Equivalent Circuits

Definition: A circuit that has the same electrical behavior outside as some other circuit, but possibly a different internal structure.
Equivalent circuits are used to simplify complex circuits and analyze their behavior.
In this lecture, we will learn about equivalent circuits and how to analyze them.
Topics to be covered include:
- Resistors in series and parallel
- Kirchhoff’s laws
- Voltage and current division
- Thevenin’s and Norton’s theorems

Resistors in Series

When resistors are connected in series:
- The total resistance is the algebraic sum of the individual resistances.
- The same current flows through each resistor.
Formula for total resistance ($R_{\text{Total}}$):
- $R_{\text{Total}} = R_1 + R_2 + R_3 + \ldots$
Example:
- $R_1 = 10 , \Omega, R_2 = 20 , \Omega, R_3 = 30 , \Omega$
- $R_{\text{Total}} = 10 , \Omega + 20 , \Omega + 30 , \Omega = 60 , \Omega$

Resistors in Parallel

When resistors are connected in parallel:
- The reciprocal of the total resistance is equal to the algebraic sum of the reciprocals of the individual resistances.
- The same voltage is applied across each resistor.
Formula for total resistance ($\frac{1}{R_{\text{Total}}}$):
- $\frac{1}{R_{\text{Total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$
Example:
- $R_1 = 10 , \Omega, R_2 = 20 , \Omega, R_3 = 30 , \Omega$
- $\frac{1}{R_{\text{Total}}} = \frac{1}{10 , \Omega} + \frac{1}{20 , \Omega} + \frac{1}{30 , \Omega} = \frac{1}{6} , \text{S}^{-1}$

Kirchhoff’s Laws

Kirchhoff’s Current Law (KCL):
- The total current entering a junction (or a node) is equal to the total current leaving the junction.
Kirchhoff’s Voltage Law (KVL):
- The algebraic sum of the potential differences in any closed loop of a circuit is zero.
KCL and KVL are fundamental laws used to analyze electrical circuits.

Voltage Division

Voltage division is a method used to determine the voltage across a particular resistor in a series circuit.
Formula for voltage across a particular resistor ($V_{\text{Resistor}}$):
- $V_{\text{Resistor}} = \frac{R_{\text{Resistor}}}{R_{\text{Total}}} \times V_{\text{Total}}$
Example:
- $V_{\text{Total}} = 12 , \text{V}, R_{\text{Total}} = 60 , \Omega, R_{\text{Resistor}} = 30 , \Omega$
- $V_{\text{Resistor}} = \frac{30 , \Omega}{60 , \Omega} \times 12 , \text{V} = 6 , \text{V}$

Current Division

Current division is a method used to determine the current flowing through a particular resistor in a parallel circuit.
Formula for current through a particular resistor ($I_{\text{Resistor}}$):
- $I_{\text{Resistor}} = \frac{\frac{1}{R_{\text{Resistor}}}}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots} \times I_{\text{Total}}$
Example:
- $I_{\text{Total}} = 4 , \text{A}, R_{\text{Resistor}} = 30 , \Omega, R_1 = 10 , \Omega, R_2 = 20 , \Omega$
- $I_{\text{Resistor}} = \frac{\frac{1}{30 , \Omega}}{\frac{1}{10 , \Omega} + \frac{1}{20 , \Omega}} \times 4 , \text{A} \approx 1.5 , \text{A}$

Thevenin’s Theorem

Thevenin’s theorem states that any linear electrical network with voltage and current sources can be replaced by an equivalent circuit consisting of a single voltage source ($V_{\text{Th}}$) in series with a single resistor ($R_{\text{Th}}$).
Thevenin’s equivalent circuit simplifies complex circuits for analysis.
Formula for Thevenin’s voltage ($V_{\text{Th}}$):
- $V_{\text{Th}} = V_{\text{Open Circuit}} = V_{\text{Terminal}}$
Formula for Thevenin’s resistance ($R_{\text{Th}}$):
- $R_{\text{Th}} = \frac{V_{\text{Th}}}{I_{\text{Short Circuit}}}$

Norton’s Theorem

Norton’s theorem states that any linear electrical network with voltage and current sources can be replaced by an equivalent circuit consisting of a single current source ($I_{\text{No}}$) in parallel with a single resistor ($R_{\text{No}}$).
Norton’s equivalent circuit simplifies complex circuits for analysis.
Formula for Norton’s current ($I_{\text{No}}$):
- $I_{\text{No}} = I_{\text{Short Circuit}}$
Formula for Norton’s resistance ($R_{\text{No}}$):
- $R_{\text{No}} = R_{\text{Th}}$

Example: Equivalent Circuit 1

Given circuit:
- Total voltage: $V_{\text{Total}} = 24 , \text{V}$
- Total resistance: $R_{\text{Total}} = 10 , \Omega$
Find the Thevenin’s voltage ($V_{\text{Th}}$) and Thevenin’s resistance ($R_{\text{Th}}$) of the circuit.

Example: Equivalent Circuit 2

Given circuit:
- Total current: $I_{\text{Total}} = 2 , \text{A}$
- Total resistance: $R_{\text{Total}} = 20 , \Omega$
Find the Norton’s current ($I_{\text{No}}$) and Norton’s resistance ($R_{\text{No}}$) of the circuit.

Resistor color code
- Each resistor has a pattern of colored bands to indicate its resistance value and tolerance.
- The color code is based on the following colors:
  - Black, Brown, Red, Orange, Yellow, Green, Blue, Violet, Gray, White
Use the color code to determine the resistance value of a resistor.
Example:
- A resistor with the colors: Red, Violet, Yellow, Gold
- The resistance value is: 27kΩ with a tolerance of ±5%

Capacitors
- Capacitors store electrical energy in an electric field.
- They consist of two conductive plates separated by a dielectric material.
- Capacitance is the ability of a capacitor to store charge.
Formula for capacitance (C):
- C = Q / V, where Q is the charge stored on the plates and V is the voltage across the capacitor.
Example:
- A capacitor has a charge of 10μC and a voltage of 5V.
- The capacitance is: C = 10μC / 5V = 2μF

Inductors
- Inductors store electrical energy in a magnetic field.
- They consist of a coil of wire.
- Inductance is the ability of an inductor to store magnetic energy.
Formula for inductance (L):
- L = Φ / I, where Φ is the magnetic flux and I is the current through the coil.
Example:
- An inductor has a magnetic flux of 20mWb and a current of 10A.
- The inductance is: L = 20mWb / 10A = 2mH

Diodes
- Diodes are electronic components that allow current to flow in one direction.
- They have a forward voltage drop and a reverse breakdown voltage.
- Diodes are commonly used to convert AC to DC and regulate voltage.
Example:
- A silicon diode has a forward voltage drop of 0.7V and a reverse breakdown voltage of 50V.
- It is commonly used for rectification and voltage regulation.

Transistors
- Transistors are semiconductor devices that amplify or switch electronic signals.
- They consist of three layers of semiconductor material: emitter, base, and collector.
- Transistors come in different types: NPN, PNP, and Field-Effect Transistors (FETs).
Example:
- A common NPN transistor has a base-emitter voltage of 0.7V and a collector-emitter voltage of 20V.
- It is commonly used in amplifiers and switching circuits.

Op-Amps
- Operational Amplifiers (Op-Amps) are high-gain voltage amplifiers with differential inputs and a single-ended output.
- They are commonly used in analog circuits to perform mathematical operations such as amplification, integration, differentiation, etc.
- Op-Amps have two inputs: non-inverting and inverting, and one output.
Example:
- An Op-Amp with a gain of 100,000 and an input voltage of 10mV.
- The output voltage will be: 10mV * 100,000 = 1V

AC Circuits
- Alternating Current (AC) circuits are circuits that have changing voltage and current over time.
- AC circuits use sinusoidal waveforms to represent the voltage or current.
- AC circuits have parameters such as frequency, amplitude, and phase.
Example:
- An AC circuit has a voltage waveform with a frequency of 50Hz and an amplitude of 10V.
- The voltage waveform will repeat 50 times per second with a maximum value of 10V.

Transformers
- Transformers are electrical devices used to transfer electrical energy from one circuit to another.
- They work on the principle of electromagnetic induction.
- Transformers have two coils: primary and secondary, and they can step up or step down voltage.
Example:
- A transformer has a primary voltage of 120V and a turns ratio of 1:2.
- The secondary voltage will be: 120V * 2 = 240V

Superposition Theorem
- The Superposition Theorem states that the total response in a linear circuit is the sum of the responses due to each individual source acting alone.
- It is used to analyze complex circuits with multiple sources.
Example:
- In a circuit with two voltage sources, V1 and V2, the total response can be calculated by considering the effect of V1 alone, and then the effect of V2 alone, and then adding the results.

Maxwell’s Equations
- Maxwell’s Equations are a set of four fundamental equations that describe how electric and magnetic fields interact.
- They unify the laws of electricity and magnetism.
- Maxwell’s equations include Gauss’s Law, Gauss’s Law for Magnetism, Faraday’s Law, and Ampere’s Law.
Example:
- Gauss’s Law states that the electric flux through a closed surface is proportional to the net electric charge enclosed by the surface. ``