Part I
Circuits at DC
The three laws are stated and exercised on resistive networks: Kirchhoff's current law, Kirchhoff's voltage law, and the element law, all carried out in real numbers. Every later part reuses this machinery unchanged.
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Lecture 1
Voltage, Current, Power, and the Sign Convention
A reference polarity and a reference direction are chosen before the answer is known, and neither can be chosen wrongly: an assumption that disagrees with the physics is reported by a negative number rather than by an error. Fixing the two references relative to each other is what allows a single product to report absorbed power throughout a circuit, and therefore what makes the powers of a whole circuit sum to zero.
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Lecture 2
KCL, the Branch-Current Method, and Equivalence
Kirchhoff's current law states that charge does not accumulate, at a single node or inside any closed surface drawn around several of them. From that one statement follows a systematic method that solves any resistive network, together with a set of equivalences by which the network is shrunk before the method is applied.
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Lecture 3
Nodal Analysis and the Supernode
Taking the node voltages as the unknowns costs one KCL equation per node, which normally yields a smaller system than the branch-current method requires. A voltage source between two nodes carries no current expression of its own, and is absorbed by drawing the KCL surface around both of its terminals.
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Lecture 4
Linearity, Superposition, and the Source Equivalents
Linearity has two consequences that govern the remainder of the course. Sources may be applied one at a time and their separate effects added, and any linear network, viewed from two terminals, is indistinguishable from a single source behind a single resistance.
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Lecture 5
Equivalents with Dependent Sources and Maximum Power Transfer
A dependent source cannot be deactivated, so the Thevenin resistance of such a circuit is no longer obtained by series and parallel combination. It is measured instead, either from the open-circuit and short-circuit pair or with a test source applied at the port, and the equivalent so obtained settles which load receives the most power.
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Lecture 6
Operational Amplifiers and the Ideal Model
An operational amplifier is treated as an ideal element obeying two rules: no current enters its inputs, and negative feedback holds those inputs at the same voltage. The rules reduce every op-amp circuit to nodal analysis, provided that KCL is never written at the output and never at ground.
Part II
Circuits in Time
A capacitor or an inductor is introduced, and the element law acquires a derivative. The same node and loop equations now describe a transient that decays from one steady state toward the next.
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Lecture 7
Energy Storage Devices and the Instant of Switching
The capacitor and the inductor replace Ohm's law with a derivative, so their behavior depends on history rather than on the present value of the source alone. Two consequences carry the whole of Part II: at DC steady state the capacitor becomes an open circuit and the inductor a short circuit, and neither the capacitor voltage nor the inductor current may change instantaneously.
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Lecture 8
Deriving the First-Order Response
A single storage element makes the network a first-order differential equation, whose solution always separates into a transient that decays and a steady state that remains. The rate of the decay is fixed by one number, the time constant, formed from the storage element and the resistance the rest of the network presents to it.
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Lecture 9
Applying the Universal First-Order Formula
Once the form of the first-order response is known, the differential equation need not be written at all. The answer is determined by three numbers alone: the value just after the switch, the value long afterwards, and the time constant, each of which is read from an ordinary DC circuit.
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Lecture 10
Setting Up the Second-Order Differential Equation
A second storage element raises the differential equation to second order, and two initial conditions are then required rather than one. This page is concerned only with assembling that equation and its boundary values from KVL and KCL; the solution is deferred to the lecture that follows.
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Lecture 11
Solving Second-Order DC Switched Circuits
The response of a second-order circuit is fixed by the roots of its characteristic equation, which determine its shape before any constant has been evaluated. Two distinct real roots give a response that decays without overshoot, a repeated root marks the boundary case, and a complex pair oscillates while decaying.
Part III
Circuits at One Frequency
The source is made sinusoidal and held at a single frequency. Differentiation becomes multiplication by , so the network is solved by the methods of Part I once more, now in complex numbers.
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Lecture 12
Sinusoids, Phasors, and Complex Impedance
In the sinusoidal steady state every voltage and current shares the frequency of the source, so only an amplitude and a phase remain to be found. Carrying that pair as a complex number turns differentiation into multiplication by , after which every method of Part I applies unchanged, now in complex arithmetic.
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Lecture 13
AC Power and Maximum Power Transfer
Instantaneous power in an AC circuit oscillates at twice the frequency of the source, so the quantity of interest is its average. Separating the part that is consumed from the part merely exchanged with the storage elements produces real and reactive power, the power factor, and the conjugate-match condition for maximum power transfer.
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Lecture 14
The Power Triangle, Power Factor Correction, and Balanced Three-Phase
Reactive power draws current through the line, and therefore causes loss, while delivering nothing to the load, which is the reason a lagging load is corrected by a shunt capacitor. Complex power adds directly across the loads sharing one bus, and the lecture closes by introducing the balanced three-phase system.
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Lecture 15
Balanced Three-Phase Circuits, Y and Δ
A balanced three-phase system is solved as a single-phase circuit, because balance makes the two remaining phases copies of the first, shifted by . Returning to the full system requires only the relations that connect line quantities to phase quantities, and the Δ connection to the Y.
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Lecture 16
Power in Balanced Three-Phase Systems
Power in a balanced three-phase load takes a single form, times the line voltage times the line current times the power factor, whether the load is connected in Y or in Δ. The work therefore lies in recognizing which quantities the given data are, and in reducing the system to its per-phase circuit.
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Lecture 17
Magnetically Coupled Circuits and the Dot Convention
A current changing in one coil induces a voltage in a second coil linked to it magnetically, although no conducting path joins the two. The dot convention fixes the sign of that induced voltage, after which the coupling is replaced by a dependent source in series with each coil and ordinary phasor analysis finishes the problem.
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Lecture 18
Ideal Transformers, the Dot Convention, and Impedance Matching
An ideal transformer scales voltage by the turns ratio and current by its inverse, so that the power entering equals the power leaving. A load seen through a transformer appears divided by the square of the turns ratio, and that reflected impedance brings source and load into a match that a direct connection cannot achieve.
Part IV
Circuits Across Frequency
The frequency is released and allowed to vary. The same phasor solution, read as a function of , becomes the transfer function, the Bode plot, and the filter.
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Lecture 19
Transfer Functions and the Frequency Response
Writing the element laws in terms of the operator replaces the differential equation of a circuit with algebra, and the ratio of output to input then defines the transfer function. That one function carries the entire behavior of the network, and the response to a sinusoid of any frequency is read from it by the substitution .
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Lecture 20
Frequency Response and Bode Plots
Plotting magnitude in decibels against a logarithmic frequency axis converts a product of factors into a sum of curves. Since every transfer function factors into four standard forms, and each form has a straight- line sketch learned once, a response of any complexity is drawn by adding a small number of known shapes.
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Lecture 21
Adding the Bode Factors, and Reading a Plot Backwards
The addition promised by the logarithmic axes is carried out here in full: each factor is drawn on its own axes, and the curves are then summed ordinate by ordinate, for the magnitude and for the phase. The inverse problem follows, in which a transfer function is recovered from the slopes and breakpoints of a sketch that is given.
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Lecture 22
Filters, First Order and Second Order
A filter is a transfer function chosen for what it passes and what it rejects, and four quantities describe it: type, order, passband gain, and cutoff frequency. The cutoff is located by one condition, that the power delivered has fallen by half, and that condition is applied first to the first-order forms and then to the second-order standard form.
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Lecture 23
Cutoff Frequencies of Second-Order Filters
The half-power condition that located the first-order cutoff by inspection becomes, at second order, a quadratic equation in the square of the frequency. One definition therefore still serves every case: the low-pass, the high-pass, and a numerator that matches no prototype in the table.
Review
Across All Four Parts
The cumulative material, which draws on every part and therefore belongs to none of them.
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Lecture 24
The Cumulative Examination Review, Worked Through
One example is worked for each topic of the course, in the order the review takes them, from superposition and the source equivalents, through the switched circuits and the sinusoidal steady state, to the three-phase load and the transformer match. The rules that these examples apply are stated on the companion reference page.
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Review
The Cumulative Examination Reference
Every rule, procedure, and formula the final examination covers, condensed onto one page and arranged in the order of the course. Nothing is worked here: each section states what is to be done, and points to the lecture that develops it and to the corresponding example in Lecture 24.
Acknowledgment
The author thanks Jennifer Marley and Elena Veety for their help with the preparation of the course.