Alternating Current (AC) Circuits

• This chapter looks at Alternating Current (AC) circuits and the mathematics behind them.
• After reading the chapter, you will be able to work with sinusoidal functions, determine amplitude, frequency, and phase shift; utilize degrees and radians, and compute RMS and average voltage and current values.
• You will also be able to define root-mean-squared amplitude, angular velocity, and phase angle; move between time domain and phasor notation, convert between polar and rectangular forms, perform operations on phasors, discuss phase relationships in different loads, use circuit analysis with phasors, define power components, realize power factor, and grasping connections in three-phase circuits.
• This chapter focuses on applying math to AC circuits and covers complex numbers, vectors, and phasors which follow certain rules.

Introduction to AC

• Most electrical power worldwide is generated, distributed, and consumed as 50 or 60 Hz sinusoidal AC voltage.
• AC is generally used for household and industrial applications like TVs, computers, microwave ovens, electric stoves, and large motors.
• AC can be transformed, unlike direct current (DC).
• A transformer adjusts voltage levels for transmission by stepping voltage up or down to transmit high voltage needing less current, which allows for smaller transmission wires.

Sinusoidal Waveforms

• AC flows in alternating directions, unlike DC.
• Sine waves or sinusoidal waveforms are common AC waveforms and remain unaltered by linear circuits, making them ideal test signals.
• AC signals can be expressed using maximum/peak, peak-to-peak, effective, average, or instantaneous values.
• In a sinusoidal wave plot, v(t) = Vp cos(wt + θ):
  • Vp is peak voltage,
  • w = 2πf is angular speed in radians per second (rad/s),
  • f is frequency in Hertz (Hz),
  • t is time in seconds (s),
  • θ is phase in degrees.
• A full cycle starts at 0° with a value of 0, peaks at 1 at 90°, falls back to 0° at 180°, reaches a negative peak at 270°, and returns to 0° at 360°.

Radians and Degrees

• A degree (° or deg) measures angles, with 360 degrees in a full revolution.
• A radian is the central angle where the arc length equals the radius (AB = A0).
• The relationship between circumference (C) and radius (r) is 2π = C/R.
• 360° equals 2π radians, so 1 radian ≈ 57.3° and 1 degree ≈ 0.017453 radians.

Peak and Peak-to-Peak Values

• Each AC signal cycle has two peak values: one positive and one negative.
• Peak value is the maximum value from zero in either direction.
• Peak-to-peak value is the difference between the peak positive and negative values, often used in AC voltage measurements, and is twice the peak value.

Instantaneous Value

• Instantaneous value is the voltage or current at a specific moment in time.
• It can be zero when voltage polarity changes or it can be the peak value when voltage/current stops increasing/decreasing.
• There are infinite instantaneous values between zero and peak.

Average Value

• Average value of AC current or voltage is the mean of all instantaneous values during one alternation and is also DC quantity.
• It is the voltage that a DC voltmeter would show across a load resistor.
• For one sine wave alternation, the average value is 0.636 times the peak value.
• Vav = 0.636Vmax (average voltage), Iav = 0.636Imax (average current)

Effective Value

• Effective value of an AC signal is equivalent to a comparable DC voltage or current when applied to the same resistance.
• It is computed by finding the square root of the average of the sum of the squared instantaneous current values, known as the root-mean-square (RMS) value.
• RMS value (Ieff) of current sine wave = 0.707 × Imax; Imax = 1.414 × Ieff.
• When a sinusoidal voltage is applied to a resistance, the current is also sinusoidal, following Ohm's law: Ieff = Veff / R.
• All AC voltage and current values are effective values.

Frequency

• The number of complete AC cycles per second is the frequency (f), measured in hertz (Hz).
• Angular velocity (ω) relates to frequency and period (T) as: ω = 2πf = 2π/T radians/second.
• Frequency is essential for AC electrical equipment operation.

Period

• Period is the time for one full waveform cycle, measured in seconds.
• The relationship is: T = 1/f.

Phase

• In-phase sinusoidal waves reach maximum and minimum points at the same time.
• Phase relationships are described using "lead" and "lag," measured in degrees.
• The phase difference between two sine waves is found by looking at where the waves cross the time axis in the same direction.

Sine and Cosine

• Sine and cosine are the same function with a 90° phase difference: sin(ωt) = cos(ωt – 90°).

Phasors

• Linear circuits with resistors, capacitors, and inductors alter signal amplitude and phase - not shape/frequency.
• Amplitude and phase are key in determining how circuits affect signals.
• Signals are expressed as complex sinusoids.
• Phase and magnitude define a phasor (vector/complex number), similar to vectors in math.
• Representing signals by phasors allows for DC circuit analysis laws (KVL, KCL) to be applied in the phasor domain by using series/parallel equivalence, voltage/current division without new techniques.
• Circuit responses are phasors (complex numbers), not DC signals (real numbers).
• Complex numbers use polar/rectangular notations.

Polar Form

• Polar form denotes complex numbers by the length (magnitude) and angle of its vector.
• Standard orientation of angles uses 0° to the right, 90° straight up, 180° to the left, and 270° straight down.
• Downward angled vectors can be positive (over 180°) or negative (less than -180°).

Voltage Representation

• Sinusoidal voltage in circuits is V = Vrms ∠ θ where uppercase V is the phasor, including magnitude and phase, and the magnitude is usually RMS.
• Polarity is important: + means "leads," – means "lags."

Rectangular Form

• Horizontal and vertical components creates a complex number; the hypotenuse is the angled vector.
• Includes horizontal/adjacent and vertical/opposite sides that are distinguished by a lower-case "j" (electronics).

Complex Algebra

• Complex number sum of real and imaginary numbers, A = Real(A) + j Imaginary(A).
• Imaginary numbers is the square root of minus one (√−1), is j.
• Electrical engineers use j as i is for instantaneous current.

Transforming Forms

• V = C∠θ = A + jB.
• To convert polar to rectangular, convert C∠0 into A and B using trigonometry.
• cos θ = Adjacent / Hypotenuse = A / C
• sin θ = Opposite / Hypotenuse = B / C
• To convert from rectangular to polar form, use trigonometric realtionships:
• C = √(A² + B²)
• tan θ = B / A
• θ = tan⁻¹ (B/A)
• Any rectangular load converts to polar form as:
  • Z = R + jXL
  • Z = √(R² + XL²) ∠ tan⁻¹ (XL/R)

Euler's Identity

• Forms the basis of phasor notation and is named after Leonard Euler.
• Defines the complex exponential as a point in the complex plane: e^(jθ) = cos θ + j sin θ.

Adding Phasors

• To add, the quantities, express each in rectangular form and complete the following steps:
  • Add the real parts of the phasors.
  • Add the imaginary parts of the phasors.
  • Form the sum as a phasor written in rectangular form.

Subtracting Phasors

• Can only be achieved with the rectangular form of the quantity and by first changing the sign of both the real and the imaginary part of the phasor to be subtracted; then the phasors are added.

Multiplying Phasors

• Rectangular form requires the form of two binomials:
  • Distribute the real part of the first complex number over the second complex number.
  • Distribute the imaginary part of the first complex number over the second complex number.
  • Replace j² with −1
  • Combine like terms.
  • Form the product as a phasor written in rectangular form.
• In Polar Form, the magnitudes are multiplied, the angles are added, and a product of a phasor is written in polar form.

Dividing Phasors

• In rectangular form the numerator and denominator are multiplied by the complex conjugate of the denominator, the real number and then the imaginary number of the numerator and denominator are divided The quotient is then created as a phasor written in rectangular form.
• The polar form requires that the magnitudes are divided and from that the angle of the denominator from the angle of the numerator is subtracted; the quotient is expressed as a phasor written in polar form.

Power of a Phasor

• Raising a phasor to a power requires that you express the phasor in polar form complete the folowing steps:
  • Raise the magnitude to the specified power.
  • Multiply the angle by the exponent.
  • Form the solution.

Electrical Systems of Phasors

• Phasors can be added, subtracted, multiplied, and divided with the most insightful operation being addition:
  • Vtotal = V1 + V2
• To add manually, the phasors must be converted into rectangular form:
  • V₁ = A₁ + jB₁
  • V2 = A2 + JB2
• To complete the process, add the real parts together and then the imaginary parts together -Vtotal = (A₁ + A₂) + j (B₁ + B₂)

Simple Vector Addition

• if the current has a description, such as, "50 mA at -60°," then it means that the current waveform has an amplitude of 50mA and it lags 60° behind the reference waveform source.

Complex Vector Addition

• Vectors with uncommon angles that are added yield magnitudes that are quite different from those of scalar magnitudes.
• If in series and out of phase, their magnitudes do not directly add, rather, they add as complex quantities through trignometry.
• Notation with these complex elements becomes difficult in relation to AC circuit analysis.

Resistive Loads

• In DC, one load type exists: resistive, whereas in AC, loads are resistive, inductive, and capacitive which creates different circuit conditions.
• Impedance rather than resistance, the element offers to a sinusoidal current, a phasor quantity.
• Pure resistance in a circuit means AC moves through it and voltage climbs as AC falls and rises.
• Voltage and current are in phase here, and heat could be produced.
• Ohm's law defines impedance: Z = V/I.

Inductive Loads

Inductance

• An inductors (L) inductance is measured in henries (H).
• Depends upon the physical make up of the coil (length, cross sectional area, turns of wire) and the permeability of the core material.
• L = N^2 uA / l
• Inductance is a primary load in AC circuits and is found in all AC circuits due to the changing magnetic field because of loads with coils such as motors, transformers and chokes.

Inductive Reactance

• Alternating magnetic field induces Voltage in coils: 180° out of phase with the applied Voltage.
• The limitation of current flow is Reactance (X).
• Inductive Reactance Xl, is measured in Ohms.
• X₁ = ω L = 2πfL
• VL = L(di/dt)
• VL = (ωL) Ip sin (ωt + 90°)
• ZL = (XL ∠90°)
• ZL = (0 + j XL) Ω

Power in Inductive Load

• True Power = Product of Voltage & Current.
• A pure inductive circuit in which voltage and current are 90° out of phase, has no true power produced.

Capacitive Loads

• Capacitors opposes a change in voltage.
• The current through a capacitance is ic = C(dv/dt).
• Xc = 1/(ωC)
• Ic = (ωC) × (Vrms∠90°)
• Zc = Xc∠-90°
• The impedance of the capacitor is a frequency -dependent, complex quantity, the function of impedance operates as an inverse proportional, but the current shifts voltage by 90°.

AC Circuit Analysis

• Impedance parameters defined in previous sections are useful in solving AC circuit analysis problems with most of the network theorems developed for DC circuits by replacing resistances with complex-valued impedances.
• All rules/laws learned in DC circuit study apply to AC circuits including Ohm's/Kirchhoff's laws, network analysis methods.
• Variables are in complex form, accounting for phase/magnitude.
• Voltages/currents share a frequency that allows for the same constant phase relationships.

Power and Power Factor

• Understanding electrical load involves both concepts.
• Power consumed has multiple components: apparent, reactive, and active/real.

Power Components

• The load's real work portion is active/real power.
• Supplying magnetic/electrical field is what reactive power provides.

Power Factor

• The ratio for real vs apparent power offers an important quantity; expressed as PF=P/S = cos(0)
• The load used to convert power consumed is the reactive factor.

Leading and Lagging Power Factor

• Lagging power factor occurs when a load lags an applied voltage, whereas leading is the opposite.