Notes on AC Circuits

Generation of Alternating Voltages

• Alternating voltages are generated by generators, functioning similarly to bicycle dynamos.

• Generators convert kinetic energy into electrical energy.

• When a conductive loop moves through a magnetic field, a voltage is induced.

• In generators, the conductive loop turns, causing the induced voltage's value and direction to change constantly, exhibiting a sinusoidal dynamic characteristic.

• Generators have a rotor with numerous conductive loops; more loops increase the induced voltage.

• The magnetic field is produced by magnetic poles in the stator; large generators use electromagnets instead of permanent magnets.

• When a conductor moves through a magnetic field, free electrons move, and the Lorentz force deflects them, creating an electric potential between the conductor's ends.

Key Variables in AC Technology

• Identifiers for voltage curves:

  • Amplitude (û): Maximum value of the curve.

  • Peak-to-peak value (upp): Difference between the maxima of the curve.

  • Period (T): Time for a curve to repeat.

  • Frequency (f): Number of oscillations in one second.

  • Rate of propagation (c): Speed at which a fixed point on the wave travels, dependent on the medium (e.g., $299792$ km/s for electromagnetic waves in a vacuum).

  • Wavelength (λ): Distance travelled by an electromagnetic wave in one period.

• Angular frequency (ω): Indicates the rotational speed in electrical engineering.

  • Expressed in radians, indicating the angle of rotation (α) of an AC or DC vector.

  • The angular frequency (ω) indicates the angle by which a vector rotates in one second.

Ohmic Resistance in an AC Circuit

• Experiment: Investigate current, voltage, and power characteristics in an ohmic resistance in an AC circuit.

• Current (i) is measured indirectly as uR2 via the current-sensing resistor R2.

• Total voltage (u) is measured directly at ohmic resistances R1 and R2.

• Parameters are set on a function generator and oscilloscope.

• Instantaneous values of u and i are calculated, and electrical power (p) is the product of voltage and current.

• Current and voltage curves intersect the t-axis simultaneously and reach their maximum together, indicating they are in phase.

• Power (p) is represented by the area below the average value.

• RMS values:

  • RMS value of an alternating variable in an ohmic resistance equals the direct variable that would generate the same power.

  • Average power (P) corresponds to RMS values.

  • The RMS value of the voltage is calculated as: $U_{rms} = \frac{û}{\sqrt{2}} = 0.7û$

• Frequency response: Ohmic resistance remains constant with varying frequencies; power consumption remains the same.

Coil in an AC Circuit

• Experiment: Investigate voltage, current, and power characteristics in a coil in an AC circuit.

• Current (i) is measured indirectly as uR1 by the current-sensing resistor R1.

• Voltage (uL1) is measured directly across coil L1.

• Parameters are set on the function generator and oscilloscope.

• Instantaneous values of uL1 and i are calculated, and electrical power (qL1) consumed by coil L1 is the product of voltage and current.

• Curves of voltage (uL1) and current (i): Represented by:

  • $uL1(t) = ûL1 \cdot cos(ωt)$

  • $i(t) = î \cdot sin(ωt)$

• Power curve (qL1): Represented by:

  • $qL1(t) = uL1(t) \cdot i(t) = û \cdot î \cdot sin(ωt) \cdot cos(ωt) = \frac{1}{2} \cdot û \cdot î \cdot sin(2ωt)$

• Power values determined as the product of voltages and currents in AC circuits can have negative components. Power in AC circuits is expressed in units of var (Volt-Ampère-réactif) instead of watts (W).

• The current and voltage have a 90° phase-displacement; current lags behind the voltage, and power oscillates at twice the frequency.

Inductive Resistance

• Experiment: Determine the inductive resistance of a coil and its response in an AC circuit at various frequencies.

• Inductive resistance: Electrical resistance generated by a coil in an AC circuit; calculated using the quotient UL / IL.

• Voltage (U) across coil L and current (I) are measured with a multimeter.

• Inductive resistance is determined using: $XL = \frac{UL}{IL}$, an extension of Ohm's law for alternating variables.

• Frequency response: Inductive resistance depends on frequency; inductance is an intrinsic characteristic of the coil and is frequency-independent. The inductance is calculated at different frequencies.

• Response in a DC circuit: The coil acts like a wire-wound resistor.

• Inductance (L) does not depend on frequency.

• N = Number of windings of the coil

• AL = Material-dependent coil constant

Series Connection of R and L I

• Experiment objective: Determine voltages across resistor (R) and inductor (L1), and use them to form the voltage triangle. Determine resistances to form the resistance triangle.

  • Voltage triangle: A vector diagram that allows voltages to be added graphically.

  • Resistance triangle: A vector diagram that allows resistances to be added graphically.

• $\hat{u}<em>R = \hat{u}</em>{R1} + \hat{u}_{R2}$

• Voltages ûL and ûR1 are measured and displayed using the oscilloscope.

• Voltage values are inserted into equations to calculate RMS values U, UL, UR1, and phase displacement φ.

  • $U<em>R = U</em>{R1} + U_{R2}$

• The voltage triangle, an extension of Kirchhoff's second law, graphically adds voltage vectors; UR points in the same direction as the current, while UL exhibits a 90° phase displacement.

• The determined resistance values R, Z and XL are used to generate corresponding resistance vectors to form the resistance triangle.

• The resistance triangle calculates the total resistance of the series connection; resistances are represented as vectors in AC technology.

Series Connection of R and L II

• Experiment aim: Determine phase displacement φ between û and î, voltage û across R and L, current î, and power S in the circuit; also, determine the power characteristic and power triangle.

  • Power triangle: A vector diagram which allows powers to be added graphically.

• Voltages ûL and ûR1+R2 are measured with the oscilloscope; ûL is across coil L, and ûR1+R2 calculates the circuit current.

• Voltage and current values determine instantaneous values, which are used to plot the power characteristic.

• Plotting the power characteristic: The power characteristic is plotted on the basis of the measurement data. An analysis of this characteristic reveals areas in which the power is negative. These areas represent the inductive reactance.

• RMS values of voltage and current are calculated to determine P (active power), QL (reactive power), and S (apparent power).

  • Active power: The actual power consumed by the load.

  • Reactive power: Power that shuttles between the coil and AC source without being consumed.

  • Apparent power: Power apparently consumed, calculated from the product of effective current and voltage.

• The determined power values P, Q and S are used to generate corresponding power vectors to form the power triangle.

• The curve no longer oscillates symmetrically about the x axis. The purely reactive power comprises the negative component. If it is subtracted from the area of the power curve above the zero line, the remaining area represents the active power P at any point in time. The apparent power S and active power P oscillate about the average value of P.

Parallel Connection of R and L

• Experiment goal: Determine currents IR and IL through resistor R and inductor L, using values with total current I to form the current triangle; calculate conductivities for the conductivity triangle; draw vector diagram and line chart.

  • Current Triangle: a vector diagram for adding currents graphically.

  • Conductivity Triangle: a vector diagram for adding conductivities graphically.

• Currents in line paths with ohmic and inductive resistance are measured, as are voltages across current-sensing resistors.

• Determined current values are inserted into equations to calculate RMS values of IL and IR, and the phase displacement φ.

• The determined RMS values of IL, I and IR are used to generate corresponding current vectors and form the current triangle. The current triangle constitutes an extension of Kirchhoff's first law (current law) for alternating currents. Instead of the current values, the current vectors are added together here. IR points in the same direction as the voltage. IL exhibits a phase displacement of 90°.

• The adjacent formulae are used to calculate the rms value U as well as the conductivities BL, Y and G.

• The determined conductivities BL, Y and G are used to generate corresponding vectors and form the conductivity triangle.

• The characteristics of the currents îR, îL, î and the voltage û are displayed in a line chart. The line chart is based on the following vector diagram:

Capacitor in an AC Circuit

• Experiment: To investigate the current, voltage, and power characteristics in a capacitor in an AC circuit.

• Voltage across capacitor C and current î flowing in the circuit are measured; current is measured indirectly by current-sensing resistor R1.

• Parameters are set using a function generator and oscilloscope.

• Instantaneous values of uC and i are calculated, and electrical power qC consumed by the capacitor is the product of voltage and current.

• Curves of voltage (uC) and current (i): Represented by:

  • $uC(t) = ûC \cdot sin(ωt)$

  • $iC(t) = îC \cdot cos(ωt)$

• Power curve (qC): Oscillates at twice the frequency:

  • $qC(t) = ûC(t) \cdot i(t) = ûC \cdot î \cdot sin(ωt) \cdot cos(ωt) = \frac{1}{2} \cdot ûC \cdot î \cdot sin(2ωt)$

• Current and voltage have a mutual phase displacement of 90°, i.e. the voltage lags behind the current. The power curve oscillates at twice the frequency. In the positive range, energy is received from the mains in order to generate the required electrical field in the capacitor. In the negative range, an ideal capacitor supplies the stored energy back to the mains.

Capacitive Resistance

• Experiment: Determine capacitive resistance of a capacitor and its response in an AC circuit at various frequencies.

  • Capacitive resistance: Electrical resistance generated by a capacitor in an AC circuit.

• Voltage (ûC) across capacitor C and current (î) flowing through the circuit are measured with a multimeter.

• Capacitive resistance is determined using: $XC = \frac{UC}{IC}$, an extension of Ohm's law applying to alternating variables.

• The capacitive resistance depends on the frequency in the AC circuit. The capacitance, however, is an intrinsic characteristic of the capacitor and does not depend on the frequency. The adjacent formula is used to calculate the capacitance at different values of the frequency f.

• Capacitive resistance produces a current wave which leads the voltage wave by 90°. The capacitive resistance decreases as the frequency increases, and has a hyperbolic characteristic. Increasing C and f causes the charging current to increase as well. At very high frequencies, the capacitor has practically the same effect as a short-circuit.

Series Connection of R and C I

• Experiment Objective: Determine voltages across resistor R and capacitor C to form the voltage triangle; determine resistances to form the resistance triangle.

  • Voltage triangle: A vector diagram which allows voltages to be added graphically.

  • Resistance triangle: A vector diagram which allows resistances to be added graphically.

• Voltages across capacitor C and resistor R are measured and displayed using the oscilloscope.

• Voltage values are inserted into equations to calculate RMS values U, UC, UR1, and phase displacement φ.

• The voltage triangle, an extension of Kirchhoff's second law, graphically adds voltage vectors; UR1 points in the same direction as the current, while UC exhibits a 90° phase displacement.

• Values of resistances R, Z, and XC are calculated to form the resistance triangle; the resistance triangle calculates the total resistance of the series connection; resistances are represented as vectors.

Series Connection of R and C II

• Experiment: Voltages across R and C and current in the circuit are measured. The measured values are used to determine the instantaneous values, phase displacement as well as the powers for recording the power characteristic. These values are to be used as a basis for forming the power triangle.

  • Power triangle: A vector diagram which allows powers to be added graphically.

• Voltages û(R1+C1) and ûR2 are measured with the oscilloscope; û(R1+C1) is across the series connection of R1 and C1, and ûR2 calculates the circuit current.

• Voltage and current values determine instantaneous values, used to plot the power characteristic.

• The power characteristic is plotted on the basis of the measurement data. An analysis of this characteristic reveals areas in which the power is negative. These areas represent the capacitive reactance.

• RMS values of voltage and current are calculated to derive P (active power), QC (reactive power), and S (apparent power).

  • Active power: Actual power consumed by the load.

  • Reactive power: Power between the coil and AC source without consumption.

  • Apparent power: Power apparently consumed, calculated from the product of effective current and voltage.

• Power values P, QC, and S are used to generate power vectors to form the power triangle.

• The curve no longer oscillates symmetrically about the x axis. The purely reactive power comprises the negative component. If it is subtracted from the area of the power curve above the zero line, the remaining area represents the active power P at any point in time. The apparent power S and active power P oscillate about the average value of P.

Parallel Connection of R and C

• Experiment: Determine currents through resistor IR and capacitor IC; use values with total current I to form the current triangle. Calculate conductivities to form the conductivity triangle; draw vector diagram and line chart.

  • Current triangle: A vector diagram for adding currents graphically.

  • Conductivity triangle: A vector diagram for adding conductivities graphically.

• Currents in line paths with ohmic and capacitive resistance are measured, as are voltages across current-sensing resistors.

• Determined current values are inserted into the specified equations in order to calculate the rms values of IC and IR as well as the phase displacement φ.

• The determined values of IC, I and IR are used to generate corresponding current vectors and form the current triangle. The current triangle constitutes an extension of Kirchhoff's first law (current law) for alternating currents. Instead of the current values, the current vectors are added together here. IR points in the same direction as the voltage. IC exhibits a phase displacement of 90°.

• The adjacent formulae are used to calculate the rms value U as well as the conductivities BC, Y and G.

• The determined conductivities BC, Y and G are used to generate corresponding vectors and form the conductivity triangle.

• The characteristics of the currents îR, îC, î and the voltage û are displayed in a line chart. The line chart is based on the following vector diagram:

Series Connection of R, L and C I

• Experiment: Determine voltages across R, L, and C, and the phase displacement. Use measured values to calculate RMS values and form the voltage triangle and resistance triangle.

  • Voltage triangle: A vector diagram allows voltages to be added graphically.

  • Resistance triangle: A vector diagram allows resistances to be added graphically.

• As the oscilloscope only has two channels, voltages across L and R are measured first, followed by C and R, to establish a reference.

• Voltage values are inserted into equations to calculate RMS values of UL, UC, UR, and total voltage U; these values are used to form the voltage triangle.

• Rms values of UC, UL, UR and U are used to generate corresponding vectors and form the voltage triangle. UR points in the same direction as the current. UC and UL each exhibit a phase displacement of 90°, but in different directions.

• Values of resistances Z, XC, XL, and the current I in the circuit are calculated to form the resistance triangle.

• Calculated resistance values R, XL, XC and Z are used to generate corresponding resistance vectors and form the resistance triangle.

• As the voltages across the capacitor and coil are phase-opposed, they are subtracted mutually, whereby the larger of the two values determines whether the resulting voltage leads or lags behind the current. The resistance triangle represents the values of the resistors in the form of vectors, in accordance with the phase angles of the individual voltages.

Series Connection of R, L and C II

• Experiment: calculate apparent power and plot its characteristic, prepare a vector diagram for the circuit, and use it as a basis for plotting a line chart. The oscilloscope is triggered externally in this experiment.

  • Vector diagram: A vector diagram represents instantaneous voltages and currents in the form of vectors.

  • Line chart: displays voltages and currents as functions of time.

• Total voltage U and voltage are measured across the resistor R; measurements are taken in two steps to determining U and UR, due to the common COM terminal on the oscilloscope channels.

• Voltage U already measured is superimposed on the voltage UR. This makes it possible to represent U and UR together. These voltages are evaluated with respect to their reference point, and the phase displacement φ is determined.

• Phase displacement φ is determining using the oscilloscope recordings, and converted into an angular value, and current Î in the circuit is calculated.

• For various angles α the apparent power S is determined as the product of the instantaneous voltage and current; these results are used to represent the apparent power in a diagram.

• The characteristics of the voltages ûR, ûL, ûC and û as well as the current î are displayed in a line chart. The vector diagram for this circuit is illustrated in the following:

• Phase displacement between resistor voltage and total voltage depends on the difference between capacitor voltage and coil voltage; if these values are equal, there is no phase displacement between current and voltage.

Parallel Connection of R, L and C

• Experiment: Determine currents through resistor IR, capacitor IC, and inductor IL; use the values with total current I to form the current triangle. Prepare a vector diagram for the circuit, and use it as a basis for drawing a line chart.

  • Current triangle: A vector diagram which allows currents to be added graphically.

  • Vector diagram: A vector diagram represents instantaneous voltages and currents in the form of vectors.

  • Line chart: A line chart displays voltages and currents as functions of time.

• RMS values of currents IR, IL, IC and I are determined to form the current triangle.

• Determined RMS values of IC, IL, IR and I are used to generate current vectors to form the voltage triangle. IR points in the same direction as voltage; IC and IL each exhibit a 90° phase displacement but in opposite directions.

• The characteristics of the currents îR, îL, îC and î as well as the voltage û are displayed in a line chart. The vector diagram for this circuit is illustrated in the following:

Series Compensation

• Experiment: Measure total voltage U and voltage UR before and after compensation; form the voltage triangle and power triangle.

  • Voltage triangle: A vector diagram which allows voltages to be added graphically.

  • Power triangle: A vector diagram which allows powers to be added graphically.

• An equivalent circuit diagram for a typical motor load is shown here. The load consists of an ohmic component (R1) and an inductive component (L1). Series compensation is activated by the capacitor C1.

• Calculated rms values of UR21, U, and UL, and phase displacements in the circuit without the capacitor.

• Determined rms values of UR21, UL and U are used to generate corresponding voltage vectors and form the voltage triangle.

• Switch is opened, series compensation is activated by capacitor C1, and RMS values of UR22, UL, UC, and U and the phase displacement in the circuit with compensation are calculated.

• The determined rms values UR22, UL, UC and U are used to generate corresponding voltage vectors and form the voltage triangle.

• Calculate the rms values P, QL, QC, and S as well as the phase displacement with the capacitor connected in the circuit.

• The determined rms values P, QL, QC and S are used to generate corresponding power vectors and form the power triangle.

• The reactive power consumed by the capacitor is displaced by 180° with respect to the reactive power consumed by the coil. The reactive power component received from the mains is reduced, thus relieving the mains. This is also indicated by the reduction in the apparent power S.

Parallel Compensation

• Experiment: Analyze the equivalent circuit diagram of a typical motor load. In this case, compensation is performed by a capacitor connected in parallel with the total load. The voltages UR1 and UR2 are to be measured before and after compensation. Furthermore, the current triangle and power triangle are to be formed.

  • Current triangle: A vector diagram which allows currents to be added graphically.

  • Power triangle: A vector diagram which allows powers to be added graphically.

• Total voltage (across R1) and the voltage across the two current-sensing resistors R2 and R3. Measure the current in two steps, starting with the current in the inductive branch.

• Determine the rms values of IR, IL and I for the purpose of forming the current triangle without the capacitor connected in the circuit.

• The determined rms values of I, IL and IR are used to generate corresponding current vectors and form the current triangle.

• Now determine the rms values of IR, IL, IC and I to form the current triangle with the capacitor connected in the circuit, i.e. with the compensation active.

• The determined rms values of I, IR, IL and IC are used to generate corresponding current vectors and form the current triangle.

• Subsequently, determine the powers P, QL, QC and S for forming the power triangle.

• The determined powers S, P, QL and QC are used to generate corresponding power vectors and form the power triangle.

• The reactive power consumed by the capacitor is phase-displaced by 180° with respect to the reactive power consumed by the coil. A parallel-connected capacitor can help reduce the apparent power consumption, given an active power of an equal magnitude. The feeding mains network and lines are thus relieved.

Voltage (Series) Resonance

• Experiment: Determine the voltage resonance of a series-connected coil, resistor, and capacitor, and find the resonance frequency of this circuit.

  • Resonance frequency: When a resonant circuit oscillates at its resonance frequency, the capacitive and inductive components cancel each other out, allowing the values of the circuit elements to be determined as such.

• Total voltage (U) and current (I) in the circuit are measured using one multimeter each in order to display frequency as a function of the total resistance.

• The apparent resistance of the circuit can be calculated using the adjacent equation. Due to its characteristic, this circuit is called a series-resonant circuit.

  • Series-resonant circuit: In AC technology, a series-resonant circuit is used to produce a low resistance at a particular frequency; for example, it serves as a tuning element in transmitters and receivers.

• The inductance of the coil and the capacitance of the capacitor can be used to determine the resonance frequency fres using the adjacent formula.

  • $f_{res} = \frac{1}{2\pi \sqrt{LC}}$

• At the resonance frequency, the voltages across the inductive and capacitive resistances are equal but aligned in opposite phase. The series-resonant circuit acts purely as an ohmic resistance in this case. At the resonance frequency, the series-resonant circuit has the lowest possible resistance.

Current (Parallel) Resonance

• Experiment: calculate the current resonance of a coil, resistor and capacitor connected in parallel, and determine the resonance frequency of this circuit.

  • Resonance frequency: When a resonant circuit oscillates at its resonance frequency, the capacitive and inductive components cancel each other out, allowing the values of the circuit elements to be determined as such.

• Partial current IL is measured to display the frequency as a function of the total current and resistance.

• Partial current IC is measured to display the frequency as a function of the total current and resistance.

• The total impedance Z and the total current I in the circuit are to be determined using the specified equations. Due to its characteristic, this circuit is termed a parallel-resonant circuit.

  • Parallel-resonant circuit: In AC technology, a parallel-resonant circuit is used to produce a high resistance at a particular frequency; for example, it acts as a tuning element in transmitters and receivers.

• The inductance of the coil and the capacitance of the capacitor can be used to determine the resonance frequency fres 1 using the adjacent formula.

  • $f_{res} = \frac{1}{2\pi \sqrt{LC}}$

• At the resonance frequency, the partial currents through the inductance and capacitance are equal. The capacitive reactance is equal to the inductive reactance, as the same voltage is present across the coil and capacitor. The same resonance condition applies as in the series-resonant circuit: equal but aligned in opposition to one another. At the resonance frequency, the total current in a parallel-resonant circuit is at its lowest, while the total resistance is at its highest.

LD Didactic GmbH Team finishes the COM3LAB course, requesting feedback via COM3LAB@ld-didactic.com.