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Questions and Answers
What characteristic defines resistors connected in series within a resistive network?
What characteristic defines resistors connected in series within a resistive network?
- The same current flows through each resistor. (correct)
- Each resistor has a different current flowing through it.
- The equivalent resistance is the reciprocal of the sum of individual resistances.
- The voltage drop across each resistor is identical.
How is the equivalent resistance calculated for resistors connected in parallel?
How is the equivalent resistance calculated for resistors connected in parallel?
- The reciprocal of the sum of the reciprocals of the individual resistances. (correct)
- The square root of the sum of the resistances.
- The sum of the reciprocals of the individual resistances.
- The product of the resistances divided by their sum.
Consider a complex resistive network. Under what condition is it impossible to simplify the entire network into a single equivalent resistance between two terminals?
Consider a complex resistive network. Under what condition is it impossible to simplify the entire network into a single equivalent resistance between two terminals?
- When the network contains only parallel resistances.
- When the network includes dependent sources. (correct)
- When the network can be fully reduced using repeated series and parallel combinations.
- When the network contains only series resistances.
A resistive cube is constructed with identical resistors of value R along each edge. What is the equivalent resistance between two diagonally opposite corners of the cube?
A resistive cube is constructed with identical resistors of value R along each edge. What is the equivalent resistance between two diagonally opposite corners of the cube?
What is the primary purpose of applying Δ-Y (Delta-Wye) transformations in circuit analysis?
What is the primary purpose of applying Δ-Y (Delta-Wye) transformations in circuit analysis?
In the context of the node-voltage method, what constitutes a 'node'?
In the context of the node-voltage method, what constitutes a 'node'?
Consider a delta network with resistors $R_{ab} = 3\Omega$, $R_{bc} = 6\Omega$, and $R_{ca} = 9\Omega$. What is the value of the equivalent wye resistor $R_2$ connected to the node between $R_{bc}$ and $R_{ab}$?
Consider a delta network with resistors $R_{ab} = 3\Omega$, $R_{bc} = 6\Omega$, and $R_{ca} = 9\Omega$. What is the value of the equivalent wye resistor $R_2$ connected to the node between $R_{bc}$ and $R_{ab}$?
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Study Notes
- A resistive network is an electrical circuit consisting only of resistors and voltage and current sources.
- Resistive networks are analyzed to determine all the voltages and currents in the circuit.
- These networks are linear if the independent source values are constant.
- Key methods of simplifying and analyzing resistive networks include series and parallel combinations, source transformations, and the node-voltage and mesh-current methods.
Series and Parallel Combinations
- Resistors in series have the same current flowing through them.
- The equivalent resistance of series resistors is the sum of their individual resistances: Requivalent = R1 + R2 + ... + Rn.
- Resistors in parallel have the same voltage across them.
- The equivalent resistance of parallel resistors is calculated by the reciprocal of the sum of the reciprocals: 1/Requivalent = 1/R1 + 1/R2 + ... + 1/Rn.
- For two resistors in parallel, the equivalent resistance simplifies to: Requivalent = (R1 * R2) / (R1 + R2).
- Series and parallel combinations are repeatedly applied to simplify a network and find a single equivalent resistance.
Voltage and Current Division
- Voltage division applies to series resistors; the voltage across a resistor is proportional to its resistance relative to the total series resistance.
- The voltage across resistor Ri in a series of n resistors is Vi = (Ri / Requivalent) * Vtotal, where Vtotal is the total voltage across the series combination.
- Current division applies to parallel resistors; the current through a resistor is inversely proportional to its resistance relative to the total parallel resistance.
- The current through resistor Ri in a parallel combination of n resistors is Ii = (G_i / G_equivalent) * Itotal, where Itotal is the total current entering the parallel combination, and G is the conductance (1/R).
- For two resistors in parallel, the current through resistor R1: I1 = (R2 / (R1 + R2)) * Itotal.
Source Transformations
- Source transformation is a technique to replace a voltage source in series with a resistor by a current source in parallel with the same resistor, or vice versa.
- A voltage source V in series with a resistor R can be transformed into a current source I = V/R in parallel with the resistor R.
- Conversely, a current source I in parallel with a resistor R can be transformed into a voltage source V = IR in series with the resistor R.
- Source transformations are useful for simplifying circuits before applying other analysis techniques.
Δ-Y (Delta-Wye) Transformations
- Δ-Y (Delta-Wye) transformation is a technique used to convert a delta (Δ) or pi (π) network of resistors into a wye (Y) or tee (T) network, and vice versa.
- These transformations are useful when series or parallel simplification isn't possible.
- For a Δ to Y transformation:
- R1 = (Rab * Rac) / (Rab + Rac + Rbc)
- R2 = (Rbc * Rab) / (Rab + Rac + Rbc)
- R3 = (Rac * Rbc) / (Rab + Rac + Rbc)
- Where Rab, Rac, and Rbc are the resistors in the delta network, and R1, R2, and R3 are the corresponding resistors in the wye network.
- For a Y to Δ transformation:
- Rab = (R1R2 + R2R3 + R3R1) / R3
- Rbc = (R1R2 + R2R3 + R3R1) / R1
- Rac = (R1R2 + R2R3 + R3R1) / R2
- Where R1, R2, and R3 are the resistors in the wye network, and Rab, Rac, and Rbc are the corresponding resistors in the delta network.
Node-Voltage Method
- The node-voltage method is a technique that uses node voltages as circuit variables.
- A node is a point in a circuit where two or more circuit elements are connected.
- One node is chosen as the reference node (ground), and all other node voltages are measured with respect to this reference.
- Kirchhoff's Current Law (KCL) is applied at each non-reference node, summing the currents entering and leaving the node.
- Ohm's law (I = V/R) is used to express the branch currents in terms of the node voltages.
- The resulting equations are solved for the unknown node voltages.
- Once node voltages are known, branch currents can be easily found using Ohm's law.
Mesh-Current Method
- The mesh-current method is a technique that uses loop currents as circuit variables.
- A mesh is a loop that does not contain any other loops within it.
- Kirchhoff's Voltage Law (KVL) is applied to each mesh, summing the voltages around the loop.
- Ohm's law (V = IR) is used to express the branch voltages in terms of the mesh currents.
- The resulting equations are solved for the unknown mesh currents.
- Once mesh currents are known, branch currents and element voltages can be found.
- The mesh-current method is generally applicable to planar circuits (circuits that can be drawn on a plane without any wires crossing).
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Description
Explore resistive networks consisting of resistors, voltage, and current sources. Learn how to analyze these linear networks using series and parallel combinations. Understand how to calculate equivalent resistance in both series and parallel configurations.
