Core Principles of Electric Circuit Theory
Thevenin’s Theorem
Thevenin’s theorem states that any linear, bilateral network with voltage sources and resistances can be replaced by an equivalent circuit consisting of a single voltage source (called Thevenin voltage, Vth) in series with a single resistance (called Thevenin resistance, Rth) when viewed from the terminals of the load.
Steps to Find Thevenin’s Equivalent
- Remove the load resistor (the part across which you want to find the equivalent).
- Find the open-circuit voltage across the load terminals. This is Thevenin’s voltage (Vth).
- Deactivate all independent sources:
- Replace voltage sources with short circuits.
- Replace current sources with open circuits.
- Calculate the equivalent resistance seen from the load terminals. This is Thevenin’s resistance (Rth).
- Draw the equivalent circuit: Vth in series with Rth connected to the load.
Norton’s Theorem
Norton’s theorem states that any linear, bilateral network with voltage or current sources and resistances can be replaced by an equivalent circuit consisting of a single current source (called Norton current, In) in parallel with a single resistance (called Norton resistance, Rn) when viewed from the load terminals.
Steps to Find Norton’s Equivalent
- Remove the load resistor.
- Find the short-circuit current across the load terminals. This is Norton’s current (In).
- Find the equivalent resistance (Rn) just like in Thevenin’s theorem (with sources deactivated).
- Draw the equivalent circuit: In in parallel with Rn connected to the load.
Passive Filters
Passive filters are electrical circuits that use only passive components like resistors (R), capacitors (C), and inductors (L) to allow signals of certain frequencies to pass while attenuating (blocking) unwanted frequencies.
Key Characteristics
- They do not require any external power supply to operate.
- They rely entirely on passive elements (no active components like transistors or op-amps).
- They control the frequency response of a signal.
Types of Passive Filters
Low-Pass Filter (LPF)
- Allows low-frequency signals to pass through.
- Attenuates high-frequency signals above a certain cut-off frequency (fc).
High-Pass Filter (HPF)
- Allows high-frequency signals to pass.
- Blocks low-frequency signals below the cut-off frequency (fc).
Band-Pass Filter (BPF)
- Allows signals within a specific frequency band to pass.
- Blocks frequencies outside this band.
Band-Stop Filter (BSF) or Notch Filter
- Attenuates signals within a certain frequency band.
- Allows frequencies outside this band to pass.
Star-Delta Transformation
In electrical circuit analysis, we often encounter complex resistive networks that cannot be simplified using simple series-parallel combinations. To overcome this, Star-Delta (Y-Δ) transformations provide a systematic way to convert a three-terminal star (Y) network into an equivalent delta (Δ) network and vice versa.
Network Configurations
Star (Y) Network
A Star (Y) network consists of three resistors (R1, R2, R3) connected to a common central node. Each resistor connects this central node to an external terminal (A, B, C).
Delta (Δ) Network
A Delta (Δ) network forms a closed loop of three resistors (RAB, RBC, RCA). Each resistor connects two terminals directly, with no central node.
Purpose of Star-Delta Transformation
- To simplify resistive networks that are neither purely series nor parallel.
- To make analysis easier using basic circuit laws (Ohm’s and Kirchhoff’s).
- Widely used in three-phase power systems and bridge circuits.
Advantages
- Simplifies network analysis by reducing complex configurations.
- Makes it easier to calculate current, voltage, and power in different parts of the circuit.
- A fundamental tool in AC and DC circuit analysis.
Conclusion
Star-Delta transformations are essential in electrical circuit theory. By converting star networks to delta and vice versa, we can analyze and solve circuits that otherwise seem complex. They are particularly useful in three-phase systems and balanced load analysis, making them a vital part of any electrical engineer’s toolkit.
Classification of Systems
A system in circuits and systems refers to any arrangement that takes an input and gives an output. Systems can be classified based on different properties:
Linear vs. Non-Linear Systems
Linear systems follow the superposition principle, meaning the response to multiple inputs is the sum of the responses to each input. Circuits with only resistors, capacitors, and inductors are linear. Non-linear systems do not follow this principle, like circuits with diodes and transistors.
Time-Invariant vs. Time-Variant Systems
Time-invariant systems have properties that do not change with time. For example, a fixed resistor or an RLC circuit. In time-variant systems, the system properties change over time, like circuits with time-varying components such as switches.
Causal vs. Non-Causal Systems
Causal systems depend only on present and past inputs; they do not rely on future inputs. Physical electrical circuits are causal. Non-causal systems depend on future inputs, like ideal filters.
Static vs. Dynamic Systems
Static or memoryless systems have an output that depends only on the present input, like a resistor where V=IR. Dynamic systems (with memory) have an output that depends on past inputs, like a capacitor, whose voltage depends on the integral of past currents.
Stable vs. Unstable Systems
Stable systems produce a bounded output for a bounded input, known as BIBO stability, like an RLC circuit in a steady state. Unstable systems can have outputs that grow indefinitely even for small inputs.
Continuous-Time vs. Discrete-Time Systems
Continuous-time systems have input and output signals defined at every instant of time, like analog circuits. Discrete-time systems have input and output signals defined only at specific time intervals, like digital filters.
These classifications help in analyzing and designing circuits and understanding their behavior in various applications.
Laplace and Z-Transforms
The Laplace Transform
The Laplace Transform is a mathematical tool used to convert a time-domain signal, f(t) (where t ≥ 0), into a complex frequency-domain function, F(s), where s is a complex variable (s = σ + jω).
The Laplace Transform helps to simplify the analysis of linear time-invariant (LTI) systems by converting differential equations in the time domain into algebraic equations in the s-domain. It is widely used in circuit analysis, control systems, and signal processing because it makes solving linear differential equations easier and helps in determining the system behavior in the frequency domain.
The Z-Transform
The Z-Transform is a similar mathematical tool used for discrete-time signals. It converts a discrete-time sequence, x[n], into a complex frequency-domain representation, X(z), where z is a complex variable.
The Z-Transform helps to analyze discrete-time systems, such as digital filters and digital signal processing applications. It converts difference equations in the time domain into algebraic equations in the z-domain. This makes it easier to analyze the stability and frequency response of digital systems.
Conclusion
Both Laplace and Z-Transforms are important tools in engineering. The Laplace Transform is used for continuous-time signals and systems, while the Z-Transform is used for discrete-time signals and systems. They simplify the analysis of linear systems by converting complex differential or difference equations into simpler algebraic equations in their respective domains.