Chapter 14. AC Circuit Theory
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 below)
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 ∠0
② Ground current (lagging current): Inductor
○ Also known as inductive impedance or inductive reactance
○ 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 10.** [Structure of a Transformer](https://jb243.github.io/pages/315)
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 11.** Dot Convention
> ① VAB = VA - VB > ② i1: Current flowing from A to B > ③ i2: Current flowing from D to C ⑹ Transformer Analysis
 **Figure 12.** 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 13.** 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 15.** 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](https://jb243.github.io/pages/1091) > 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 16.** 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 17.** 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 18.** 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 19.** 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_