Sources and References

• Nilsson, J. W. and Riedel, S. A. Electric Circuits. 12th edition. Pearson.
• Irwin, J. D. and Nelms, R. M. Basic Engineering Circuit Analysis. Wiley.
• Alexander, C. K. and Sadiku, M. N. O. Fundamentals of Electric Circuits. McGraw-Hill.
• Hayt, W. H., Kemmerly, J. E. and Durbin, S. M. Engineering Circuit Analysis. McGraw-Hill.
• MIT OpenCourseWare 6.002 Circuits and Electronics and 6.302 Feedback Systems.
• Stanford EE 101 Linear Systems and Circuits.
• Cambridge Part IA Electrical and Information Engineering.

Chapter 1: Charge, Current, Voltage, and the Passive Sign Convention

Electric circuit theory begins with a careful accounting of the two conserved quantities that flow through every wire and every device: electric charge and electric energy. Charge is measured in coulombs, and the rate at which it crosses a reference surface in a conductor is the current, \( i(t) = \mathrm{d}q/\mathrm{d}t \), measured in amperes. Because charge is conserved, the current entering any closed region of space must equal the current leaving it, a statement that survives intact in the lumped-element limit as Kirchhoff’s current law. Voltage, by contrast, is a measure of energy per unit charge. The work done on a test charge as it is moved from one node to another is \( w = \int v \, \mathrm{d}q \), which gives the instantaneous power delivered to a device as \( p(t) = v(t)\,i(t) \).

The passive sign convention fixes the sign of that product. When the reference arrow for current is drawn entering the terminal marked with the positive reference for voltage, positive values of \( p \) indicate power absorbed by the element, and negative values indicate power delivered. This convention is not a physical law; it is a bookkeeping discipline that makes energy audits unambiguous. A resistor obeying Ohm’s law, \( v = R\,i \) with \( R > 0 \), always absorbs power, since \( p = i^{2}R = v^{2}/R \ge 0 \). Batteries and active sources, on the other hand, routinely produce negative values of \( p \) under this convention, revealing that they are delivering energy to the rest of the network.

The lumped-element idealization underlying all of this deserves explicit notice. A wire has no resistance, a resistor has no inductance, and signal propagation is instantaneous. These approximations hold whenever the physical dimensions of the circuit are small compared with the wavelength of the highest frequency of interest, which is comfortably the case at audio frequencies and across most of the radio-frequency band used in introductory laboratories. Once the approximations fail, transmission-line and full-field treatments are required, a transition that belongs to later courses.

Chapter 2: Kirchhoff’s Laws and Simple Resistive Networks

Two laws suffice to describe any lumped circuit built from two-terminal elements. Kirchhoff’s current law (KCL) asserts that the algebraic sum of currents leaving any node is zero, expressing charge conservation at that node. Kirchhoff’s voltage law (KVL) asserts that the algebraic sum of voltage drops around any closed loop is zero, expressing the fact that the electrostatic potential is single-valued. Together with the constitutive equations of the individual elements, the two laws yield a determined system.

The simplest corollaries are the series and parallel rules. Resistors in series carry a common current and add, so that

\[ R_{\mathrm{eq}} = R_{1} + R_{2} + \cdots + R_{n}. \]

Resistors in parallel share a common voltage, and their conductances add:

\[ \frac{1}{R_{\mathrm{eq}}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \cdots + \frac{1}{R_{n}}. \]

From these follow the voltage divider, \( v_{k} = v_{\mathrm{total}} \, R_{k}/R_{\mathrm{eq}} \), and the current divider, \( i_{k} = i_{\mathrm{total}} \, G_{k}/G_{\mathrm{eq}} \), which should be used whenever possible to shortcut formal analysis. The divider formulas also clarify what loading means. Connecting a finite-resistance instrument across two nodes changes the Thévenin equivalent seen by the network and therefore perturbs the very quantity the instrument is intended to measure. A good voltmeter has \( R_{\mathrm{in}} \gg R_{\mathrm{Th}} \); a good ammeter has \( R_{\mathrm{in}} \ll R_{\mathrm{Th}} \).

Circuits also contain independent and dependent sources. An independent voltage source maintains a prescribed \( v(t) \) regardless of the current drawn, and an independent current source maintains a prescribed \( i(t) \) regardless of the voltage across it. Dependent sources, four types in all (VCVS, VCCS, CCVS, CCCS), tie an output variable to a control variable elsewhere in the network and are indispensable for modelling transistors and operational amplifiers.

Chapter 3: Nodal and Mesh Analysis

Once the network grows beyond a handful of elements, ad hoc combinations give way to systematic methods. Nodal analysis chooses a reference node (ground) and writes KCL at each of the remaining \( n - 1 \) nodes, expressing every branch current as a conductance times a node-voltage difference. The result is a linear system \( \mathbf{G}\mathbf{v} = \mathbf{i}_{s} \) in which the conductance matrix \( \mathbf{G} \) is symmetric and positive-definite for networks of positive resistors. Its diagonal entries are sums of conductances incident on each node; its off-diagonal entries are negative shared conductances. This structure is no accident: it is the discrete analogue of a Laplacian and guarantees a unique solution.

Mesh analysis is the dual procedure. For a planar network one identifies the independent loops, assigns a mesh current to each, and writes KVL around each loop. This yields \( \mathbf{R}\mathbf{i} = \mathbf{v}_{s} \), again symmetric and positive-definite for networks of positive resistors. Which method is preferable depends on counting. A network with many nodes but few meshes favours mesh analysis, while one with many meshes but few nodes favours nodal analysis. Non-planar networks must use nodal analysis or the more general cutset formulation.

Voltage sources complicate nodal analysis because the current through an ideal source is not determined by its voltage. The remedy is the supernode: a Gaussian surface enclosing both ends of the source, across which KCL is applied directly, while the source itself contributes an algebraic constraint between the two node voltages. Current sources play the analogous role in mesh analysis, giving rise to supermeshes. These techniques preserve the matrix structure at the cost of a slight reorganization of the unknowns.

Chapter 4: Linearity, Superposition, and Source Transformation

A network composed entirely of linear elements and independent sources obeys the superposition principle: the response to the sum of several source excitations equals the sum of the responses to each excitation applied alone, with all other independent sources deactivated. Deactivating a voltage source replaces it with a short circuit; deactivating a current source replaces it with an open circuit. Dependent sources remain intact throughout, since they are not independent excitations. Superposition decomposes a hard problem into several easy ones and clarifies how each source contributes to a given branch variable, though it rarely reduces overall arithmetic unless one exploits symmetry.

Source transformation converts a Thévenin form (an ideal voltage source \( v_{s} \) in series with a resistor \( R \)) into the equivalent Norton form (an ideal current source \( i_{s} = v_{s}/R \) in parallel with the same resistor), or vice versa. Applied iteratively, it collapses chains of series and parallel combinations into a single source, often revealing the answer without a full system solve. It is particularly effective on ladder networks and on bias circuits for transistors.

Chapter 5: Thévenin and Norton Equivalents, and Maximum Power Transfer

Any linear two-terminal network, no matter how intricate, can be replaced at a chosen port by a Thévenin equivalent consisting of a single voltage source \( v_{\mathrm{Th}} \) in series with a single resistance \( R_{\mathrm{Th}} \), or by the dual Norton equivalent \( i_{\mathrm{N}} = v_{\mathrm{Th}}/R_{\mathrm{Th}} \) in parallel with \( R_{\mathrm{Th}} \). The open-circuit voltage at the port gives \( v_{\mathrm{Th}} \); the short-circuit current gives \( i_{\mathrm{N}} \); and their ratio is \( R_{\mathrm{Th}} \). When the network contains dependent sources, \( R_{\mathrm{Th}} \) cannot be found by deactivating sources and simplifying, because dependent sources must remain active. One then drives the port with a test source and computes the ratio of port voltage to port current directly.

The power delivered to a resistive load \( R_{L} \) connected to a Thévenin source is

\[ P_{L} = \left(\frac{v_{\mathrm{Th}}}{R_{\mathrm{Th}} + R_{L}}\right)^{2} R_{L}. \]

Differentiating with respect to \( R_{L} \) and setting the derivative to zero gives the maximum-power-transfer theorem: the load that extracts the greatest power from a fixed source has \( R_{L} = R_{\mathrm{Th}} \), and the maximum power is \( v_{\mathrm{Th}}^{2}/(4R_{\mathrm{Th}}) \). Efficiency at that operating point is only fifty percent, a warning that maximum power transfer and maximum efficiency are distinct design goals. Signal-processing stages routinely sacrifice efficiency to match impedances; power-distribution systems do the opposite.

Chapter 6: Capacitance, Inductance, and Energy Storage

Resistors dissipate energy; capacitors and inductors store it, and because they do, they introduce time derivatives into the governing equations. A capacitor stores energy in an electric field between two conductors and obeys

\[ i(t) = C\,\frac{\mathrm{d}v(t)}{\mathrm{d}t}, \]

so a sudden jump in voltage across an ideal capacitor demands infinite current and is therefore forbidden in any physical circuit: capacitor voltage is a continuous function of time. The stored energy is \( w_{C} = \tfrac{1}{2} C v^{2} \).

An inductor stores energy in the magnetic field generated by its current and satisfies the dual relationship

\[ v(t) = L\,\frac{\mathrm{d}i(t)}{\mathrm{d}t}, \]

so inductor current is a continuous function of time; a sudden interruption of inductor current produces a voltage spike, which is why switching an inductive load without a freewheeling diode destroys switches. Stored energy is \( w_{L} = \tfrac{1}{2} L i^{2} \). Capacitors in parallel add capacitances, while capacitors in series combine as conductances do (the reciprocals add). Inductors obey the resistor-like rules: series inductors add, parallel inductors combine by reciprocal sums.

Chapter 7: First-Order Transient Response

A circuit containing one energy-storage element and any number of resistors and independent sources is governed, at every port, by a single first-order linear differential equation. For an RC circuit driven from a constant source, the capacitor voltage satisfies

\[ \tau\,\frac{\mathrm{d}v_{C}}{\mathrm{d}t} + v_{C} = v_{\infty}, \qquad \tau = R_{\mathrm{Th}}C, \]

where \( R_{\mathrm{Th}} \) is the Thévenin resistance seen from the capacitor terminals with all independent sources deactivated, and \( v_{\infty} \) is the final (steady-state) capacitor voltage. The general solution is

\[ v_{C}(t) = v_{\infty} + \left[v_{C}(0^{+}) - v_{\infty}\right] e^{-t/\tau}, \]

an expression often remembered as “final plus (initial minus final) times a decaying exponential.” Continuity of capacitor voltage gives \( v_{C}(0^{+}) = v_{C}(0^{-}) \). The dual formula for an RL circuit uses \( \tau = L/R_{\mathrm{Th}} \) and the continuity of inductor current. After five time constants the transient has decayed to under one percent of its initial deviation, a rule of thumb useful for sizing measurement windows and settling times.

Chapter 8: Sinusoidal Steady State, Phasors, and Impedance

When all independent sources are sinusoids of a common angular frequency \( \omega \), the forced response of any linear circuit settles into a sinusoidal steady state at that same frequency, with possibly different amplitudes and phases at each node. Rather than solving the differential equations anew for every new frequency, one introduces phasors: complex amplitudes that encode magnitude and phase. The real time-domain signal \( v(t) = V_{m}\cos(\omega t + \phi) \) is represented by the complex constant \( \mathbf{V} = V_{m} e^{j\phi} \), and the time-derivative operator becomes multiplication by \( j\omega \).

Under this transformation the differential equations of resistors, capacitors, and inductors become algebraic relations of the form \( \mathbf{V} = Z\,\mathbf{I} \), where the impedance \( Z \) is \( R \) for a resistor, \( 1/(j\omega C) \) for a capacitor, and \( j\omega L \) for an inductor. Impedances combine in series and parallel exactly as resistances do, and the network equations retain their nodal and mesh structure with \( \mathbf{G} \) and \( \mathbf{R} \) replaced by their complex counterparts \( \mathbf{Y}(j\omega) \) and \( \mathbf{Z}(j\omega) \). Every technique from the resistive chapters (nodal analysis, mesh analysis, superposition, Thévenin and Norton equivalents, source transformation, maximum power transfer) carries over without modification to the phasor domain, provided one remembers that real physical quantities are recovered by taking the real part of \( \mathbf{V}e^{j\omega t} \).

Average power in sinusoidal steady state is

\[ P_{\mathrm{avg}} = \tfrac{1}{2}\, \mathrm{Re}\!\left[\mathbf{V}\,\mathbf{I}^{*}\right] = \tfrac{1}{2} V_{m} I_{m} \cos\theta, \]

where \( \theta \) is the phase of \( \mathbf{V} \) relative to \( \mathbf{I} \). The factor \( \cos\theta \) is the power factor, and the capacitive and inductive reactive power it hides is a significant practical concern in power systems. Maximum average power is transferred to a load when the load impedance is the complex conjugate of the Thévenin impedance seen at the port.

Chapter 9: The Ideal Operational Amplifier

The operational amplifier is the workhorse of analog electronics, and its idealization is among the most useful caricatures in engineering. An ideal op amp has infinite differential gain, infinite input impedance, and zero output impedance, with zero offset voltage. When placed in a circuit with negative feedback, these assumptions force the two input terminals to the same voltage (the “virtual short”) and require zero current to flow into either input. Together these two rules suffice to analyze a vast catalogue of circuits without ever writing a differential equation for the amplifier itself.

Applied to the inverting configuration with input resistor \( R_{1} \) and feedback resistor \( R_{f} \), the rules give \( v_{\mathrm{out}} = -(R_{f}/R_{1})\,v_{\mathrm{in}} \), a scaling by a ratio of resistors. The non-inverting configuration yields \( v_{\mathrm{out}} = (1 + R_{f}/R_{1})\,v_{\mathrm{in}} \), always with gain of at least unity. The unity-gain buffer is the limiting case and is used as an impedance isolator between a high-impedance source and a low-impedance load. Summing amplifiers, difference amplifiers, and instrumentation amplifiers generalize these ideas to multiple inputs and differential measurements.

Replacing resistors with capacitors or inductors yields integrators, differentiators, and active filters whose cutoff frequencies and pole locations are set by RC products rather than by the amplifier’s intrinsic bandwidth. The price of negative feedback is paid in reduced closed-loop gain, but the rewards are large: linearity, predictability, immunity to device-to-device variations, and the ability to shape frequency response at will. The ideal-op-amp framework will be refined in later courses by the frequency-dependent open-loop gain, finite input and output impedances, and noise, but the cascade of simple rules introduced here remains the scaffold on which all analog design is built.

Chapter 10: Frequency Response and Filters

A transfer function \( H(j\omega) = \mathbf{V}_{\mathrm{out}}/\mathbf{V}_{\mathrm{in}} \) summarizes how a linear circuit weights the phasor amplitude and phase at each frequency. Its magnitude and phase plotted against \( \log\omega \) (Bode plots) reveal the circuit’s character at a glance. A first-order low-pass section built from a single RC pair has

\[ H(j\omega) = \frac{1}{1 + j\omega/\omega_{c}}, \qquad \omega_{c} = \frac{1}{RC}, \]

with a passband at low frequencies, a \( -3\,\mathrm{dB} \) corner at \( \omega_{c} \), and a roll-off of twenty decibels per decade above. A high-pass section swaps the capacitor and resistor and has the dual behaviour. Band-pass and band-reject filters combine the two, and second-order sections add a quality factor \( Q \) that controls the sharpness of the transition between passband and stopband.

Passive RLC filters are cheap and robust but cannot provide gain, suffer from loading between stages, and require inductors that are bulky at audio frequencies. Active filters replace inductors with op-amp feedback networks, yielding Sallen-Key and multiple-feedback topologies that realize Butterworth, Chebyshev, and Bessel responses with resistors and capacitors alone. The choice of topology trades flatness in the passband for sharpness at the corner or for linearity of phase, decisions guided by the intended application: audio reconstruction, control-loop anti-aliasing, or biomedical signal conditioning.

Chapter 11: Laboratory Practice and Electronic Waste

The laboratory component translates analytical results into working hardware. Standard instruments include bench power supplies, function generators, digital multimeters, and oscilloscopes, supplemented by breadboards, discrete components, and op-amp prototyping kits. Measurement discipline matters: probe loading and ground loops can dominate the quantities being observed, and a systematic approach (estimate before measure, identify the Thévenin source at each port, and verify with a sanity check) catches most errors before they become puzzles.

The course also situates circuit practice within an environmental frame. Electronic waste is one of the fastest-growing municipal waste streams, and printed circuit boards in particular concentrate copper, tin, gold, and a cocktail of potentially hazardous materials including lead, brominated flame retardants, and rare-earth elements. Responsible recycling recovers valuable metals through mechanical shredding, magnetic and eddy-current separation, and pyrometallurgical or hydrometallurgical refining, while design-for-disassembly and modularity can reduce waste at the source. Understanding these end-of-life considerations is increasingly part of what it means to practice engineering competently.