Chapter 14. AC Circuit Theory (frequency-domain)
Recommended post : 【Circuit Theory】 Circuit Theory Table of Contents
1. Overview
5. Power
7. Transformers
1. Overview
⑴ Advantages of AC
① Easy voltage conversion
② Excellent long-distance transmission efficiency (∵ high voltage)
⑵ Average and RMS Values
① Average Value : Considering the average value of the absolute value because the average value of a periodic function is 0
② RMS Value : Equivalent to the effect (mainly power) produced by AC power sources, like DC power value
⑶ Typical AC circuits have sinusoidal power sources with a single frequency
① Counterexample : Non-sinusoidal waves (see reference 5)
⑷ Commonalities between AC and DC circuits
① Series-parallel composition
② Kirchhoff’s laws
③ Principle of superposition
④ Equivalent circuit
2. Laplace Transform
3. Concept of Phasors
⑴ Laplace Transform : Easily interprets all systems
① 1st. Problems related to instantaneous voltage, current, and power are transformed into expressions for V, I, P, etc., using Laplace transforms
○ Instantaneous Voltage : Voltage measured at a specific moment. Represented as v(t)
○ Instantaneous Current : Current measured at a specific moment. Represented as i(t)
○ Instantaneous Power : Product of voltage and current measured at a specific moment. Represented as p(t)
② 2nd. Solve for V, I, P, etc., separately
③ 3rd. Obtain solutions in the desired form by calculating the inverse Laplace transform of V, I, P, resulting in v(t), i(t), p(t)
④ 4th. Equations involving ‘s’ correspond to Laplace transforms ↔ Equations involving ‘jω’ correspond to phasor equations
○ Terminal characteristics of resistors
○ Terminal characteristics of capacitors
○ Terminal characteristics of inductors
⑵ Phasors (Complex form, phasor) : Representation of steady-state sinusoidal quantities using complex numbers
① Introduction : Complex numbers have a powerful visualization tool called the complex plane
② Premise : For v(t) = A cos(ωt + θ),
○ Consider it as the real part of a complex number called phasor
○ ωt is common, so it can be ignored
③ Representation of complex quantities
○ Rectangular coordinate system : z = x + jy
○ Polar coordinate system : z = r∠θ○ Equation of a circle : z = rejθ
④ Operations with Phasors
⑶ Laplace transforms can be applied to all systems, but phasors can only be used in AC circuits
① Complex forms are not applicable, and there are examples where only Laplace transforms are used
② Example : Circuits using both DC and AC voltages in electromagnetic wave oscillators
**4. Applications of Phasors **
**⑴ Impedance (Z, impedance) **: Impedance can be treated as resistance for complex forms
① Overview
○ Resistance, capacitor, and inductor can be treated as resistance in complex forms
○ Important reason : Differential operations can be replaced by arithmetic operations
② Resistance : Has an impedance of R (Ω)
③ Capacitor : Has an impedance of 1/jωC (Ω)
④ Inductor : Has an impedance of jωL (Ω)
⑤ Impedance = Resistance + j × Reactance
○ Resistance : Real part of impedance
○ Reactance : Imaginary part of impedance. Represented as X. Originates from the resonant reaction of capacitors and coils
○ Admittance : Reciprocal of impedance
○ Admittance = Conductance + j × Susceptance
○ Conductance : Real part of admittance
○ Susceptance : Imaginary part of admittance
⑥ In AC circuits based on the premise of complex forms, there are cases where ‘j’ is omitted in the representation
⑵ DC circuit theory still holds in complex forms
① Example 1. Node voltage analysis
Figure. 1. Example of node voltage analysis
② Example 2. Mesh current analysis
Figure. 2. Example of mesh current analysis
⑶ Phasor Diagram
① Definition : Graph representing voltage, current, power, etc., on the complex plane
○ Green arrow is faster than the red arrow (comparison of speeds after ensuring a phase difference less than 180 degrees)
Figure. 3. Phasor diagram
○ Phasor points ωt’s phase to 0° in the phasor diagram○ Example : I = 50 sin(ωt + ∠0) → I = 50 ∠0② Ground current (lagging current) : Inductor ○ Also known as inductive impedance or inductive reactance
○ Ground current phasor diagram
Figure. 4. Ground current phasor diagram
③ True current (leading current) : Capacitor
○ Capacitor as capacitive impedance or capacitive reactance
○ True current phasor diagram
Figure. 5. True current phasor diagram
5. Power
⑴ Overview
① Power is not a simple control quantity, just as in DC circuit theory
② According to the law of energy conservation, the sum of power of all components within the circuit is 0.
⑵ RMS (root-mean-square) : Introducing RMS due to similarity to DC circuit theory
① RMS Voltage
○ Standard in South Korea
② RMS Current
③ Effective Value : Physical quantity obtained according to the definition of RMS
○ Background : Since power = voltage × current naturally, effective value is defined as representative values of voltage and current
○ Effective value generally used in phasors, where sinusoidal voltage and current are represented as arrows on the complex plane
○ Form factor = effective value / average value ○ Crest factor = maximum value / effective value
④ Power in Resistors, Coils, and Capacitors
○ Energy consumed in resistors
○ Energy stored in coils
○ Energy stored in capacitors
⑶ Power Calculation
① Time-domain power calculation
○ Assume v and i are expressed as follows : Vm = Vpeak = maximum value
○ Then, the expected value of power is as follows.
② Apparent power
○ Definition : Power value that considers complex values. Product of voltage phasor and current phasor
○ Unit : VA (volt-ampere)
○ Formulation
○ Reason for multiplying by the complex conjugate
○ Complex conjugate can be applied to voltage or current
③ Real power
○ Power actually consumed by resistance (P)
○ Unit : W (Watt)
○ Real power is related to power in resistors and a necessary and sufficient condition
④ Reactive power
○ Power stored in coils and capacitors, denoted as Q
○ Unit : VAR (volt-ampere reactive)
○ Formulation
○ Energy stored in coils and capacitors is not actually used
○ Reactive power is the amount of energy that doesn’t get fully utilized by the circuit as it returns to the power source while the electrons circulate
○ Positive reactive power in an inductor : Electrical energy is converted to magnetic field energy
○ Negative reactive power in an inductor : Magnetic field energy is converted to electrical energy
○ Reactive power increases real power
○ Reactance generates reactive power and reduces the power factor
○ Reduced power factor requires more current to perform the same work
○ More current flowing leads to increased energy loss (real power) through resistors
⑤ Power factor : cos θ (where θ : phase difference between current and voltage) represents it
○ Length of the hypotenuse × cos θ = Length of the base
○ Magnitude of a complex number × cos θ = Real value
○ Magnitude of the apparent power × cos θ = Magnitude of the effective value
○ Effective power ÷ Apparent power = cos(θv - θi)
○ (Note) Reactive power factor : Reactive power ÷ Apparent power = sin(θv - θi)
○ Negative resistance does not exist in circuits, so cos(θv - θi) is never negative.
○ Indicated as lagging/leading p. f
⑷ Load impedance and maximum effective power
Figure. 6. Load impedance and maximum effective power
(Note: VS is represented as RMS power)
⑸ Phase compensation
Figure. 7. Phase compensation
(Note: VS is represented as RMS power)
① Calculation
○ VL is constant : Power source’s voltage continues to be fed back to keep consumer’s voltage constant
○ P is a decreasing function of L2 : Most efficient when L2 becomes ∞ (⇔ eliminating the effect of load coil)
② Introducing capacitors : Can eliminate the effect of the load coil through resonance
Figure. 8. Introducing capacitors in phase compensation
③ (Note) Resonance phenomenon
○ Resonance : The phenomenon where capacitive reactance and inductive reactance cancel each other out, resulting in zero reactive power.
○ RLC Series
○ RLC Parallel
⑹ Example 1: The resistive load has only a pure resistance and consumes 50 kW. The apparent power magnitude of the motor is 100 kVA, and the power factor is 0.8 lagging.
Figure. 9. Example 1
(Note: VS is represented in rms power)
① Calculate the current flowing through the resistive load and the motor, as well as the total current Itot.
② Determine the apparent power and power factor of the load resistance and the motor when connected in parallel.
6. Non-Sinusoidal Wave
⑴ Overview
① All waveforms with a constant period can be expressed as a sum of sinusoidal waves.
② These sinusoidal waves can be divided into harmonics with frequencies that are integer multiples of the fundamental frequency.
③ Fourier series representation
⑵ Interpretation
① Interpreted through the principle of superposition.
② Note the impedance variation with frequency.
⑶ Voltage
⑷ Apparent power
⑸ Power factor
7. Transformer
⑴ Structure : Structure with coils wound around a core in the shape of a closed loop made of thin metal plates.
Figure. 9. Structure of a Transformer
⑵ Function
① Impedance matching
② Allows AC to pass through to the load while isolating DC.
③ Harmoniously connects balanced and unbalanced circuits, feed systems, and loads.
④ Power is conserved in a transformer.
⑶ Conditions for Transformer Core
① High transformation ratio
② High electrical resistance
③ Made of laminated core
④ Low hysteresis loss coefficient
⑷ Mutual Inductance
① Introduction of mutual inductance coefficient
○ Situation : There are n coils, and each coil carries current ip.
○ ΦB, p, q : Magnetic flux produced at terminal p of coil 1 when current iq flows through coil q only.
○ Maxwell’s equations can determine constants as follows
○ When p = q : Lpq is called self-inductance. Inductance produced by the same coil.
○ When p ≠ q : Lpq is called mutual inductance. Inductance produced by different coils.
○ Total magnetic flux produced at terminal p : Faraday’s law can be applied.
○ εq : A constant that becomes -1 or 1 depending on the direction chosen for the current’s positive direction.
○ For two coils:
⑸ Dot Convention : If current increases in the direction of the dot, flux is added.
Figure. 10. Dot Convention
① VAB = VA - VB means
② i1 : Current flowing from A to B
③ i2 : Current flowing from D to C
⑹ Transformer Analysis
Figure. 11. Transformer Analysis
① Step 1: M12 = M21
○ Situation : Two coils wound on different parts of a conductor.
○ i1 : Current flowing through coil 1
○ Φ1 : Flux through coil 1 due to its own current
○ Φ1 = Φ11 + Φ12
○ Φ11 : Leakage flux through coil 1 due to its own current
○ Φ12 : Flux through coil 2 due to coil 1’s current
○ i2 : Current flowing through coil 2
○ Φ2 : Flux through coil 2 due to its own current
○ Φ2 = Φ21 + Φ22
○ Φ21 : Flux through coil 1 due to coil 2’s current
○ Φ22 : Leakage flux through coil 2 due to its own current
○ Maxwell’s law : Φ ∝ 1/N, Φ ∝ I → Derive L
○ Final conclusion using symmetry
② Step 2: M = √(L1L2)
○ Coupling factor : Indicates the degree of mutual coupling between two coils
○ k = 0 : No mutual inductance (no coupling)
○ k = 1 : Perfect coupling (no leakage)
○ In case of ideal solenoid, k = 1
③ Step 3: V2 = a * V1
○ Since L ∝ N^2, prove the proposition
○ Same conclusion holds for reverse-connected transformer
④ Step 4: Apply KVL (Kirchhoff’s Voltage Law).
○ If ZS = 0 on the load side of the power source, I1 = a * I2 holds. In this case, P = V1I1 = V2I2, showing power conservation.
○ Discover composite impedance
○ Transformer can be considered as composed of a load impedance reduced by a factor of a^2 and the parallel connection of the first circuit’s coil
○ Since the coils themselves have resistance, the composite impedance needs modification
○ Due to the large core resistance, composite impedance can be approximated as shown (see Note: ⑵)
○ Conditions for maximum power transfer in ideal transformer circuits are as follows (see Note: 3.4.)
⑤ Step 5: Actual apparent power and composite impedance apparent power are the same.
⑺ Impedance Matching Example: Design for maximum power transfer to the load.
Figure. 12. Impedance Matching
① Step 1. Input impedance is 1 + j2π * 10^-1 Ω.
② Step 2. Load impedance of 1 - j2π * 10^-1 Ω results in maximum power transfer.
③ Step 3. Adjust the resistance using a transformer with a turns ratio of 1:10.
Figure. 14. Impedance Matching Step 3
④ Step 4. Adjust the imaginary part with the insertion of a capacitor.
Figure. 13. Impedance Matching Step 4
⑻ Campbell Bridge
① Interpretation of Campbell Bridge
② Equilibrium condition: Condition where the current in the secondary circuit is zero.
⑼ Transformer Percentage Impedance
① Impedance voltage of the transformer: Product of transformer impedance and rated current.
② % Decrease in resistance
③ % Decrease in reactance
④ % Decrease in impedance
⑤ Copper loss (Impedance Watt)
⑽ Applications
① Application 1: Substation Transformer
② Application 2: Soldering iron, glue gun
○ If the output voltage decreases n times compared to the input voltage, the output current increases roughly n times.
○ Load heating is proportional to the square of the current, allowing this transformer circuit to generate n^2 times the heat output.
○ In practice, soldering irons and glue guns use a coil ratio of 1 on the output side to maximize heat output.
③ Application 3: Wireless charging for mobile phones
○ 1st. Generating an upward magnetic field from the wireless charger.
○ 2nd. Induced electromotive force and current are generated in the coil within the mobile phone due to changing magnetic flux.
○ 3rd. The induced current generates a current in the opposite direction that opposes the increasing magnetic flux within the coil.
○ 4th. Current flows in the a direction.
○ 5th. Charge is stored in the capacitor within the mobile phone (charging).
Figure. 14. Wireless Charging for Mobile Phones
④ Application 4: Induction Cooker
○ 1st. AC current flows through the coil in the induction cooker: AC with a frequency of over 20,000 Hz.
○ 2nd. Cookware serves as the secondary coil and induces a current within it.
○ 3rd. The induced current heats the cookware.
Figure. 15. Induction Cooker
○ 3-1. Dielectric Heating
○ Target: Dielectric materials like wood, rubber, fabric, etc.
○ Principle: Heating due to dielectric loss
○ Feature: Uniform heating from within the material
○ 3-2. Inductive Heating
○ Target: Metals
○ Principle: Joule heating due to eddy currents induced in the metal
⑤ Application 5: Traffic card terminal
○ 1st. Contacting the traffic card with the AC terminal.
○ 2nd. Changing magnetic field induces current in the coil inside the traffic card.
Figure. 16. Traffic Card Terminal
⑥ Application 6: Metal Detector
○ 1st. AC current flows through the transmitting coil of the metal detector.
○ 2nd. Eddy currents induced in the metal due to electromagnetic induction.
○ 3rd. Detecting coil detects changes in the magnetic field generated by the metal.
Figure. 17. Metal Detector
⑦ Application 7: High-voltage discharge device used for igniting fuel in cars.
○ 1st. Pass current through the primary coil of the transformer and suddenly cut it off.
○ 2nd. Induce overcurrent in the secondary coil.
○ 3rd. Sparks occur between two metal parts connected to the secondary coil.
Input: 2016.01.20 23:54
Modified: 2018.12.12 11:56