Introduction to AC Resistive and Capacitive Circuits and Basic Circuits

AC Resistive Circuits

• Definition: An AC resistive circuit is an alternating current circuit containing only resistance ($R$). It does not contain inductive components (coils, inductors) or capacitive components (capacitors).

• Behavior and Function:     * The electrical behavior of these circuits is relatively simple and predictable compared to other AC circuits.     * They serve as the foundational learning guide for more complex circuits involving inductance ($L$) and capacitance ($C$).

• Voltage and Current Relationship:     * The relationship is straightforward: current through the circuit changes direction proportional to the voltage.     * When the AC voltage increases toward its positive peak, the current follows and increases toward its positive peak.     * When the voltage decreases back toward zero and becomes negative, the current follows the exact same pattern.     * Voltage and current rise and fall together.

• Phase Relationship:     * In a purely resistive circuit, voltage and current are said to be in phase.     * In phase means the peaks, zero crossings, and negative peaks of both the voltage and current waveforms occur at the exact same instant in time.     * There is zero phase shift between voltage and current.

• Energy and Power Dissipation:     * Resistors do not store energy. Unlike capacitors (which store energy in an electric field) and inductors (which store energy in a magnetic field), resistors oppose current flow and convert electrical energy directly into heat.     * This process is known as power dissipation.     * Since no energy storage occurs, there is no delay or displacement between the voltage and current waveforms.

• Applying Ohm’s Law:     * Ohm’s Law applies to AC resistive circuits in the same way it does to DC circuits.     * Formula: $I = \frac{V}{R}$.     * When analyzing AC circuits, technicians typically refer to RMS (Root Mean Square) or effective values, which represent the effective value of the AC waveform. This allows for AC calculations to be handled similarly to DC calculations.

• Real-World Examples:     * Electric heating elements (e.g., toasters, space heaters).     * Incandescent light bulbs.     * In a toaster, the red heating elements are resistive loads that convert energy to heat. In bulbs, the resistive element converts energy to light and heat.

Characteristics of Resistive AC Waveforms

• Visual Representation (Figure 14-2):     * A graph of a purely resistive circuit shows the voltage waveform (often purple) and current waveform (often green) perfectly aligned in time.     * The sine waves both cross zero at the same instant and reach maximum positive/negative peaks simultaneously.

• Amplitude/Magnitude:     * While the waveforms are in phase, they do not usually have the same height or amplitude on a graph because they represent different quantities (Volts vs. Amperes).     * The relationship between the magnitudes is determined by the resistance value through Ohm’s Law.

• Phase Shift: A purely resistive circuit is unique because it features a zero phase shift. Current neither leads nor lags the voltage.

Series and Parallel AC Resistive Circuits

• Effective Value Definition: The amount of AC voltage that produces the same degree of heat as the equivalent DC voltage value. It is essentially the DC equivalent.

• Calculations in Parallel Circuits:     * In a parallel AC circuit, the total voltage ($E_T$) is equal across all resistors ($E_R1 = E_R2 = E_T$).     * Example Case:         * $E_T = 120\,V$         * $R_1 = 470\,\Omega$         * $R_2 = 1000\,\Omega$         * $I_{R1} = \frac{120}{470} = 0.26\,A$ (or $260\,mA$).         * $I_{R2} = \frac{120}{1000} = 0.12\,A$ (Note: Transcript mentions $0.2\,A$ and then $120\,mA$, however, mathematically $\frac{120}{1000} = 0.12\,A$).

• Kirchhoff’s Voltage Law: The total applied voltage is equal to the sum of the individual voltage drops across resistors in series.

• Power Characteristics:     * Power formula: $P = I \times E$ (measured in Watts).     * Power represents the rate at which electrical energy is converted into other forms (heat).     * Because power is a product of current and voltage, its value reaches its peak when voltage/current reach their peaks and hits zero when they are at zero.     * The Power Curve: In a resistive circuit, the power curve never goes below the reference line (never becomes negative). Even during the negative half of the AC cycle, the product of a negative voltage and a negative current is positive. Thus, the resistor is always dissipating energy as heat.     * Real Power Formula: $P = V_{RMS} \times I_{RMS}$.

Capacitive AC Circuits

• Definition: An alternating current circuit containing a capacitor.

• Energy Storage: Capacitors store electrical energy in an electrical field. Unlike resistors, they do not dissipate energy as heat; they store it and release it back into the cycle.

• Phase Relationship (ICE):     * In a capacitive circuit, current leads voltage by $90^{\circ}$ (one-quarter of a cycle).     * Current reaches its peak value before the voltage does.     * Phase difference occurs because the capacitor must charge before voltage builds across it.     * ICE Acronym: Current ($I$) leads Voltage ($E$) in a Capacitive ($C$) circuit.

• Current Flow Appearance:     * Electrons do not actually pass through the dielectric of the capacitor.     * The movement of electrons from one plate to the other as the capacitor charges and discharges creates the appearance of current flow in the circuit.

• Capacitive Resistance (Capacitive Reactance):     * Represented by the symbol $X_C$.     * Formula: $X_C = \frac{1}{2\pi fC}$.     * $X_C$: Capacitive resistance in Ohms.     * $\pi$: Constant (approx. $3.14$).     * $f$: Frequency in hertz ($Hz$).     * $C$: Capacitance in farads ($F$).

• Relationship with Frequency:     * There is an inverse relationship between frequency and capacitive reactance.     * As frequency increases, $X_C$ decreases (higher frequencies pass easier).     * As frequency decreases, $X_C$ increases (lower frequencies are restricted/blocked).

• Applications: Electronic filters, timing circuits, coupling signals, amplifier stages, and power factor correction in medical equipment.

RC Networks and Filters

• Resistor-Capacitor (RC) Networks: Used for filtering, decoupling, DC blocking, or phase shift circuits.

• Filter Definition: A circuit that discriminates between frequencies, allowing some to pass while attenuating (weakening) others.

• Cutoff Frequency ($f_c$):     * The point where the output voltage is approximately $70.7\%$ of the input voltage (correlating to $-3\,dB$).     * Formula: $f_c = \frac{1}{2\pi RC}$.

• Low Pass Filter:     * Allows low frequencies to pass with little opposition while attenuating high frequencies.     * Consists of a resistor and capacitor in series, with the output taken across the capacitor.     * At high frequencies, $X_C$ is low, causing most voltage to drop across the resistor, leaving little at the output.

• High Pass Filter:     * Allows high frequencies to pass while attenuating low frequencies.     * Output is taken across the resistor ($R$).     * At low frequencies, $X_C$ is high, blocking the signal from reaching the resistor/output.

• RC Decoupling Network: Used when AC is superimposed over DC. A low-pass filter allows the low-signal DC to pass while eliminating AC noise (oscillations, transient spikes).

RC Phase Shift Networks

• Function: Shifts the phase of an AC signal so the output waveform occurs slightly earlier or later than the input.

• Leading Phase Shift Network:     * Input is applied across the capacitor; output is taken across the resistor.     * Because current leads voltage in capacitors, and the resistor voltage follows the current, the output voltage leads the input voltage.

• Lagging Phase Shift Network:     * Input is applied across the resistor; output is taken across the capacitor.     * Capacitors resist sudden voltage changes (they must charge/discharge), delaying the output voltage relative to the input.

• Cascading RC Networks:     * A single RC network is typically limited to a phase shift of less than $60^{\circ}$.     * Multiple stages can be connected ("cascaded") to achieve larger shifts (e.g., three $60^{\circ}$ stages for a total of $180^{\circ}$).     * Limitation: Each stage reduces signal amplitude through voltage division. An amplifier is often added after cascading stages to restore signal strength.

AC Measurement Instruments

• Analog Meters:     * Moving Coil / D'Arsonval Meter: Originally designed for DC. Measuring AC directly is difficult as the needle would bounce with the oscillations.     * Bridge Rectifier: To measure AC on a DC-based movement, a bridge rectifier (four diodes in a diamond shape) converts AC to pulsating DC (rectification).     * Iron Vane Meter: Designed specifically for AC. Uses a fixed and a movable iron vane. AC through a coil creates a magnetic field that magnetizes both vanes with the same polarity, causing them to repel. This repulsion rotates the pointer regardless of current direction.

• Digital Multimeters (DMMs):     * Modern replacement for analog meters.     * Internally rectifies AC to DC for measurement.     * Displays RMS or Average values.     * Formulae:         * $V_{RMS} = V_{Peak} \times 0.707$         * $V_{Peak} = V_{Peak-to-Peak} \times 0.5$         * $V_{RMS} = V_{Peak-to-Peak} \times 0.3535$         * $V_{Average} = V_{Peak} \times 0.637$

• Clamp-on Meters:     * Measures current through a wire without breaking the circuit by detecting the magnetic field around the conductor.     * Uses a split-core transformer. The magnetic field from the AC wire induces voltage in the meter's internal coil.     * Used primarily for high-power AC applications (e.g., power lines).

• Oscilloscope:     * The most versatile BMET diagnostic tool. It displays signals as waveforms on a graph (Voltage vs. Time).     * Provides data on frequency, duration, phase relationship, waveform shape, and noise.     * Cathode Ray Tube (CRT) Components:         1. Electron Gun: Fires a stream of electrons forward.         2. Phosphor Screen: Glows when struck by electrons, creating a visible trace.         3. Vertical Deflection Plates: Moves the beam up/down based on signal amplitude.         4. Horizontal Deflection Plates: Moves the beam left/right (controlled by the Sweep Generator) to represent time.         5. Power Supply: Provides voltages for the gun and amplifiers.     * Graticule: The grid pattern on the screen, typically measured in centimeters or divisions, used to calculate voltage and period.

• Frequency Counter:     * Measures repetition rate in hertz ($Hz$).     * Internal Process: Input Signal -> Signal Conditioner (cleans waveform) -> Main Gate/Counter (counts cycles) -> Display.     * The Time Base (internal clock) determines the length of the counting window.

• Bode Plotter:     * Named after HW Bode.     * Produces a graph of frequency response (Gain and Phase Shift vs. Frequency).     * Highly useful for analyzing filter circuits in software like MultiSIM or Electronic Workbench.

Questions & Discussion

• Student (Mike): Suggested the correlation of "Eli for ICE."

• Instructor Note: Acknowledged Mike’s input regarding the "ICE" mnemonic (Current leads voltage in a capacitive circuit) and its counterpart "ELI" (Voltage leads current in an inductive circuit).