AC Resistive Circuits: Sinusoidal Voltage & Current

• Alternating Current (AC) sinusoidal voltage applied to a resistor results in a sine wave current flow.
• In a resistive circuit, when the AC voltage is zero, the current is zero.
• In a resistive circuit, when the AC voltage is at its maximum, the current is at its maximum.
• Voltage polarity direction changes cause the current flow direction to reverse.
• Peak and RMS values of AC current and voltage can be used in accordance with Ohm's law.
• Phasor diagrams represent the magnitude and phase angle of sine waves.
• Voltage and current phasors have a phase angle relationship of 0°.
• Current and voltage are in phase in a purely resistive AC circuit.
• Ohm's and Kirchhoff's laws apply to purely resistive AC circuits.
• Circuit parameter values must be in the same terms before calculations in a resistive AC circuit.
• Understanding the rates of change in a sine wave is essential for inductive and capacitive AC circuits.
• The rate of change varies along a sinusoidal waveform, described as the 'Gradient' of the curve.
• The rate of change is fastest at the zero crossing of a sine wave.

Lead and Lag

• 'Phase shift', 'lead', and 'lag' describe LCR circuits.
• Phase shift comparison:
• 90 degrees: A leads B.
• 90 degrees: B leads A.
• 180 degrees: A and B are mirror-images.
• 0 degrees: A and B are in perfect step.

AC and Inductive Circuits

• Inductance in DC circuits is revisited before AC inductance.
• An inductor (L), resistor (R), and DC voltage source are connected in series with a switch (S1).
• No current flows the instant switch (S1) is closed (A-B); the inductor opposes current change initially.
• The rapid rate of change induces a voltage across the inductor, equal in magnitude but opposite in polarity to the applied voltage.
• After five time constants, the induced voltage across the inductor nears zero and the current flowing in the circuit is at its maximum.
• When the switch is changed to position B-C, disconnecting the power source, the inductor opposes this change.
• The inductor induces voltage of the same polarity as source.
• The inductor acts as a power source, supplying current to the resistor through contacts 'B-C'.
• The current flows in the same direction as the normal circuit.
• After five time constants, the energy stored in the inductor dissipates as heat through the resistor and the current falls to zero.
• Understanding the AC circuit effects is important to show the phase relationship between voltage and current.

AC Phase Relationships

• Voltage induced across an inductor (coil) is proportional to current change rate (Faraday's law).
• Coil voltage is at its maximum when the current sine wave has its greatest rate of change which occurs crossing the zero reference line.
• The voltage is at its minimum value, occurring when current changes at its minimum rate (zero at the peak).
• In a purely inductive AC circuit, current lags voltage by 90°, or voltage leads current by 90°.
• In an LR circuit, the relationship between current and voltage is altered.
• The acronym CIVIL is a helpful mnemonic.
• V (Voltage) is the midpoint.
• C represents capacitive loads or circuits.
• L represents inductive loads or circuits.
• Capacitive loads: I (current) comes before V, that is, current leads voltage.
• Inductive loads: I comes after V, that is, current lags voltage.

Inductive Circuit Phase Angle

• The phasor representing current (I) is drawn 90° clockwise from the voltage (V) phasor.
• The inductive circuit phase relationship shows the current lagging the voltage by 90°.

AC and Capacitive Circuits

• Capacitance in DC circuits is reviewed before AC circuits.

• A capacitor (C), resistor (R), and DC voltage source are connected in series with a switch.

• The circuit current is at its maximum when the switch (S1) is closed (A-B): the current is limited by wiring's resistance.

• The Capacitor starts charging, therefore acting as a short circuit.

• When switched to position B-C, current through the resistor will be maximum, but in opposite direction to the original flow.

• After five time constants, the capacitor fully discharges.

• After five time constants, current no longer flows through the resistor.

• The rate at which charge moves (Q/t) determines the rate of voltage changes.

• When the current is changing at its max (zero crossing), the voltage is at its max (peak).

• When the current is changing at its min (zero at the peak), the voltage is at its min (zero).

• The phase relationship shows current I leading voltage V by 90°.

Capacitive Circuits

• Current peaks one quarter cycle before the voltage sine peaks.
• The current leads the voltage by 90° in a capacitive AC circuit (or voltage lags current by 90°.)
• Using CIVIL, in a CR circuit, current leads voltage by 90°.
• The current (I) phasor is drawn 90° anti-clockwise from the voltage (V) phasor.
• The phase represents the current leading the voltage by 90°.
• Circuit components cause a phase shift with constantly changing AC values.
• A 90° phase angle exists between current and voltage when some electrical components cause the current to reach its max 90° before the voltage.
• Current will lead the voltage by 90° for purely capacitive loads.
• Current will lag the voltage by 90° for purely inductive loads.

Inductive Reactance

• The inductor has on the circuit, when an alternating current is linked across the inductor, has an effect similar to a resistor.
• Recall that resistance opposes current flow in a resistive circuit and follows Ohm's law.
• Opposition to current by pure inductance in an AC circuit is given by VL/IL.
• Impedance is expressed in ohms (Ω) and refers to the total opposition to current flow across an AC circuit.
• The voltage/current relationship must be expressed in ohms (Ω) without being called resistance.
• Instead, 'inductive reactance' is used, represented as XL.
• Use this formula to find inductive reactance: XL = VL / IL.

Factors Affecting Inductive Reactance

• Inductive reactance is the opposition to current flow because of inductance.
• Inductance: the change in current causes the inductor to produce opposing voltage (EMF).
• If inductor value remains constant and inductor's current increases, then the voltage across the inductor will increase, opposing the current.
• Increases to CEMF must increase the inductive reactance.
• VL = L * ΔIL
• L = the inductance in henrys
• ΔIL = the rate of change in current through the inductor.
• If L increases, V₁ must increase, and inductive reactance will too (XL).
• XL is proportional to L: changing the inductor (L) size will change the inductive reactance.
• Reactance is affected by frequency: Change in current through inductor (Δl) relates to frequency.
• A higher frequency accelerates current changes and shortens the time for current to reach its max with Δl higher.
• If inductance is constant but frequency increases, inductive reactance also increases.
• XL is proportional to frequency: A change in freq will change the reactance (XL).
• Inductive reactance is proportional to inductance and to frequency. XL is proportional to fL.

Calculating Inductive Reactance

• The formula to find XL when combining the effects of frequency and inductance: XL = 2πfL.
• XL is the inductive reactance in ohms.
• 2π (6.28) relates sine wave derivation from circle.
• f is the frequency in hertz,
• L is the inductance in henrys.
• Valid for sine wave applications.
• XL doesn't depend on the amplitude signal. XL remains constant if inductance/frequency remain and the applied signal amplitude changes.

Capacitive Reactance

• Opposition to current flow is capacitive reactance (XC) in purely capacitive AC circuits.
• Capacitive reactance opposes the current flow due to capacitance.
• A major difference exists between the effects of inductance and capacitance in AC circuits.
• When a capacitor's capacitance increases, the amount of charge it holds increases, increasing the current.
• By applying Ohm's law: Xc = Vc / Ic.
• From this formula, if VC is held constant, and value of Ic increases due to size of capacitor increase, then Xc must decrease.
• "Xc is proportional to 1/C" means any change in C will adjust Xc.
• In capacitive circuit, if rate (frequency) at which voltage is changing increases, the amount of charge moving in a given period of time must also increase.
• Amount of current for a fixed amount of voltage increase indicates the opposition to the current has decreased.
• Opposition to current (Xc) changes inversely based frequency.
• When capacitance is constant:
• Xc approaches a short circuit (low reactance) at high frequencies.
• Xc approaches an open circuit (high reactance) at low frequencies.
• Capacitive reactance is inversely proportional to both capacitance and frequency.
• Xc is proportional to 1/fc

Calculating Capacitive Reactance

• The formula for Xc when combining the effects of frequency and capacitance: Xc = 1 / 2πfc
• Xc is the capacitive reactance in ohms,
• 2π (6.28) is a constant related to the sine waves derivation from a circle.
• f is the frequency in hertz.
• C is the capacitance in farads.
• A capacitance decrease causes capacitive reactance to increase.
• Xc is independent of the applied signal's amplitude.
• Increasing or decreasing the amplitude has no effect to capacitive reactance.

Impedance

• Impedance is the total opposition to current flow across an AC circuit.
• Impedance's abbreviation is Z and is expressed in ohms (Ω).
• In a purely inductive circuit, equivalent to the total inductive reactance.
• In a purely resistive circuit, impedance equals to total circuit resistance.
• In a purely capacitive circuit, the it is equivalent to the value of total capacitive reactance.
• Z = V / I.
• Most electrical circuits aren't purely inductive/capacitive and instead hold some resistance.
• the circuit impedance for component combinations must be able to be determined for resistors, capacitors, and inductors.

Impedance in Series LCR Circuits

• The AC creates changes to reactance and capacity by supplying voltage at certain frequency.
• This frequency adjusts the total impedance (Z) as well as angle across total current (IT) and the voltage applied (VGEN).
• At some frequencies, the circuit will act capacitively and current will lead the voltage applied.
• At others, the circuit will act inductively, and current will lag the applied voltage.
• A single path exists in series for current flow (all 3 components have this). Each component experiences a unique voltage drop.
• The voltage drops on the circuit current will have diverse phase connections as caused by the components that they occur across.
• Ohm's can be used to find voltage drops from individual circuit components, if impedance and current for each component are known.

Determining Impedance of Series LCR Circuits

• For a typical series LCR circuit connected to AC, the reactance of the inductor and capacitor are calculated using the applied input frequency.
• Reactance can be shown graphically on a phasor diagram.

Equivalent Reactance

• Opposite phases that can be subtracted to make a single equivalent reactance value.
• Capacitive/Inductive which acts can be made (capacitively leads voltage), by a resulting reactance which occurs across the circuit.
• If capacitive reactance is larger than the inductive reactance, it cancels out a smaller inductor, and the current leads the circuit's voltage.
• To see id a circuit is inductive or capacitive note larger reactance.
• Any equivalent circuit has the ability to be represented if a phasor is derived from a series.
• Change to frequency results to change inductive and capacitive reactance which de-validates circuits.

Total Circuit and Current

• Total circuit impedance (Z) is the total vector addition of the reactance and resistance phasors.
• The magnitude of the total impedance is determined by how much resistance and reactance there is.
• "Z = √ R2 + XEQ"
• After the total component of a magnitude is known, the determination of the applied voltage can be found by the total current using Ohm's law.
• "IT = VGEN / Z"
• Individual resistance and the drop across the component, can be identified via application via Ohm's Law.
• Total current in formulas via:
  • Resistor: VR1 = IT × R.
• Series LCR is found (capacative and inductive in a circuit) by measurement.

Resistor Component Voltage

• Total current in series through the components.
• Voltage resistor is drawn parallel - in phase.

Diagram

• Mnemonic CIVIL recalls determine phase relationships, across voltage.
• The smaller the impedance, the total resistance in the circuit.
• The phase angle results in a degree via use a combination of polarity y axis.
• The reactance and total values are valid across frequency kHz.

Summary of Series LCR Circuits

• The equations that we can determine:
• Impedance: the total difference between value of X₁ and Xc and the total of the whole value of Z.
• Z = √ R2 + XEQ
• Formulas to determine the total current.
• Іт = V/Z
• Component determination and use.
• VR = IT × R -VL = IT × XL
• Vc = IT × XC
• "0 = tan-1 Vc - VL/VR"

Parallel LCR Circuits

• The circuits and components need the correct parameters (capacitive and inductive).
• As they need voltage, which has many functions.
• The main parameters:
• The phase angle across various branch currents.
• "For components for the supplied inductor, Ohm's Law is imperative."
• From the 3 branch currents , can define all applied power from all generators.
• Total current also can be from Ohm's law.
• Once the phasor can be can be used to make a capacitive or inductive measurement.
• The diagram can help see the phase relationship also can apply to a supply of voltage.

Parallel LCR Circuits Determination

"applied voltage = is defined via frequency/reactance". "XL = 2πfl" "Xc = 1/ 2πfc "

Parallel LCR Circuits - Current Determination

The voltage, resistance is the components for equations.

Formulas is applied:

• "IR = VGEN/R"
• "IC = VGEN/XC"
• "IL = VGEN/XL"

Parallel LCR Circuits - Total Current Determination

• When the amount of current/voltage is known can identify components through:
  • Pythagoras' equation
  • reactive current
• "IT = √ IR + (IC - IL)2"
• inductive - parallel LR- and capacitive - parallel CR - circuits.
• Frequency in Xc and X₁: Xc equals X₁, the total circuit current cancel with a result with of resonant angle.

Parallel LCR Circuits - Relationships

• In a the current that shows the various shifts is defined:
• Diagram and applied Phasors "current phase = applied voltage."
• Drawn degrees across inductor current by phase for applied. formula with current and capacitance.

Formulas: "0 ="tan"-1""IL""IC /" IR""" Changes can result depending in : branch, frequency and impedance. - total.