\[ p = vi = i^2 R = v^2/R, \]

always positive for passive resistors. Physically the resistance of a uniform conductor of length \(\ell\), cross section \(A\), and resistivity \(\rho\) is \(R = \rho\ell/A\). Copper at room temperature has \(\rho \approx 1.7 \times 10^{-8}\) Ω·m.

2.2 Capacitors

A capacitor stores energy in an electric field between two conductors separated by a dielectric. Its constitutive relation is

\[ i = C\frac{dv}{dt}, \]

where \(C\) is capacitance in farads. Integrating,

\[ v(t) = v(0) + \frac{1}{C}\int_0^t i(\tau)\,d\tau. \]

The stored energy is \(w = \tfrac{1}{2}Cv^2\). Capacitor voltage cannot change instantaneously in response to a bounded current, a fact that underlies every RC transient.

2.3 Inductors

An inductor stores energy in a magnetic field. Its constitutive relation is

\[ v = L\frac{di}{dt}, \]

where \(L\) is inductance in henries. Stored energy is \(w = \tfrac{1}{2}Li^2\). Inductor current cannot change instantaneously under bounded voltage; an attempt to interrupt an inductor carrying current produces a large voltage spike, which is why relay coils need flyback diodes.

Chapter 3: Kirchhoff’s Laws and Network Analysis

3.1 KCL and KVL

Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering any node of a circuit is zero: \(\sum i_k = 0\). It is a consequence of charge conservation combined with the quasi-static assumption that charge does not accumulate at a node. Kirchhoff’s voltage law (KVL) states that the algebraic sum of voltages around any closed loop is zero: \(\sum v_k = 0\). It follows from the conservative nature of the electrostatic field in the lumped-element approximation.

3.2 Series and Parallel Combinations

Resistors in series add: \(R_{eq} = R_1 + R_2 + \cdots\). Resistors in parallel add as conductances: \(1/R_{eq} = 1/R_1 + 1/R_2 + \cdots\). Capacitors in parallel add; in series they add reciprocally. Inductors behave like resistors in this regard.

The voltage divider for two resistors \(R_1\) and \(R_2\) in series driven by \(V_s\) gives

\[ v_2 = V_s\frac{R_2}{R_1 + R_2}. \]

The current divider for two resistors in parallel driven by \(I_s\) gives

\[ i_1 = I_s\frac{R_2}{R_1 + R_2}. \]

3.3 Nodal Analysis

Choose a reference node (ground). At each of the remaining \(n-1\) nodes, write KCL using the node voltages as unknowns. For a network of linear resistors and independent sources, the resulting system is \(\mathbf{G}\mathbf{v} = \mathbf{i}\), where \(\mathbf{G}\) is the symmetric nodal conductance matrix.

3.4 Mesh Analysis

For planar circuits the dual method assigns a mesh current to each independent loop and writes KVL around each mesh. The system becomes \(\mathbf{R}\mathbf{i} = \mathbf{v}\). For most small circuits one method is more convenient than the other depending on whether sources are predominantly voltage or current.

3.5 Thevenin and Norton Equivalents

Any linear network seen from two terminals can be replaced by a voltage source \(V_{th}\) in series with a resistance \(R_{th}\) (Thevenin) or by a current source \(I_N = V_{th}/R_{th}\) in parallel with the same \(R_{th}\) (Norton). \(V_{th}\) is the open-circuit terminal voltage; \(R_{th}\) is the ratio of open-circuit voltage to short-circuit current, or equivalently the resistance seen from the terminals with independent sources deactivated.

Chapter 4: First- and Second-Order Transients

4.1 The RC Circuit

A capacitor \(C\) discharging through a resistor \(R\) obeys

\[ RC\frac{dv}{dt} + v = 0, \]

with solution \(v(t) = V_0 e^{-t/\tau}\), where the time constant \(\tau = RC\). After one time constant the voltage has fallen to about 37% of its initial value; after five time constants the response is within 1% of its final value.

4.2 The RL Circuit

Dually, an inductor \(L\) with resistance \(R\) has time constant \(\tau = L/R\), and current decays as \(i(t) = I_0 e^{-t/\tau}\).

4.3 The RLC Circuit

A series RLC circuit with no source satisfies

\[ L\frac{d^2i}{dt^2} + R\frac{di}{dt} + \frac{1}{C}i = 0. \]

Defining the damping factor \(\alpha = R/(2L)\) and natural frequency \(\omega_0 = 1/\sqrt{LC}\), the behaviour depends on the ratio. Overdamped (\(\alpha > \omega_0\)) gives a sum of two decaying exponentials; critically damped (\(\alpha = \omega_0\)) gives the fastest non-oscillatory decay; underdamped (\(\alpha < \omega_0\)) gives a decaying sinusoid at frequency \(\omega_d = \sqrt{\omega_0^2 - \alpha^2}\).

Chapter 5: Sinusoidal Steady State and Impedance

5.1 Phasors

For a linear circuit driven at a single angular frequency \(\omega\), a sinusoidal voltage \(v(t) = V_m\cos(\omega t + \phi)\) is represented by the complex phasor \(\mathbf{V} = V_m e^{j\phi}\). Differentiation in the time domain becomes multiplication by \(j\omega\) in the phasor domain.

5.2 Impedance

Each element has a complex impedance \(Z = \mathbf{V}/\mathbf{I}\):

• Resistor: \(Z_R = R\)
• Inductor: \(Z_L = j\omega L\)
• Capacitor: \(Z_C = 1/(j\omega C) = -j/(\omega C)\)

Impedances combine like resistances: series adds, parallel adds reciprocally. All DC network theorems (KCL, KVL, Thevenin, Norton, nodal, mesh) carry over with impedances replacing resistances.

5.3 Power in AC Circuits

The average power delivered to a load with phasor voltage \(\mathbf{V}\) and phasor current \(\mathbf{I}\) is

\[ P = \tfrac{1}{2}\operatorname{Re}\{\mathbf{V}\mathbf{I}^*\} = V_{rms}I_{rms}\cos\theta, \]

where \(\theta\) is the angle between voltage and current and \(\cos\theta\) is the power factor. The apparent power \(S = V_{rms}I_{rms}\) in volt-amperes exceeds the real power unless the load is purely resistive, a fact of enormous consequence for power billing and conductor sizing.

Chapter 6: Operational Amplifiers and Instrumentation

6.1 The Ideal Op-Amp

An ideal operational amplifier has infinite open-loop gain, infinite input impedance, zero output impedance, and zero offset. In negative feedback its two input terminals are at the same voltage (virtual short) and draw no current (virtual open). These two rules let us analyze most op-amp circuits by inspection.

The inverting amplifier with input resistor \(R_1\) and feedback resistor \(R_f\) has gain \(v_o/v_i = -R_f/R_1\). The non-inverting amplifier has gain \(1 + R_f/R_1\). The voltage follower (unity gain buffer) isolates a source from a load.

6.2 The Instrumentation Amplifier

When measuring a small differential signal riding on a large common-mode voltage (thermocouples, strain gauge bridges, biomedical electrodes), a three-op-amp instrumentation amplifier provides high input impedance on both inputs, a single resistor \(R_g\) that sets gain, and large common-mode rejection:

\[ v_o = \left(1 + \frac{2R_1}{R_g}\right)\frac{R_3}{R_2}(v_+ - v_-). \]

Commercial parts (AD620, INA128) integrate the network with matched resistors, achieving common-mode rejection ratios above 100 dB.

Chapter 7: Transducers and Data Acquisition

7.1 Transducers

A transducer converts a physical quantity into an electrical one. Common types include:

Transducer	Measures	Principle
Strain gauge	Strain	Change in resistance of a foil under deformation
Thermocouple	Temperature	Seebeck voltage at a junction of dissimilar metals
RTD	Temperature	Resistance of platinum varies linearly with \(T\)
Thermistor	Temperature	Semiconductor resistance varies exponentially with \(T\)
Photodiode	Light	Reverse current proportional to photon flux
LVDT	Displacement	Mutual inductance between primary and two secondaries
Piezoelectric	Force, pressure, acceleration	Charge generated by crystal deformation

A Wheatstone bridge converts small resistance changes into differential voltage. For a single active arm with nominal resistance \(R\) and fractional change \(\Delta R/R\), and three identical passive arms, the bridge output with excitation \(V_{ex}\) is approximately

\[ V_{out} \approx \frac{V_{ex}}{4}\,\frac{\Delta R}{R}. \]

7.2 Resolution, Precision, Accuracy

Resolution is the smallest change a measurement system can detect. Precision (repeatability) is the spread of readings of the same quantity. Accuracy is closeness to the true value; it combines systematic (bias) and random errors. An n-bit analog-to-digital converter spanning full-scale voltage \(V_{FS}\) has resolution \(V_{FS}/2^n\). A 12-bit ADC on a 5 V span resolves 1.22 mV per count.

7.3 Signal Conditioning and Sampling

A typical data acquisition chain is transducer → instrumentation amplifier → low-pass anti-aliasing filter → sample-and-hold → ADC → digital processing. The Nyquist criterion requires sampling at more than twice the highest frequency component in the signal; aliasing below Nyquist corrupts reconstructed data and cannot be undone after sampling. Anti-aliasing filters are therefore non-negotiable.

Chapter 8: Uncertainty and Practice

The final output of any measurement carries an uncertainty. For a computed quantity \(y = f(x_1, x_2, \ldots, x_n)\) whose inputs have standard uncertainties \(u_i\), the combined standard uncertainty under the assumption of independent inputs is

\[ u_y = \sqrt{\sum_{i=1}^{n}\left(\frac{\partial f}{\partial x_i}\right)^2 u_i^2}. \]

Engineering practice requires not only computing a number but stating its uncertainty honestly, traceable to national standards. This discipline separates measurement engineering from guesswork.