Sources and References

Primary textbook — W.H. Hayt, J.E. Kemmerly & S.M. Durbin, Engineering Circuit Analysis, 10th International Edition, McGraw-Hill, 2019. Classic alternative — C.K. Alexander & M.N.O. Sadiku, Fundamentals of Electric Circuits, 7th ed., McGraw-Hill, 2021. Online resources — [MIT OpenCourseWare 6.002 Circuits and Electronics (Agarwal & Lang); Stanford EE101A open course notes; AllAboutCircuits.com (open reference, CC-BY); Paul Falstad interactive circuit simulator (falstad.com/circuit)]

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Chapter 1: Foundations — Voltage, Current, and Power

1.1 The Language of Circuits

Before any calculation is possible, you must be comfortable with the vocabulary that circuit engineers use to describe the movement of charge and the exchange of energy. This vocabulary is precise in a way that everyday language is not: every term has a quantitative definition tied directly to SI units, and the definitions determine exactly how you set up every equation in the course.

An electric circuit is a closed path through which charge can flow. The components of the circuit — resistors, capacitors, inductors, sources, and so on — constrain how that charge flow occurs. The circuit engineer’s job is to predict, from the properties of the components and their interconnections, what voltages and currents will appear throughout the network under any prescribed excitation.

1.1.1 Charge and Current

The fundamental quantity of electrostatics is electric charge, measured in coulombs (C). One coulomb corresponds to the aggregate charge carried by roughly \( 6.24 \times 10^{18} \) protons (or the same number of electrons, with sign reversed). In a metal conductor, electrons are free to drift under an applied electric field; their motion constitutes an electric current.

A few conventions deserve immediate attention. First, we always define current as the flow of positive charge — even though in a metal wire the actual charge carriers (electrons) flow in the opposite direction. This conventional current convention is universal in circuit analysis and matches the sign convention used on ammeters and in Kirchhoff’s laws. Second, when we draw an arrow on a circuit element indicating a current \( i \), we are asserting that positive charge flows in the direction of the arrow. If the solved value of \( i \) turns out to be negative, it simply means the actual current flows opposite to the labeled direction — there is nothing wrong with a negative current.

1.1.2 Voltage

Current tells us how fast charge moves; voltage tells us the energy cost of moving it.

Voltage is always defined between two points. Saying “the voltage at node \( a \)” is meaningless without specifying a reference. In practice, one node of the circuit is declared the ground or reference node and assigned voltage zero; all other node voltages are then measured with respect to ground.

The notation \( v_{ab} \) means “voltage of node \( a \) with respect to node \( b \)”, so \( v_{ab} = -v_{ba} \). This antisymmetry is crucial when you write KVL equations.

1.1.3 Power and Energy

The rate of energy delivered to (or extracted from) a circuit element is its instantaneous power:

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

measured in watts (W), where \( v(t) \) is the voltage across the element and \( i(t) \) is the current through it. Energy delivered over a time interval \( [t_1, t_2] \) is

\[ w = \int_{t_1}^{t_2} p(t)\,dt = \int_{t_1}^{t_2} v(t)\,i(t)\,dt. \]

The sign of \( p \) carries physical meaning. When \( p > 0 \), energy flows into the element — the element is absorbing power. When \( p < 0 \), energy flows out of the element — the element is delivering power to the rest of the circuit. The sign depends critically on the choice of reference directions for \( v \) and \( i \), which is where the passive sign convention becomes essential.

Every element in this course is drawn with a voltage polarity (+ and − signs) and a current arrow. Before writing \( p = vi \), always check whether those references satisfy the PSC. If the current arrow enters the minus terminal instead of the plus terminal, the absorbed power is \( p = -vi \).

1.1.4 Independent and Dependent Sources

Circuit sources supply the energy that drives currents through the network.

An independent voltage source maintains a prescribed terminal voltage \( v_s(t) \) regardless of the current drawn from it. An independent current source forces a prescribed current \( i_s(t) \) regardless of the voltage across its terminals. Neither type has any internal constraint linking its output to voltages or currents elsewhere in the circuit.

Dependent (controlled) sources are different: their output depends on a voltage or current measured at another location in the circuit. The four varieties are:

• Voltage-controlled voltage source (VCVS): \( v = \mu\,v_x \)
• Current-controlled voltage source (CCVS): \( v = r_m\,i_x \)
• Voltage-controlled current source (VCCS): \( i = g_m\,v_x \)
• Current-controlled current source (CCCS): \( i = \beta\,i_x \)

Dependent sources appear extensively in transistor and op-amp models. Their presence does not change the analysis method — KCL and KVL still apply — but they do require that the controlling variable be expressed in terms of the node voltages or mesh currents you are solving for.

1.2 Kirchhoff’s Laws

Gustav Kirchhoff stated his circuit laws in 1845. They follow directly from two physical conservation principles: conservation of charge (KCL) and conservation of energy (KVL). Together they are the foundation on which all systematic circuit analysis rests.

Before stating the laws, two topological terms are needed. A node is any point in a circuit where two or more elements are connected. A loop is any closed path through the circuit that starts and ends at the same node without passing through any element more than once.

1.2.1 Applying KVL — Sign Convention

When traversing a loop for KVL, choose a direction (clockwise or counterclockwise — it does not matter which). As you pass through each element:

• If you enter the positive (+) terminal and exit the negative (−) terminal, assign the voltage a negative sign.
• If you enter the negative (−) terminal and exit the positive (+) terminal, assign the voltage a positive sign.

Sum all signed voltages to zero. This approach is consistent with the definition of voltage as potential difference.

1.3 Resistance and Ohm’s Law

Ohm’s Law relates the voltage across a resistor to the current through it. For a resistor with resistance \( R \) (measured in ohms, \( \Omega \)):

\[ v = Ri \]

when \( v \) and \( i \) are referenced according to the passive sign convention. Resistance is always non-negative for a physical resistor. The conductance \( G = 1/R \) (measured in siemens, S) is sometimes more convenient: \( i = Gv \).

The power dissipated by a resistor is always non-negative and appears as heat:

\[ p = vi = i^2 R = \frac{v^2}{R} \geq 0. \]

1.3.1 Series and Parallel Combinations

Two resistors are in series when the same current flows through both. Their equivalent resistance is:

\[ R_{\text{eq}} = R_1 + R_2 + \cdots + R_N. \]

Two resistors are in parallel when the same voltage appears across both. Their equivalent resistance satisfies:

\[ \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_N}. \]

For exactly two resistors in parallel this simplifies to the product-over-sum formula:

\[ R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}. \]

1.3.2 Voltage and Current Division

Voltage division applies to a series string. When a voltage \( v_s \) is applied across resistors \( R_1, R_2, \ldots, R_N \) in series, the voltage across the \( k \)-th resistor is:

\[ v_k = v_s \cdot \frac{R_k}{R_1 + R_2 + \cdots + R_N}. \]

Current division applies to a parallel bank. When a current \( i_s \) is fed into two resistors \( R_1 \) and \( R_2 \) in parallel, the current through \( R_1 \) is:

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

Note the cross-ratio: the current through \( R_1 \) is proportional to \( R_2 \), not \( R_1 \), because a smaller resistance attracts more current. This is frequently a source of sign or index errors — always double-check which resistor appears in the numerator.

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Chapter 2: Systematic Circuit Analysis

2.1 Node-Voltage Method

The node-voltage method (also called nodal analysis) is a systematic procedure for finding all voltages in a circuit with \( N \) nodes. Choose one node as ground; the remaining \( N - 1 \) nodes have unknown voltages \( v_1, v_2, \ldots, v_{N-1} \) measured with respect to ground. Apply KCL at each non-reference node, expressing every branch current in terms of the node voltages using Ohm’s Law. The result is a system of \( N - 1 \) equations in \( N - 1 \) unknowns.

For resistive branches connecting node \( j \) to node \( k \), the current leaving node \( j \) is:

\[ i_{jk} = \frac{v_j - v_k}{R_{jk}}. \]

2.1.1 Supernodes

When an independent voltage source connects two non-reference nodes, neither node voltage is immediately known. The technique is to define a supernode encompassing both nodes and the voltage source, write KCL for the combined supernode (summing currents leaving the entire boundary), and supplement with the constraint equation \( v_j - v_k = V_s \) that the voltage source imposes.

A dependent voltage source is handled identically, except that the constraint equation involves the controlling variable, which must be expressed in terms of node voltages before solving.

2.2 Mesh-Current Method

The mesh-current method (also called mesh analysis or loop analysis) is the dual of nodal analysis. A mesh is a loop that encloses no smaller loops — i.e., a window in the circuit graph. For a planar circuit with \( M \) meshes, assign a mesh current \( i_1, i_2, \ldots, i_M \) circulating around each mesh (by convention, clockwise). Apply KVL around each mesh, expressing every voltage in terms of the mesh currents. The result is a system of \( M \) equations.

For a resistor shared by two adjacent meshes \( j \) and \( k \), the voltage across it as seen from mesh \( j \) is \( R(i_j - i_k) \), where the sign accounts for the opposing current directions.

2.2.1 Supermeshes

When an independent current source is shared by two adjacent meshes, neither mesh KVL equation can be written without knowing the current source voltage, which is unknown. The supermesh technique merges the two meshes into one and writes KVL around the combined outer boundary, then adds the constraint equation relating the two mesh currents to the source current.

2.2.2 Choosing Between Nodal and Mesh

Both methods yield correct results for any circuit. As a rough guide: choose nodal analysis when the circuit has many current sources or voltage sources tied to ground (which directly set node voltages), and choose mesh analysis when the circuit is planar and has many voltage sources (which directly set mesh current sums). Neither method is universally superior; experience and circuit topology drive the choice.

2.3 Linearity and Superposition

A circuit is linear if it satisfies two properties: (1) homogeneity — scaling all sources by a constant \( k \) scales all responses by \( k \); and (2) additivity — the response to a sum of sources equals the sum of responses to each source acting alone. Any circuit composed entirely of resistors, capacitors, inductors, and independent/dependent sources (with linear constitutive relations) is linear.

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Chapter 3: Circuit Theorems — Thevenin, Norton, and Maximum Power Transfer

3.1 Thevenin’s Theorem

In many practical problems you need to know how a complex circuit will behave when a particular load element is connected to its output terminals. Redoing the full circuit analysis for every possible load would be tedious. Thevenin’s theorem says that, from the perspective of those two terminals, any linear circuit can be replaced by a single voltage source in series with a single resistor.

Finding \( R_{\text{Th}} \) requires care when dependent sources are present. In that case, deactivating only independent sources and computing the resistance by inspection does not work, because dependent sources change the effective resistance. Instead, apply a test voltage \( v_T \) at the terminals (with all independent sources deactivated) and measure (or calculate) the resulting current \( i_T \); then

\[ R_{\text{Th}} = \frac{v_T}{i_T}. \]

Alternatively, find both the open-circuit voltage \( V_{\text{oc}} \) and the short-circuit current \( I_{\text{sc}} \), then:

\[ R_{\text{Th}} = \frac{V_{\text{oc}}}{I_{\text{sc}}}. \]

3.2 Norton’s Theorem

Norton’s theorem is the dual of Thevenin’s theorem. The same two-terminal linear circuit can also be replaced by a current source \( I_N \) in parallel with \( R_N = R_{\text{Th}} \).

The two equivalents are related by source transformation:

\[ V_{\text{Th}} = I_N \, R_{\text{Th}}, \qquad I_N = \frac{V_{\text{Th}}}{R_{\text{Th}}}. \]

Source transformation in general allows you to convert any ideal voltage source in series with a resistor into a current source in parallel with the same resistor (and vice versa), leaving the terminal behavior unchanged. This is a powerful simplification technique applicable within a larger circuit, not just at the output terminals.

3.3 Maximum Power Transfer

Given a source with Thevenin equivalent \( (V_{\text{Th}}, R_{\text{Th}}) \) driving a load \( R_L \), what value of \( R_L \) maximizes the power delivered to the load?

The load current is \( i_L = V_{\text{Th}} / (R_{\text{Th}} + R_L) \) and the load power is:

\[ p_L = i_L^2 R_L = \frac{V_{\text{Th}}^2\,R_L}{(R_{\text{Th}} + R_L)^2}. \]

Differentiating with respect to \( R_L \) and setting the derivative to zero gives:

At this matched condition the source and load each dissipate half the total power, so the efficiency is only 50%. This is acceptable in signal-processing and communication systems (where maximum signal transfer is paramount) but unacceptable in power-delivery systems (where efficiency matters far more than load matching).

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Chapter 4: Operational Amplifiers

4.1 The Ideal Op-Amp Model

The operational amplifier (op-amp) is a differential-input, high-gain voltage amplifier packaged as an integrated circuit. Its importance in ECE 140 stems from the fact that, with appropriate feedback networks, op-amps implement virtually any linear signal-processing function — amplification, summation, integration, differentiation, filtering — using nothing more than resistors and capacitors.

Real op-amps (e.g., the LM741 or the TL071) have finite gain, finite input impedance, and nonzero output impedance. The ideal op-amp model simplifies these parameters to their limiting values and is accurate enough for most first-level analysis:

The output voltage of an ideal op-amp is \( v_o = A(v_+ - v_-) \). Because \( A \to \infty \) and the output remains finite in a properly designed circuit, we must have \( v_+ - v_- \to 0 \), which yields the virtual-short-circuit principle:

These two rules — \( i_+ = i_- = 0 \) and \( v_+ = v_- \) — are the only tools needed to analyze every ideal op-amp circuit in this course.

4.2 Classic Op-Amp Configurations

4.2.1 Inverting Amplifier

The inverting amplifier connects the input signal through \( R_1 \) to the inverting input (−), and the feedback resistor \( R_f \) from the output to the inverting input. The non-inverting input (+) is grounded.

By the virtual short, \( v_- = v_+ = 0 \) (a virtual ground). KCL at the inverting node (no current into the op-amp):

\[ \frac{v_{\text{in}} - 0}{R_1} + \frac{v_o - 0}{R_f} = 0 \implies v_o = -\frac{R_f}{R_1}\,v_{\text{in}}. \]

The closed-loop gain is \( A_v = -R_f/R_1 \). The negative sign means the output is 180° out of phase with the input — hence “inverting”. The magnitude of the gain is set entirely by the resistor ratio, independent of the op-amp’s open-loop gain.

4.2.2 Non-Inverting Amplifier

Here the input signal drives the non-inverting terminal (+) directly, and a voltage divider formed by \( R_1 \) (to ground) and \( R_f \) (to the output) provides feedback to the inverting terminal (−).

Virtual short: \( v_- = v_+ = v_{\text{in}} \). The voltage at the inverting terminal through the divider:

\[ v_- = v_o \cdot \frac{R_1}{R_1 + R_f} = v_{\text{in}} \implies v_o = v_{\text{in}}\left(1 + \frac{R_f}{R_1}\right). \]

The gain \( A_v = 1 + R_f/R_1 \geq 1 \) is positive, so the output is in phase with the input. Setting \( R_f = 0 \) and \( R_1 \to \infty \) (or removing \( R_1 \)) gives the voltage follower (unity-gain buffer) with \( v_o = v_{\text{in}} \), extremely high input impedance, and essentially zero output impedance — ideal for impedance matching.

4.2.3 Summing Amplifier

The summing amplifier extends the inverting topology to multiple inputs \( v_1, v_2, \ldots, v_N \), each connected through its own input resistor \( R_k \) to the virtual ground node:

\[ v_o = -R_f\left(\frac{v_1}{R_1} + \frac{v_2}{R_2} + \cdots + \frac{v_N}{R_N}\right). \]

When all input resistors are equal (\( R_1 = R_2 = \cdots = R_N = R \)), the output is simply:

\[ v_o = -\frac{R_f}{R}(v_1 + v_2 + \cdots + v_N). \]

This circuit forms the core of many audio mixing consoles and digital-to-analog converters.

4.2.4 Difference Amplifier

The difference amplifier produces an output proportional to the difference of two inputs. With equal matched resistors \( R_1 = R_2 = R \) and \( R_3 = R_4 = R_f \):

\[ v_o = \frac{R_f}{R}(v_2 - v_1). \]

The common-mode rejection ratio (CMRR) characterizes how well the amplifier suppresses signals common to both inputs (e.g., noise pickup from power lines) while amplifying only the differential signal.

4.2.5 Integrating and Differentiating Amplifiers

Replacing the feedback resistor in an inverting amplifier with a capacitor \( C \) (and keeping \( R_1 \)):

\[ v_o(t) = -\frac{1}{R_1 C}\int_0^t v_{\text{in}}(\tau)\,d\tau + v_o(0). \]

This is the integrating amplifier. Conversely, placing \( C \) in the input path (and \( R_f \) in the feedback):

\[ v_o(t) = -R_f C\,\frac{dv_{\text{in}}}{dt}. \]

This is the differentiating amplifier. Both circuits are important building blocks in analog signal processing, though the differentiator is noise-sensitive in practice because differentiation amplifies high-frequency components.

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Chapter 5: Energy Storage Elements — Capacitors and Inductors

5.1 Capacitors

A capacitor is formed by two conducting plates separated by an insulating dielectric. When a voltage is applied, charge accumulates on the plates, creating an electric field that stores energy. The fundamental constitutive relation is:

\[ q = Cv, \]

where \( C \) is the capacitance measured in farads (F) and \( v \) is the voltage across the capacitor. Since \( i = dq/dt \):

Several properties follow immediately:

1. Voltage continuity: Because \( i \) must remain finite, \( v(t) \) cannot change instantaneously. \( v_C(0^+) = v_C(0^-) \) — the voltage across a capacitor is continuous even when the circuit changes abruptly (e.g., a switch opens or closes).
2. DC steady state: In DC steady state \( dv/dt = 0 \), so \( i = 0 \). A capacitor acts as an open circuit at DC.
3. Energy stored:

\[ w_C = \frac{1}{2}Cv^2. \]

5.1.1 Series and Parallel Capacitors

Capacitors combine opposite to resistors. In series (same charge on each plate):

\[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_N}. \]

In parallel (same voltage):

\[ C_{\text{eq}} = C_1 + C_2 + \cdots + C_N. \]

5.2 Inductors

An inductor is a coil of wire that stores energy in the magnetic field created by current flowing through it. The constitutive relation involves the magnetic flux linkage \( \lambda = Li \):

The dual properties to those of a capacitor:

1. Current continuity: \( i_L(0^+) = i_L(0^-) \) — the current through an inductor cannot change instantaneously.
2. DC steady state: In DC steady state \( di/dt = 0 \), so \( v = 0 \). An inductor acts as a short circuit at DC.
3. Energy stored:

\[ w_L = \frac{1}{2}Li^2. \]

5.2.1 Series and Parallel Inductors

In series (same current):

\[ L_{\text{eq}} = L_1 + L_2 + \cdots + L_N. \]

In parallel (same voltage):

\[ \frac{1}{L_{\text{eq}}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_N}. \]

5.3 Comparing Capacitors and Inductors

The mathematical duality between capacitors and inductors is profound and very useful:

Property	Capacitor	Inductor
Symbol	\( C \) (farads)	\( L \) (henries)
\( v \)-\( i \) relation	\( i = C\,dv/dt \)	\( v = L\,di/dt \)
Stored energy	\( \frac{1}{2}Cv^2 \)	\( \frac{1}{2}Li^2 \)
Continuity	\( v_C \) continuous	\( i_L \) continuous
DC behavior	Open circuit	Short circuit
Series combination	\( 1/C_{\text{eq}} = \sum 1/C_k \)	\( L_{\text{eq}} = \sum L_k \)
Parallel combination	\( C_{\text{eq}} = \sum C_k \)	\( 1/L_{\text{eq}} = \sum 1/L_k \)

Every result you prove for a capacitor circuit has a dual result for an inductor circuit — swapping \( v \leftrightarrow i \), \( C \leftrightarrow L \), short circuit \( \leftrightarrow \) open circuit.

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Chapter 6: First-Order Circuits — Transient Analysis

6.1 Natural Response

Consider a circuit containing a single energy storage element (one capacitor or one inductor) and resistors. After initial conditions are established, the independent sources are removed (or a switch opens/closes) and the circuit evolves according to its stored energy. This evolution is the natural response — it depends only on the circuit’s own properties, not on any forcing function.

6.1.1 RC Natural Response

An initially charged capacitor \( (v_C(0) = V_0) \) discharges through a resistor \( R \). KVL:

\[ v_C + Ri = 0 \implies v_C + RC\,\frac{dv_C}{dt} = 0. \]

This is a first-order linear ODE with constant coefficients. The solution is:

\[ v_C(t) = V_0\,e^{-t/\tau}, \quad t \geq 0, \]

where \( \tau = RC \) is the time constant measured in seconds. At \( t = \tau \), the voltage has decayed to \( V_0/e \approx 0.368\,V_0 \). After \( 5\tau \), it is less than 1% of the initial value — engineering convention says the transient is “essentially complete” after five time constants.

6.1.2 RL Natural Response

An inductor carrying initial current \( i_L(0) = I_0 \) connects to a resistor \( R \). KVL:

\[ L\,\frac{di_L}{dt} + Ri_L = 0 \implies i_L(t) = I_0\,e^{-t/\tau}, \quad t \geq 0, \]

with time constant \( \tau = L/R \). Note the duality: \( \tau = RC \) for RC circuits and \( \tau = L/R \) for RL circuits.

6.2 Step Response

When an independent source is suddenly applied (e.g., a voltage source is switched on at \( t = 0 \)), the circuit response is the step response — the combination of the natural response (which accounts for initial conditions) and the forced response (which the source alone would produce in steady state).

6.2.1 Complete Response of a First-Order Circuit

For any first-order circuit, the general solution has the form:

\[ x(t) = x(\infty) + \bigl[x(0^+) - x(\infty)\bigr]\,e^{-t/\tau}, \quad t \geq 0, \]

where \( x \) is the quantity of interest (voltage or current), \( x(0^+) \) is the initial value, \( x(\infty) \) is the DC steady-state value, and \( \tau \) is the time constant. This three-parameter formula is the single most important result in this chapter — it applies universally to all first-order circuits with step excitation.

6.2.2 Finding \( \tau \) from the Thevenin Equivalent

When a first-order circuit is more complex than a single R and C (or R and L), find the time constant by computing the Thevenin resistance seen by the energy-storage element with all independent sources deactivated:

\[ \tau = R_{\text{Th}}\,C \quad \text{(RC circuit)}, \qquad \tau = \frac{L}{R_{\text{Th}}} \quad \text{(RL circuit)}. \]

This reduces any first-order circuit to the standard form, regardless of how many resistors or dependent sources are present.

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Chapter 7: Second-Order Circuits

7.1 The RLC Circuit

When a circuit contains both a capacitor and an inductor, two energy storage elements are present and the governing equation is second-order. The series RLC circuit — a resistor \( R \), inductor \( L \), and capacitor \( C \) all in series with a voltage source — is the canonical example.

Applying KVL and using \( i = C\,dv_C/dt \):

\[ LC\,\frac{d^2 v_C}{dt^2} + RC\,\frac{dv_C}{dt} + v_C = v_s(t). \]

Dividing by \( LC \) and defining the natural frequency \( \omega_0 = 1/\sqrt{LC} \) and damping coefficient \( \alpha = R/(2L) \):

\[ \frac{d^2 v_C}{dt^2} + 2\alpha\,\frac{dv_C}{dt} + \omega_0^2\,v_C = \frac{v_s(t)}{LC}. \]

The damping ratio is \( \zeta = \alpha/\omega_0 \). The character of the natural response depends on whether the roots of the characteristic equation \( s^2 + 2\alpha s + \omega_0^2 = 0 \), namely

\[ s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}, \]

are real and distinct, real and equal, or complex conjugates.

7.2 Natural Response of the Series RLC Circuit

7.2.1 Overdamped Case: \( \alpha > \omega_0 \) (i.e., \( \zeta > 1 \))

Both roots \( s_1, s_2 \) are real and negative. The natural response decays without oscillation:

\[ v_C(t) = A_1\,e^{s_1 t} + A_2\,e^{s_2 t}. \]

The constants \( A_1 \) and \( A_2 \) are determined from the two initial conditions \( v_C(0^+) \) and \( \dot{v}_C(0^+) = i(0^+)/C \).

7.2.2 Critically Damped Case: \( \alpha = \omega_0 \) (i.e., \( \zeta = 1 \))

The characteristic equation has a repeated root \( s_1 = s_2 = -\alpha \). The general solution is:

\[ v_C(t) = (A_1 + A_2 t)\,e^{-\alpha t}. \]

Critical damping provides the fastest return to equilibrium without oscillation — a desirable property in many control and measurement systems.

7.2.3 Underdamped Case: \( \alpha < \omega_0 \) (i.e., \( \zeta < 1 \))

The roots are complex conjugates: \( s_{1,2} = -\alpha \pm j\omega_d \), where the damped natural frequency is

\[ \omega_d = \sqrt{\omega_0^2 - \alpha^2}. \]

The natural response oscillates while decaying:

\[ v_C(t) = e^{-\alpha t}\bigl(B_1\cos(\omega_d t) + B_2\sin(\omega_d t)\bigr). \]

This is the ringing behavior familiar from LC tank circuits and tuned filters.

7.3 Step Response of the RLC Circuit

When a DC source is applied at \( t = 0 \), the complete response is:

\[ v_C(t) = v_C(\infty) + v_{\text{natural}}(t), \]

where \( v_C(\infty) \) is the DC steady-state value (the particular solution for a step input equals the DC forced response, since \( v_C \to V_s \) as \( t \to \infty \) for a series RLC with a step source).

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Chapter 8: Sinusoidal Steady State and Phasors

8.1 Sinusoidal Signals

Sinusoidal voltages and currents are the workhorses of electrical power systems and communication systems. A general sinusoid is characterized by three parameters:

\[ v(t) = V_m\cos(\omega t + \phi), \]

where \( V_m \) is the amplitude (peak value), \( \omega = 2\pi f \) is the angular frequency in rad/s, and \( \phi \) is the phase angle in radians (or degrees). The period is \( T = 1/f = 2\pi/\omega \).

The root-mean-square (RMS) value is the equivalent DC value that delivers the same average power to a resistor:

\[ V_{\text{rms}} = \frac{V_m}{\sqrt{2}} \]

for a pure sinusoid. All AC power quantities in ECE 140 are expressed using RMS values.

8.2 Phasor Representation

Analyzing sinusoidal circuits directly in the time domain — differentiating and integrating trigonometric functions — is cumbersome. The phasor technique transforms the problem into complex algebra, which is far more tractable.

The key insight is Euler’s formula: \( e^{j\theta} = \cos\theta + j\sin\theta \). A sinusoid \( v(t) = V_m\cos(\omega t + \phi) \) is the real part of the complex exponential \( V_m e^{j(\omega t + \phi)} = V_m e^{j\phi} e^{j\omega t} \).

The crucial property of phasors is that differentiation in the time domain corresponds to multiplication by \( j\omega \) in the phasor domain:

\[ \frac{d}{dt}\bigl[V_m\cos(\omega t + \phi)\bigr] \longleftrightarrow j\omega\,\mathbf{V}. \]

This converts differential equations into algebraic equations — the same simplification the Laplace transform provides, but restricted to sinusoidal steady state.

8.3 Impedance

Applying phasor analysis to the constitutive relations of circuit elements:

Resistor: \( v = Ri \implies \mathbf{V} = R\mathbf{I} \). The impedance of a resistor is \( Z_R = R \) (real, frequency-independent).

Inductor: \( v = L\,di/dt \implies \mathbf{V} = j\omega L\,\mathbf{I} \). The impedance is \( Z_L = j\omega L \) (purely imaginary, increases with frequency).

Capacitor: \( i = C\,dv/dt \implies \mathbf{I} = j\omega C\,\mathbf{V} \), so \( \mathbf{V} = \frac{1}{j\omega C}\mathbf{I} \). The impedance is \( Z_C = \frac{1}{j\omega C} = \frac{-j}{\omega C} \) (purely imaginary, decreases with frequency).

The admittance \( \mathbf{Y} = 1/\mathbf{Z} = G + jB \) (siemens) is the phasor-domain analog of conductance.

Once voltages and currents are expressed as phasors and elements as impedances, all DC analysis techniques — KCL, KVL, nodal analysis, mesh analysis, Thevenin/Norton, superposition — apply without modification, but with complex arithmetic.

8.4 AC Circuit Analysis

The procedure for sinusoidal steady-state analysis is:

1. Express all sources as phasors at the operating frequency \( \omega \).
2. Replace resistors with \( R \), inductors with \( j\omega L \), and capacitors with \( 1/(j\omega C) \).
3. Apply any DC analysis method (nodal, mesh, Thevenin, etc.) using complex algebra.
4. Convert phasor results back to time-domain sinusoids.

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Chapter 9: AC Power Analysis

9.1 Instantaneous and Average Power

For a circuit element with voltage \( v(t) = V_m\cos(\omega t + \theta_v) \) and current \( i(t) = I_m\cos(\omega t + \theta_i) \), the instantaneous power is:

\[ p(t) = v(t)\,i(t) = V_m I_m\cos(\omega t + \theta_v)\cos(\omega t + \theta_i). \]

Using the product-to-sum identity:

\[ p(t) = \frac{V_m I_m}{2}\cos(\theta_v - \theta_i) + \frac{V_m I_m}{2}\cos(2\omega t + \theta_v + \theta_i). \]

The first term is constant; the second oscillates at twice the supply frequency. The average power (real power), obtained by averaging over one cycle:

\[ P = \frac{V_m I_m}{2}\cos(\theta_v - \theta_i) = V_{\text{rms}}\,I_{\text{rms}}\cos\phi, \]

where \( \phi = \theta_v - \theta_i \) is the power factor angle. The quantity \( \cos\phi \) is the power factor (pf).

9.2 Complex Power

The apparent power is \( S = |\mathbf{S}| = V_{\text{rms}}\,I_{\text{rms}} \) in volt-amperes (VA). The power factor is \( \text{pf} = P/S = \cos\phi \).

Physical interpretation:

• Real power \( P \) is the actual power consumed (converted to heat or mechanical work). It is the economically relevant quantity.
• Reactive power \( Q \) represents energy that oscillates between the source and the reactive elements (inductors and capacitors). Inductors absorb positive \( Q \); capacitors supply negative \( Q \) (or equivalently, absorb negative \( Q \)). Reactive power does no net work but causes extra current to flow, increasing losses in transmission lines.
• A unity power factor (\( \phi = 0 \)) means voltage and current are in phase; all power is real.

9.3 Power Factor Correction

Industrial loads are often inductive (motors, transformers), with lagging power factors (\( \phi > 0 \)). A lagging pf causes excessive current for a given real power, increasing \( I^2 R \) losses in supply lines and forcing utilities to install oversized infrastructure.

Power factor correction adds a capacitor in parallel with the load to supply reactive power locally, reducing the reactive power demand from the source. The required capacitance is:

\[ C = \frac{Q_C}{\omega\,V_{\text{rms}}^2}, \]

where \( Q_C = P\,(\tan\phi_1 - \tan\phi_2) \) is the reactive power the capacitor must supply to improve the power factor angle from \( \phi_1 \) to \( \phi_2 \).

9.4 Conservation of Power

For any circuit, complex power is conserved: the total complex power supplied by all sources equals the total complex power absorbed by all elements. This is Tellegen’s theorem applied to complex power. In particular:

\[ \sum_{\text{all elements}} P_k = 0, \qquad \sum_{\text{all elements}} Q_k = 0. \]

This conservation principle is an extremely powerful check: if your computed powers do not balance, there is a calculation error somewhere.

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Chapter 10: Frequency Response and Bode Plots

10.1 Transfer Functions and Frequency Response

The transfer function of a linear circuit is the ratio of output phasor to input phasor as a function of frequency:

\[ H(j\omega) = \frac{\mathbf{V}_{\text{out}}(j\omega)}{\mathbf{V}_{\text{in}}(j\omega)}. \]

It is a complex function of \( \omega \) with magnitude \( |H(j\omega)| \) and phase \( \angle H(j\omega) \). Plotting these as functions of frequency gives the frequency response of the circuit.

The frequency response is related to, but distinct from, the Laplace-domain transfer function \( H(s) \): the frequency response is obtained by evaluating \( H(s) \) along the imaginary axis, i.e., substituting \( s = j\omega \).

10.2 Filters

Filters select certain frequency ranges and attenuate others. The four basic types, defined by their passband location:

Low-pass filter (LPF): passes low frequencies, attenuates high frequencies. A simple first-order RC LPF has the transfer function:

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

where \( \omega_c = 1/(RC) \) is the cutoff frequency (−3 dB frequency). At \( \omega = \omega_c \), the magnitude is \( 1/\sqrt{2} \approx 0.707 \) and the phase is −45°.

High-pass filter (HPF): passes high frequencies, attenuates low frequencies. A series RC circuit with the output taken across the resistor:

\[ H(j\omega) = \frac{j\omega RC}{1 + j\omega RC} = \frac{j(\omega/\omega_c)}{1 + j(\omega/\omega_c)}. \]

Band-pass filter (BPF): passes a band of frequencies around a center frequency. A series RLC with output across R:

\[ H(j\omega) = \frac{j\omega(R/L)}{\omega_0^2 - \omega^2 + j\omega(R/L)}. \]

The bandwidth \( B = R/L = \omega_0/Q \) where \( Q = \omega_0 L/R \) is the quality factor.

Band-stop (notch) filter: attenuates a band of frequencies. Realized by taking the output across L and C in series (with R).

10.3 Bode Plots

A Bode plot is a frequency-response plot using logarithmic scales. The magnitude is plotted in decibels:

\[ |H|_{\text{dB}} = 20\log_{10}|H(j\omega)|, \]

and the frequency axis is logarithmic. The phase \( \angle H(j\omega) \) in degrees is plotted on a linear scale against logarithmic frequency.

Bode plots are drawn using asymptotic approximations — straight-line approximations that are exact at low and high frequencies and have bounded errors near the break points (poles and zeros):

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Chapter 11: Two-Port Networks (Optional / Supplemental)

11.1 Two-Port Parameters

Many practical circuits — amplifiers, filters, transmission lines — are characterized by their input-output behavior rather than their internal topology. A two-port network is described by two pairs of terminals (ports): a port is a pair of terminals through which current can enter and leave. The network’s behavior is fully characterized by a set of two-port parameters that relate the port voltages and currents.

The four common parameter sets are:

• Impedance (z) parameters: \( \mathbf{V} = \mathbf{Z}\,\mathbf{I} \)
• Admittance (y) parameters: \( \mathbf{I} = \mathbf{Y}\,\mathbf{V} \)
• Hybrid (h) parameters: used for transistor small-signal models
• Transmission (ABCD) parameters: useful for cascaded networks

Two-port theory provides a systematic framework for analyzing interconnected systems and is particularly valuable when individual blocks (an amplifier, a filter, a matching network) are characterized separately and then cascaded.

11.2 Y-Parameters and Reciprocity

The admittance (y) parameters are found by short-circuiting each port in turn:

\[ I_1 = y_{11}V_1 + y_{12}V_2, \qquad I_2 = y_{21}V_1 + y_{22}V_2. \]

For a network containing only passive elements (resistors, capacitors, inductors — no dependent sources), the network is reciprocal: \( z_{12} = z_{21} \) and \( y_{12} = y_{21} \). When active elements (transistors modeled as dependent sources) are present, reciprocity generally does not hold and all four parameters must be measured independently.

11.3 Hybrid Parameters and Transistor Models

The hybrid (h) parameters mix voltages and currents as independent and dependent variables:

\[ V_1 = h_{11}I_1 + h_{12}V_2, \qquad I_2 = h_{21}I_1 + h_{22}V_2. \]

For a bipolar junction transistor (BJT) in common-emitter configuration, the h-parameters have direct physical interpretations: \( h_{11} = h_{ie} \) is the input impedance, \( h_{12} = h_{re} \) is the reverse voltage gain, \( h_{21} = h_{fe} = \beta \) is the forward current gain, and \( h_{22} = h_{oe} \) is the output admittance. This is why h-parameters appear prominently in electronics courses that follow ECE 140.

11.4 Cascaded Two-Ports and Transmission Parameters

When two-port networks are cascaded — the output port of one feeding the input port of the next — the ABCD (transmission) parameters are the natural choice because the overall transmission matrix is simply the product of the individual matrices:

\[ \begin{pmatrix} V_1 \\ I_1 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} V_2 \\ -I_2 \end{pmatrix}. \]

For a cascade of two networks with transmission matrices \( \mathbf{T}_1 \) and \( \mathbf{T}_2 \), the overall matrix is \( \mathbf{T} = \mathbf{T}_1 \mathbf{T}_2 \). This multiplicative property makes ABCD parameters indispensable in the analysis of transmission lines, ladder filters, and multi-stage amplifiers.

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Chapter 12: Review and Integration

12.1 Connecting the Major Themes

Looking back over the course, several deep connections run through all the material.

The first connection is between DC and AC analysis. Every technique developed for resistive DC circuits — KCL, KVL, nodal analysis, mesh analysis, Thevenin/Norton, superposition — applies without modification to sinusoidal AC circuits when resistors are generalized to impedances and voltages/currents are represented as phasors. This universality is not accidental: it follows from the linearity of the circuit equations and the special properties of sinusoidal excitation.

The second connection is the time-frequency duality. The transient behavior of circuits (natural and step responses, time constants, oscillation frequencies) and the frequency-domain behavior (impedance vs. frequency, filter cutoff frequencies, resonance) are two faces of the same underlying differential equation. The time constant \( \tau = RC \) corresponds to a corner frequency \( \omega_c = 1/(RC) \) in the frequency domain. The damped natural frequency \( \omega_d \) in the transient response of an underdamped RLC circuit corresponds to the peak of the band-pass frequency response.

The third connection is energy and power. At DC, energy is either stored (in capacitors and inductors) or dissipated (in resistors). In the sinusoidal steady state, real power is dissipated, reactive power oscillates, and the ratio between them is captured by the power factor. At resonance, the reactive powers of the inductor and capacitor cancel exactly, so the circuit presents a purely resistive load to the source.

12.2 A Systematic Problem-Solving Approach

Successful circuit analysis requires discipline in applying a consistent methodology. The following approach works for virtually any problem in this course:

1. Identify the circuit type. Is it DC? First-order transient? Second-order? Sinusoidal AC? The type determines which tools to deploy.
2. Label all unknowns with consistent reference directions. Draw a clear diagram.
3. Choose an analysis method (nodal, mesh, Thevenin, superposition). Apply it systematically.
4. Enforce initial conditions (for transient problems) or boundary conditions (for AC problems).
5. Check the answer. Verify units, check limiting cases, verify power conservation.

12.2.1 Worked Strategy Example — Choosing Nodal vs. Mesh

Consider a circuit with five nodes, three voltage sources (two of which are tied to ground), and two current sources. Counting: with two ground-tied voltage sources, two node voltages are immediately known, leaving only two unknowns for nodal analysis. Mesh analysis, by contrast, would require identifying all windows and dealing with two current sources as supermeshes. Nodal analysis is clearly preferable here.

Contrast with a circuit that is a planar ladder network with five meshes and only one current source but four voltage sources in mesh branches. Each voltage source directly constrains a mesh current sum, leaving only a few genuine unknowns. Mesh analysis wins.

Developing this judgment — recognizing quickly which method will result in fewer equations — is a hallmark of an experienced circuit analyst.

12.2.2 Complex Impedance Arithmetic

Working with phasor-domain circuits requires fluency in complex arithmetic. Recall:

For addition: \( (a + jb) + (c + jd) = (a+c) + j(b+d) \).

For multiplication: \( (a + jb)(c + jd) = (ac - bd) + j(ad + bc) \).

For division (rationalize denominator by multiplying by conjugate):

\[ \frac{a + jb}{c + jd} = \frac{(a+jb)(c - jd)}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + j\frac{bc - ad}{c^2 + d^2}. \]

For series impedances: simply add. For parallel impedances use the product-over-sum (two elements) or the reciprocal formula (many elements). When doing parallel combinations of complex impedances, the arithmetic becomes involved — working in admittance form (\( \mathbf{Y} = 1/\mathbf{Z} \)) is often faster for parallel combinations since admittances add.

12.3 Key Formulas for Rapid Reference

The table below collects the most frequently used relationships in the course.

Quantity	Formula
Ohm’s Law	\( v = Ri \)
Series resistance	\( R_{\text{eq}} = \sum R_k \)
Parallel resistance	\( 1/R_{\text{eq}} = \sum 1/R_k \)
Capacitor \( i \)-\( v \)	\( i = C\,dv/dt \)
Inductor \( v \)-\( i \)	\( v = L\,di/dt \)
First-order response	\( x(t) = x(\infty) + [x(0^+) - x(\infty)]e^{-t/\tau} \)
RC time constant	\( \tau = R_{\text{Th}}C \)
RL time constant	\( \tau = L/R_{\text{Th}} \)
Second-order \( \omega_0 \)	\( \omega_0 = 1/\sqrt{LC} \)
Impedance (inductor)	\( Z_L = j\omega L \)
Impedance (capacitor)	\( Z_C = 1/(j\omega C) \)
Average power	\( P = V_{\text{rms}}I_{\text{rms}}\cos\phi \)
Complex power	\( \mathbf{S} = P + jQ = \mathbf{V}_{\text{rms}}\mathbf{I}_{\text{rms}}^* \)
Inverting amplifier gain	\( A_v = -R_f/R_1 \)
Non-inverting gain	\( A_v = 1 + R_f/R_1 \)
Thevenin resistance	\( R_{\text{Th}} = V_{\text{oc}}/I_{\text{sc}} \)
Max power to load	\( p_{\max} = V_{\text{Th}}^2/(4R_{\text{Th}}) \) at \( R_L = R_{\text{Th}} \)

12.4 Laboratory Connection

The five laboratory experiments in ECE 140 are not administrative exercises — they are designed to make the abstract formulas above viscerally concrete. In Lab 1 (Voltage and Current), you measure node voltages and branch currents in a simple resistive network and verify that your KCL and KVL equations are satisfied within measurement uncertainty. This experience calibrates your intuition: you discover that a 1% resistor tolerance can shift node voltages by a few percent, and that your oscilloscope probe’s 10 MΩ input impedance slightly loads the circuit under test.

In Lab 2 (Resistance), you measure equivalent resistance of series and parallel combinations and experimentally determine the Thevenin resistance of a loaded network by plotting the load line and finding the short-circuit current and open-circuit voltage. This is the Thevenin theorem brought to life: the equivalent circuit you sketched on paper accurately predicts the load behavior over a wide range of \( R_L \).

Lab 3 (Circuit Analysis Methods) asks you to verify the superposition theorem by measuring partial responses and checking that they add to the total. It also reinforces nodal analysis: you build a three-node resistive network, predict all node voltages by solving your nodal equations, then measure them with a multimeter.

Lab 4 (Operational Amplifiers) uses the ideal op-amp model you learned in Chapter 4. You build an inverting amplifier, measure its gain as a function of frequency, and discover the gain-bandwidth product — the real op-amp’s open-loop gain falls with frequency so that \( A_v \times f \) is roughly constant, limiting the closed-loop bandwidth. This is the clearest illustration that the ideal model has limits.

Lab 5 (AC Signals) brings together phasors, impedance, and power factor. You drive an RC or RL circuit with a sinusoidal source from a function generator, use an oscilloscope to measure the phase shift between voltage and current, and compute both the apparent power and the power factor. These measurements close the loop between the abstract phasor formalism and what you can observe on an oscilloscope screen.

12.5 Common Pitfalls and How to Avoid Them

Several categories of error appear repeatedly in student work on ECE 140 exams and assignments.

Sign errors in KVL. The most common mistake is inconsistent application of the sign convention when traversing a loop. The safest remedy is to always mark + and − polarity on every element before writing any equations, then strictly follow the rule: subtract when entering the + terminal while traversing, add when entering the − terminal. Never try to do this “in your head.”

Forgetting initial conditions. In transient problems, students often solve the differential equation correctly but evaluate the constants \( A_1 \) and \( A_2 \) using the wrong time reference. Remember: the initial conditions \( v_C(0^+) \) and \( i_L(0^+) \) must be evaluated in the post-switching circuit topology, not the pre-switching topology (though the values of \( v_C \) and \( i_L \) themselves carry over from \( t = 0^- \)).

Phasor phase errors. A sine wave \( A\sin(\omega t + \phi) \) is not the same phasor as \( A\angle\phi ^\circ \). Always convert sines to cosines before extracting the phasor. Equivalently, subtract 90° from the phase of any sine-referenced signal.

Dependent source handling in Thevenin. Deactivating only independent sources and computing resistance by inspection does not work when dependent sources are present — they change the effective resistance. Always use the test-source method or the \( V_{\text{oc}}/I_{\text{sc}} \) formula when dependent sources appear in the network.

Power is not superimposable. As noted in Chapter 2, superposition applies to voltages and currents but not to power. Exam problems sometimes ask for the power delivered by a source in a multi-source circuit. You must find the total current through and total voltage across the source (after applying superposition to get all voltages and currents), then compute \( p = vi \) once.