Circuit Analysis Demystified

 

Circuit Analysis Study Guide

Quiz

Q: Explain the relationship between charge and current.
A: The current is the rate of change of charge with respect to time. Mathematically, this is expressed as $i_{t} = \frac{dq}{dt}$, where $i_{t}$ is the current, $q$ is the charge, and $t$ is the time.

Q: What is the significance of a positive versus a negative current in circuit analysis?
A: A positive current indicates that positive charges are flowing in the direction of the arrow in the circuit diagram, while a negative current implies positive charges are moving in the opposite direction of the arrow. In effect, it is the same as a positive charge moving the other way.

Q: Describe the difference between a voltage source and a current source.
A: A voltage source is a circuit element that maintains a specific voltage across its terminals, while a current source is an element that always has a specified current flowing through it, regardless of the voltage across it.

Q: What is the difference between a branch, a node, and a loop in a circuit?
A: A branch is a single element or component in a circuit, a node is a connection point between two or more branches, and a loop is any closed path in a circuit that allows you to return to your starting point.

Q: State Kirchhoff's Voltage Law (KVL).
A: KVL states that the sum of the voltages around any closed loop in a circuit must equal zero. This law is based on the conservation of energy for circuits.

Q: State Kirchhoff's Current Law (KCL).
A: KCL states that the total current entering a node must equal the total current leaving that node. This law is based on the principle of conservation of charge for circuits.

Q: What is the relationship between resistance, voltage, and current according to Ohm's Law?
A: Ohm's Law states that the voltage across a resistor is directly proportional to the current flowing through it. It is expressed as $V = RI$ , where $V$ is the voltage, $R$ is the resistance, and $I$ is the current.

Q: What is the difference between conductance and resistance?
A: Conductance is the inverse of resistance, often denoted by $G$, and is measured in Siemens (S). It is defined as $G = \frac{1}{R}$, where $R$ is the resistance, typically measured in ohms (Ω).

Q: Explain the concept of Root Mean Square (RMS) value of a time varying signal.
A: The RMS value of a time-varying signal is calculated by taking the square root of the mean (average) of the square of the signal over one period, which gives the effective value of the signal.

Q: What is the voltage divider rule, and when is it applicable?
A: The voltage divider rule states that, in a series circuit, the voltage across a given resistor is a fraction of the source voltage, proportional to the resistor's resistance. It is applicable when you have resistors connected in series with a voltage source.

Essay Questions

Q: Compare and contrast Thevenin's Theorem and Norton's Theorem, explaining when each is most useful in circuit analysis and providing examples to illustrate each method.

Q: Discuss the concepts of steady-state analysis for sinusoidal circuits and how it differs from transient analysis. Explain how phasors are used in steady-state analysis, including the rules for adding and multiplying phasors, and provide a specific example.

Q: Explain how Laplace transforms are used in circuit analysis, emphasizing their advantages in solving complex circuits. Provide an example illustrating the use of the Laplace transform in analyzing a circuit with both initial conditions and a time-varying source.

Q: Discuss the different types of filters (low-pass, high-pass, band-pass, and band-stop) and their applications. Include an explanation of Bode plots, including the significance of corner frequencies, and use Bode plots to illustrate the response of a low-pass and a high-pass filter.

Q: Describe the concept of stability in circuit analysis, focusing on impulse response stability and bounded-input bounded-output (BIBO) stability, and explain how the poles and zeros of a transfer function influence the stability of a circuit with reference to real-world circuit examples.

Glossary of Key Terms

• Ampere (A): The SI unit of electric current, representing the flow of one coulomb of charge per second.
• Branch: A single element or component in a circuit.
• Capacitance (C): A property of a circuit element (capacitor) that stores energy in an electric field, measured in farads (F).
• Capacitor: A circuit element designed to store electrical energy in an electric field.
• Charge (q): A fundamental property of matter that can be positive or negative, measured in coulombs (C).
• Circuit: A closed path through which electric current can flow.
• Conductance (G): The reciprocal of resistance, measured in siemens (S), which reflects how easily current flows through a component.
• Current (i): The flow of electric charge, measured in amperes (A).
• Current Divider Rule: A technique for finding the current through a resistor in a parallel circuit, where the current divides proportionally based on the conductance of the branches.
• Current Source: A circuit element that supplies a specified current, regardless of the voltage across it.
• Delta (Δ) Circuit: A network of three components (e.g., resistors) connected in a triangular formation.
• Dependent Source: A source (either voltage or current) whose value depends on another voltage or current elsewhere in the circuit.
• Impedance (Z): The total opposition to current flow in an AC circuit, incorporating both resistance and reactance, measured in ohms (Ω).
• Inductance (L): A property of a circuit element (inductor) that stores energy in a magnetic field, measured in Henries (H).
• Inductor: A circuit element designed to store electrical energy in a magnetic field.
• Kirchhoff's Current Law (KCL): The law that states that the sum of currents entering a node must equal the sum of currents leaving the node.
• Kirchhoff’s Voltage Law (KVL): The law that states that the sum of the voltages around any closed loop in a circuit must equal zero.
• Laplace Transform: A mathematical technique used to convert differential equations into algebraic equations in the complex frequency domain (s-domain), simplifying circuit analysis.
• Loop: A closed path in a circuit.
• Node: A connection point between two or more branches in a circuit.
• Norton’s Theorem: A method used to simplify complex circuits by replacing part of a circuit with a current source in parallel with a resistor.
• Ohm's Law: The law that states that the voltage across a resistor is directly proportional to the current flowing through it: V = IR.
• Open Circuit: A circuit configuration where there is no path for current to flow (infinite resistance).
• Phasor: A complex number representing the magnitude and phase of a sinusoidal signal used in AC steady-state analysis.
• Pole: A value of 's' (complex frequency) that causes the magnitude of a function (e.g., transfer function) to tend to infinity. The roots of the denominator of a transfer function are the poles.
• Power (P): The rate at which energy is used or transferred, measured in watts (W).
• Reactance (X): The opposition to current flow in AC circuits due to capacitance or inductance, measured in ohms (Ω).
• Resistance (R): The opposition to current flow, measured in ohms (Ω).
• Resistor: A circuit element designed to oppose the flow of current.
• Root Mean Square (RMS) Value: The effective value of a time-varying signal.
• Short Circuit: A circuit configuration where there is no resistance to current flow (zero resistance).
• Superposition Theorem: A method that allows analysis of a linear circuit with multiple sources by examining the effects of each source independently and then summing the results.
• Thevenin's Theorem: A method used to simplify complex circuits by replacing part of a circuit with a voltage source in series with a resistor.
• Transfer Function (H(s)): A mathematical function that relates the output of a system to its input, often represented in the s-domain.
• Transient Analysis: The analysis of circuits that describes the behaviour of the circuit during the time when the source is switched on or off or any abrupt change in the circuit parameters.
• Voltage (V): The electric potential difference between two points, measured in volts (V).
• Voltage Divider Rule: A technique for finding the voltage across a resistor in a series circuit, where the voltage divides proportionally based on the resistance of the resistors.
• Voltage Source: A circuit element that provides a specified voltage.
• Watt (W): The SI unit of power, representing one joule per second (J/s).
• Wye (Y) Circuit: A network of three components (e.g., resistors) connected in a Y formation.
• Zero: A value of 's' (complex frequency) that causes the magnitude of a function (e.g., transfer function) to be equal to zero. The roots of the numerator of a transfer function are the zeros.

Briefing Document: Circuit Analysis Demystified

1. Introduction & Core Concepts

Focus: This document provides a self-teaching guide to circuit analysis, aiming to demystify complex concepts for learners.

Fundamental Units:

Charge is measured in Coulombs (C).
Current (rate of charge flow) is measured in Amperes (A), where $1A = 1 \frac{C}{s}$.
Voltage is measured in Volts (V), where $1V = 1 \frac{J}{C}$ (Joules per Coulomb).
Power is measured in Watts (W), where $1W = 1 \frac{J}{s}$.

Key Definitions:

Current ( $i_{t}$): The rate of change of charge with respect to time: $i_{t} = \frac{dq}{dt}$ .

Charge ($q$): The integral of current with respect to time: $q = \int i_{t}\ dt$. Definite integrals allow us to calculate charge passed during specific time intervals. For example: $q = \int_{a}^{b} i_{t}\ dt$ provides the charge passed between time 'a' and 'b'.

Voltage ($v_{t}$): The energy required to move a unit charge between two points. Voltage is positive from higher potential to lower potential.

Power ($p_{t}$): The rate at which energy is absorbed or delivered, calculated as $p_{t} = v_{t} i_{t}$.

Conventions:

Current Arrow: Indicates the direction of positive charge flow. If the calculated current is negative, it means the positive charge is flowing in the opposite direction of the arrow, or that it would be equivalent to a positive charge flowing in the opposite direction.

Voltage Polarity: "+" sign indicates higher potential; "-" indicates lower.

Power Absorption/Delivery: If $p = vi \gt  0$, the element absorbs power. If$p = vi \lt  0$, the element delivers power. When calculating power, current arrow relative to voltage signs is key. If arrow goes from + to - of voltage, use a + current; if it goes - to +, use a - current.

Examples:

1. Calculation of current from a time-varying charge function (Example 1-1 and 1-2).
2. Calculation of charge given a time-varying current function (Example 1-3, 1-4 and 1-5).
3. Calculation of power absorption/delivery for various circuit components, (Example 1-8).

2. Circuit Elements & Their Properties

• Voltage Sources: Maintain a specific voltage across their terminals. Can be independent (fixed voltage) or dependent (voltage depends on other elements in the circuit).
• Current Sources: Maintain a specific current flow through it. Can be independent or dependent.
• Resistors (R): Components that impede current flow.
• Ohm's Law: $V = RI$, or $v_{t} = R \cdot i_{t}$: Voltage across a resistor is proportional to current flowing through it.
• Resistance (R): Measured in ohms (Ω).
• Conductance (G): The inverse of resistance, $G = \frac{1}{R}$, measured in siemens (S).
• Power Dissipation: Resistors always absorb power, usually in the form of heat: $P = V \cdot I = \frac{V^2}{R} = R \cdot I^2$.
• Short Circuits: Have zero voltage across them.
• Open Circuits: Have zero current flowing through them.
• Capacitors (C): Store energy in an electric field.
• Capacitance (C): Measured in Farads (F).
• Current-Voltage Relationship: $i_{t} = C\cdot \frac{dV}{dt}$.
• Voltage-Current Relationship $v_{(t)} = v_{(0)} + \frac{1}{C} \int i_{(s)} ds$.
• Energy Stored: $w = \frac{1}{2} Cv^2$.
• Inductors (L): Store energy in a magnetic field.
• Inductance (L): Measured in Henries (H).
• Voltage-Current Relationship: $v = L \frac{di}{dt}$.
• Energy Stored: $w = \frac{1}{2} L\cdot i^2$.
• Mutual Inductance (M): The property where a current in one inductor causes a voltage in a second nearby inductor.

3. Circuit Analysis Techniques & Theorems

• Branches, Nodes, Loops: Fundamental circuit topology concepts.
• Branch: A single element or a set of elements with the same current.
• Node: A connection point between two or more branches.
• Loop: A closed path in a circuit.
• Kirchhoff's Laws: Fundamental laws for circuit analysis:
• Kirchhoff's Current Law (KCL): The sum of currents entering a node equals the sum of currents leaving the node.
• Kirchhoff's Voltage Law (KVL): The sum of voltages around a closed loop equals zero. (Voltage drops in the direction of current flow are positive).
• Resistor Combinations:Series Resistors: Total resistance is the sum of individual resistances: $RT = R1 + R2 + ... + Rn$.
• Parallel Resistors: The reciprocal of the total resistance equals the sum of reciprocals of individual resistances. (For two resistors: $RT = \frac{R1\cdot R2}  {R1 + R2}$)
• Voltage and Current Dividers: Rules for calculating voltage across and current through resistors in series and parallel configurations.
• Voltage Divider: The voltage across resistor j in a series circuit is $v_{j(t)} = \frac{(Rj )}{Req} \cdot  V_{s(t)}$.
• Current Divider: The current through resistor j in a parallel circuit is $Ij = \frac{Gj  }{Geq} \cdot  Is$.
• Thevenin’s Theorem: Simplifies a complex circuit to a voltage source in series with a resistor (Thevenin equivalent).

Steps:

1. Disconnect the outside network.
2. Set independent sources to zero.
3. Measure the resistance at the terminals (Rth).
4. Calculate the open-circuit voltage (Vth).

• Norton's Theorem: Simplifies a complex circuit to a current source in parallel with a resistor (Norton equivalent).
• The Norton resistance is the same as Thevenin equivalent resistance, and the Norton current is the Thevenin voltage divided by the Thevenin resistance.
• Superposition: Used in circuits with multiple sources.
• The total response of a linear circuit is the sum of individual responses due to each source acting alone.
• When analyzing the contribution of one source, other voltage sources are shorted and current sources are opened.
• Millman's Theorem: Simplifies a circuit with multiple parallel voltage sources to a single voltage source in series with a single resistance, when each voltage source is in series with a resistance. $VM = \frac{G1V1 + G2V2 + ... + GnVn}{G1 + G2 + ... + Gn}$. The equivalent resistance RM is the reciprocal of the sum of the conductances. $RM = 1 / (G1 + G2 + ... + Gn)$ .
• Delta-Wye (Δ-Y) Transformation: Enables the transformation of circuits with Δ or Y configurations into each other.
• Bridge Circuits: Used for measuring resistance. (Example: Wheatstone Bridge).

4. AC Circuit Analysis & Phasors

• Sinusoidal Sources: Characterised by frequency (f, in Hz), angular frequency ($\omega= 2\pi f$ in rad/s), and amplitude (V, in Volts).
• RMS (Root Mean Square) Values: Represent the effective value of AC voltage or current: $V_{rms} = \frac{V_{peak}}{\sqrt{2}}$.
• Phasors: A complex number that represents the amplitude and phase of a sinusoidal function.
• Used to simplify analysis of circuits with sinusoidal sources in steady-state.
• A sinusoidal function $A\cdot  sin(\omega t + \varphi)$ is represented as $A \angle  \varphi$.
• Impedance (Z): The AC equivalent of resistance, combining resistance (R) and reactance (X).
• $Z = R + jX$ , where j is the imaginary unit.
• Inductive Reactance: $X_{L} = \omega L$ .
• Capacitive Reactance: $X_{C} = \frac{-1}{(\omega C)}$.
• Ohm's Law for AC circuits: $V = ZI$.
• Admittance (Y): The inverse of impedance: $Y = \frac{1}{Z}$.
• $Y = G + jB$, where G is conductance and B is susceptance.
• Power in AC CircuitsInstantaneous Power: $p_{(t)} = v_{(t)}\cdot i_{(t)}$
• Average Power (P): $P = V_{rms} \cdot  I_{rms} \cdot  \cos(\theta)$, where θ is the phase difference between voltage and current.
• Reactive Power (Q): $Q = V_{rms} \cdot  I_{rms} \cdot  \sin(\theta)$.
• Apparent Power (S): $S = V_{rms} \cdot  I_{rms}$.
• Power Factor: $\cos(\theta) = \frac{P}{S}$. A power factor near 1 is desired (less wasted power).
• Frequency Response: How a circuit behaves over a range of frequencies.

5. Laplace Transforms

• Definition: A mathematical tool for transforming a function of time into a function of a complex variable s.
• Purpose: Converts differential equations to algebraic equations, simplifying circuit analysis.
• Properties:Linearity, time shift, differentiation, integration.
• Inverse Laplace Transform: Transforms a function from the s domain back to the time domain.
• Transfer Function: H(s) = Output(s) / Input(s). Describes how a circuit responds to different inputs.
• Poles and Zeros: Used for analysing circuit stability and behaviour.
• Zeros: The values of s for which the transfer function is zero.
• Poles: The values of s for which the transfer function goes to infinity (denominator zeros).

6. Circuit Stability

Impulse Response Stability: A circuit is impulse response stable if the transfer function remains finite, that is, if $\lim_{t \to \infty }  \left| h(t) \right| \lt  \infty$ . This is often tested by considering zero-input stability.

Bounded Input-Bounded Output (BIBO) Stability: A circuit is BIBO stable if a bounded input results in a bounded output. This can be determined by analyzing the poles of the transfer function.

A circuit is stable if the poles of H(s) are negative and real (lie in the left-hand side of the s plane). If poles are on the imaginary axis or in the right-hand side, the circuit may be unstable.

Sinusoidal Instability: Occurs when poles of H(s) are located on the imaginary axis, meaning the circuit can be driven to ever larger outputs by a sinusoidal input.

7. Filters & Bode Plots

• Filters: Circuits that pass some frequencies and attenuate others.
• Low-Pass: Pass low frequencies, attenuate high frequencies.
• High-Pass: Pass high frequencies, attenuate low frequencies.
• Band-Pass: Pass frequencies within a specific band.
• Band-Stop (Notch): Attenuate frequencies within a specific band.
• Bode Plots: Graphs of a filter's frequency response (magnitude and phase).
• Frequency is represented on a log scale.
• Magnitude is typically represented in decibels (dB). $\left| H(ω) \right|\ dB = 20 \log_{10} \left| H(ω) \right|$
• Butterworth Filters: A type of filter with a maximally flat passband.
• Order of Filter: Determines the steepness of the roll-off (higher order = steeper roll-off)
• Cutoff frequency: Defines where the filter begins to attenuate.
• Attenuation: nth-order Butterworth filters have attenuation of -6n dB/octave.

Key Takeaways

This document provides the basic concepts, definitions, theorems, and techniques used in analysing and understanding the behaviour of electric circuits.

It is a guide which gradually introduces more advanced topics, such as Laplace transforms, stability, and filters, building upon the core foundational concepts.

Emphasis is placed on understanding both the time and frequency domain behaviour of circuits, and the mathematical tools to analyze them.