Lesson 17. 4-Terminal Network and Distributed Capacitance Circuit
Recommended Article : [Circuit Theory] Circuit Theory Table of Contents
1. Inhomogeneity 1. Polarization
2. Inhomogeneity 2. Poynting Vector
3. 4-Terminal Network and Distributed Capacitance Circuit
1. Inhomogeneity 1. Polarization
⑴ Overview
① Polarization : The phenomenon where the distribution of charges within an object is rearranged when an external electric field is applied. Electric dipoles are created.
② Dielectric : A substance where polarization occurs when an electric field is applied.
③ Permittivity or Dielectric Constant : The degree of polarization occurring in a dielectric material.
○ Represents the electrical characteristic of an insulator; higher insulation results in larger permittivity.
○ In chemistry, the fact that higher polarity of a solvent leads to a greater dielectric constant is important.
○ The ratio by which the strength of an electric field decreases due to polarization.
○ (Note) Regular air has a negligible conductivity.
○ (Note) Conductivity increases on rainy days, leading to more significant attenuation.
④ Polarization Current : Temporal variation of polarized charge.
⑵ Types of Polarization
① Molecular Polarization : Also known as dipole orientation polarization.
○ When an electric field is applied to a dipolar molecule, it undergoes rotational force and becomes polarized.
○ High temperature reduces polarization due to molecular thermal vibration.
② Ionic Polarization : Also known as atomic polarization.
○ Applying an electric field to an ionic crystal induces relative changes in positive and negative ions, resulting in induced electric dipoles.
○ Depending on the structure, various displacement ranges occur, allowing for high permittivity.
○ Commonly found in ceramic compounds.
③ Electronic Polarization
○ When an atom is placed in an electric field E, opposing forces act on the nucleus and the electron cloud (because the polarity of the nucleus and the electron cloud is different).
○ The atom becomes a dipole and polarization occurs. The magnitude of this polarization is very small.
○ Induced to a certain extent in all atoms.
○ Reacts rapidly to changes in electric field of high frequency without significant phase difference.
④ Interfacial Polarization : Also known as space charge polarization.
○ Occurs when mobile charges within a polycrystalline solid receive the electric field, but their movement is hindered at interfaces.
○ The slight movement of electrons within the crystal creates polarization, mimicking the effect of having high permittivity.
○ DC Electric Field : Strongly influenced by polarization. Increases relative permittivity.
○ AC Electric Field : In low-frequency range, interfacial polarization diminishes and becomes less significant.
⑶ Bound Charges and Capacitance
① Bound Charge Density P : Polarization per unit area.
② Displacement Current Density D : Surface charge density per unit area.
③ Electric Susceptibility χe
⑷ Dielectrics in AC Electric Fields
Figure 1. Dielectrics in AC Electric Fields [Footnote 1]
① Charging Current (ic) : The current flowing to charge the dielectric-filled electrodes when the switch is closed.
② Absorption Current (id) : The current that decreases over time and is absorbed by the dielectric.
③ Leakage Current (il) : The current that flows steadily over a long period due to carriers within the dielectric.
④ Discharge Current (ic’) : The current that flows when the charge stored in the capacitor is discharged upon shorting the switch.
⑤ Residual Current (id’) : The current that decreases over time and flows due to remaining charges.
⑥ For most dielectrics, ic = ic’ and id = id’ hold true.
⑦ Dielectric After-Effect : A phenomenon where absorption and residual currents flow. Takes time for polarization to occur or disappear.
⑸ Frequency Characteristics of Polarization Rate
① At low frequencies, changes in dielectric polarization roughly correspond to changes in the electric field.
② At high frequencies, dielectric polarization lags behind frequency changes → permittivity decreases.
③ At frequencies above 1 GHz, only ion polarization and electronic polarization occur, with interfacial and molecular polarization disappearing.
Figure 2. Frequency Characteristics of Dielectric Polarization Rate [Footnote 2]
○ Interfacial Polarization : Below 1 Hz
○ Molecular Polarization : 104 ~ 109 Hz
○ Ionic Polarization : 109 ~ 1013 Hz
○ Electronic Polarization : 1014 ~ 1016 Hz
⑹ Dielectric Loss : Also known as dielectric heating.
① Definition : Energy loss due to the reversal of electric dipoles and leakage current, emitted as thermal energy.
○ Example 1. Dielectrics composed only of dipole molecules exhibit maximum energy loss at frequencies of 104 ~ 109 Hz.
○ Example 2. Microwaves maximize dielectric loss in target substances, while selecting lower frequencies for other materials.
② Loss Tangent : Dielectric loss angle, also known as loss coefficient.
○ Equivalent circuit of dielectrics in AC electric field: parallel circuit of resistance and capacitor.
○ IC : Conduction current. The current when no loss occurs.
○ IR : Resistance current. The current when loss occurs.
○ At a specific frequency f where displacement current equals conduction current, the loss angle tan δ of the dielectric is given by f tan δ = fc, where fc is the frequency where the equality holds.
③ Dissipation Factor = Energy Loss per Frequency / (2π × Stored Maximum Energy) : Equivalent to loss tangent.
④ Complex Permittivity (ε) and Complex Relative Permittivity (εr)
○ Complex concepts in AC circuits introduce complex permittivity and relative permittivity.
○ Complex Permittivity
○ Complex Relative Permittivity
⑤ Heat Energy Generated per Unit Volume of Dielectric
○ f : Frequency (Hz)
○ E : Electric field magnitude
○ ε” = ε tan δ, tan δ : Likely the loss coefficient (though not necessarily)
⑺ Measurement Method 1. Schering Bridge Method : Below 1 MHz
⑻ Measurement Method 2. Resonance Method : 1 ~ 900 MHz. Q meter, RLC meter, impedance analyzer
① Resonance : A phenomenon where the amplitude of oscillating systems rapidly increases.
○ Example: The Tacoma Narrows Bridge in Washington state collapsed due to resonance caused by a wind speed of 18.8 m/s four months after its completion in 1940.
② DC Resonant Circuit : L and C cause resonance while R acts as a shock absorber (Bumping).
③ AC Resonant Circuit : Reactances interact, causing the sequence to be zero in the composite impedance.
④ Series AC Resonant Circuit : Under resonance conditions, composite current suddenly increases.
○ Quality Factor (Q) : A measure of sensitivity to resonance in a resonant circuit, denoted as Q.
○ In a series resonant circuit, VL or VC is amplified dozens of times more than the power voltage V.
○ The ratio of VL or VC to the power voltage V is known as the voltage magnification or Q factor.
○ A measure of the energy accumulation efficacy of the resonant circuit = Accumulated Energy ÷ Average Power = Q factor.
○ Cut-off Frequency : The frequency at which reactance equals resistance, resulting in the signal being 1/√2 times the original signal.
○ Sharpness (S)
⑤ Parallel AC Resonant Circuit : Under resonance conditions, composite current suddenly decreases.
⑥ Measurement of Inductance and Capacitance Using Q Meter
○ Structure of Q Meter : Consists of RF oscillator, variable standard capacitor (Cs), voltage meter (VTVM), and high-frequency current meter (A).
Figure 3. Structure of Q Meter
Sure, here’s the translation of the Korean text in the given Markdown code:
○ Frequency of AC power : 50 ~ 75 kHz
○ Eo : Voltage drop when components Cx and Lx are not connected for measurement
○ Ec : Value obtained by adjusting variable capacitor Cs after connecting components Cx and Lx, such that the voltmeter reading is at its maximum
○ Condition for maximum voltmeter reading
○ Conclusion : Determining Lx and Rx from Ec = QEo when the voltmeter reading is at its maximum
⑦ Measurement using Nelson’s proposed Q-meter : (Note) Let’s not overthink this
○ Method used by Nelson (1965) and Stetson (1972) to measure permittivity at frequencies below 40 MHz
○ Method to exclude the influence of stray capacitance
○ 1st. Measure the capacitance Cv1 of the variable capacitor when Ex is at its maximum without connecting the sample holder
○ (Note) Similar to measuring a blank signal
○ 2nd. Measure the capacitance Cv2 of the variable capacitor when Ex is at its maximum with the empty sample holder connected in parallel
Figure. 4. Connection of sample holder to Q-meter’s variable capacitor [Footnote: 4]
○ Capacitance Cs2 of the sample holder is determined as follows
○ h : Height of the sample holder
○ do : Outer cylinder diameter of the sample holder
○ di : Inner electrode rod diameter of the sample holder
○ Cr : Stray capacitance due to the sample holder
○ 3rd. Repeat the 2nd process with the sample holder filled up to h1
○ Capacitance Cs3 of the sample holder with the sample filled up to h1 is determined as follows
○ 4th. Repeat the 2nd process with the sample holder filled up to h2
○ Capacitance Cs3 of the sample holder with the sample filled up to h2 is determined as follows
○ 5th. Determination of the dielectric constant Cx of the sample
○ 6th. Relationship between tan δ and Q
○ 7th. Dielectric loss
○ 8th. High-frequency resistance Rx of the sample
⑼ Measurement Method 3: Transmission Method : Also known as the Attenuation Method
① Measures changes in energy intensity and phase when energy passes through dielectric material to calculate permittivity and loss coefficient
② Uses waveguide
⑽ Measurement Method 4: Reflection Method
① Utilizes the phenomenon of partial reflection due to impedance mismatch on the surface of dielectric material
② Measures reflected wave’s intensity and phase to derive the wave impedance of the dielectric material
③ Obtains dielectric characteristics from this wave impedance
2. Inhomogeneity 2. Poynting Vector
⑴ Definition : Energy passing through per unit area, per unit time
⑵ Generation of electromagnetic waves
Figure. 5. Generation of electromagnetic waves [Footnote: 5]
① Expression for electromagnetic wave electric field
② Expression for electromagnetic wave magnetic field
③ Common features of electric field (E) and magnetic field (H)
○ e 3 component is 0
○ First derivative with respect to x is 0
○ First derivative with respect to y is 0
④ (Reference) Maxwell’s Third Law : Introduces displacement current
⑤ Application of Maxwell’s Third Law
⑥ Final Poynting Vector expression
⑶ Intrinsic Impedance
① Intrinsic Impedance : Z = E / H
① Relation between electric field energy density We and magnetic field energy density Wm in space (η : Intrinsic Impedance)
② Propagation coefficient of electric field : Unit of magnetic field H is AT/m
③ Reflection coefficient = Reflected electric field strength ÷ Incident electric field strength
④ Changes in medium can modify μ, ε, altering wavelength and amplitude but not frequency
⑤ Phase of electromagnetic wave lags behind displacement current by 90°
⑥ Z0 = E / H is real, so phase of E and H is the same
⑷ Vector Magnetic Potential
3. Modeling of 4-Port Network
⑴ Transmission Line Modeling
Figure. 6. Transmission line modeling for 4-port network
Assume a line with impedance Z = R + jωL per unit length in the direction of the line. Also, assume admittance Y = G + jωC per unit length between lines. In the 4-port network circuit where Z and Y are arranged along ‘ㄱ’,
In the 4-port network circuit where Z and Y are arranged along ‘Γ’,
Therefore,
⑵ Characteristic Impedance and Propagation Constant
① Characteristic Impedance (Wave Impedance)
○ L : Distributed inductance, R : Resistance, C : Capacitance, G : Leakage conductance
② Propagation Constant
○ α : Attenuation constant
○ β : Phase constant
③ 4-terminal constants of distributed network
⑶ Lossless Transmission Line
① Lossless Transmission Line : Line with R = G = 0
② Characteristic impedance Z0
③ Propagation constant γ
④ Wave velocity v
○ Conclusion 1: No signal attenuation in lossless transmission line
○ Conclusion 2: Same waveform of the same magnitude propagates at velocity v regardless of frequency
⑷ Lossy Transmission Line
① Conditions for lossy transmission line
② Characteristic impedance Z0
③ Propagation constant γ
④ Wave velocity v : Z, α, v are independent of frequency
Input: 2020.06.02 19:15