32.3
Resistor in an AC Circuit
An alternating emf or voltage source is needed to supply an alternating current (AC) to a circuit. A coil of wire rotating in a magnetic field at a co…
The alternating current source is a device that supplies a sinusoidally varying voltage or current with time.
A function can describe the sinusoidal voltage or current, where V or I are the voltage or current, respectively, at time t, V0 or I0 are the peak voltage or current, respectively, and ω is the angular frequency.
The sinusoidally varying voltages and currents are represented by rotating vector diagrams.
These vectors rotating counterclockwise at a constant angular frequency are called phasors; the diagrams depicting them are known as phasor diagrams.
Consider a resistor connected across an alternating current voltage source. Recalling Kirchhoff's loop rule, the instantaneous voltage across and the current flowing through the resistor can be determined.
Since the maxima and minima of both curves are reached simultaneously, the curves are said to be in phase.
Therefore, the relationship between current and voltage can be represented using phasor diagrams.
The projections on the vertical axis represent instantaneous voltage and current. Since they are in phase and have the same frequency, their phasors point in the same direction and rotate together.
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Q1: Why are voltage and current in phase in a resistor connected to an AC circuit?
In a resistor, voltage and current reach their maximum and minimum values simultaneously, meaning they are in phase with each other. This occurs because the resistor's behavior is independent of current direction. Using Kirchhoff's loop rule, both the instantaneous voltage across and current through the resistor follow the same sinusoidal pattern, resulting in a phase angle of zero degrees.
Q2: What is a phasor diagram and how does it represent AC quantities?
A phasor diagram uses rotating vectors called phasors to depict sinusoidally varying voltages and currents. These vectors rotate counterclockwise at a constant angular frequency, with their vertical axis projections representing instantaneous values. Phasors are geometric tools, not physical quantities like velocity or momentum, that help visualize and analyze AC circuit behavior over time.
Q3: How do you calculate total voltage in a purely resistive series AC circuit?
In a purely resistive series AC circuit, all voltage drops across individual resistors are in phase with one another. Since they share the same phase angle, you can add all voltage drops algebraically to determine the circuit's overall voltage. This direct addition works because there is no phase shift between any of the resistive elements.
Q4: What mathematical function describes sinusoidal voltage and current in AC sources?
Sinusoidal voltage or current can be described using a function where V or I represents the instantaneous value, V₀ or I₀ represents the peak amplitude, ω is the angular frequency, and t is time. This mathematical representation allows precise calculation of voltage and current at any moment in the AC cycle.
Q5: How do phasors behave when voltage and current are in phase?
When voltage and current are in phase, their corresponding phasors point in the same direction and rotate together at the same angular frequency. This alignment reflects the simultaneous occurrence of voltage and current maxima and minima, making phasor diagrams an effective tool for analyzing resistive AC circuits.
Q6: What is the relationship between branch currents in a purely resistive parallel AC circuit?
In a purely resistive parallel AC circuit, every branch current is in phase with every other branch current. Because all currents share the same phase angle, you can add all individual branch currents together to determine the total circuit current without accounting for phase differences.
Q7: How does a resistor's behavior differ between DC and AC circuits?
In AC circuits, a resistor's behavior is independent of current direction, unlike considerations in some other circuit elements. The resistor maintains the same relationship between voltage and current regardless of whether current flows forward or backward. This consistent behavior allows resistors to be analyzed using straightforward phasor diagrams in rlc series circuits introduction.