From charlesreid1

Chapter 4: Components and Circuits

Section 4.1: Current, Voltage, Power

  • an increase in power of 2x is equal to 3 dB
  • in a purely resistive parallel circuit, the total amount of current is the sum of each branch current
  • if 400 V dc is supplied to an 800 ohm load, use the formula P = \dfrac{E^2}{R} = \dfrac{400^2}{800} = 200 \text{W}
  • if a 12 V DC light bulb draws 0.2 A, use the formula P = I E = (12)(0.2) = 2.4 \text{W}
  • if 7 mA flows through 1.25 kOhm load, the amount of power dissipated can be found using P = I^2 R = (7.0 \times 10^{-3})^2 (1.25 \times 10^3) = 61 \text{mW}
  • a power transmission line loss of 1 dB: to find percent power loss, use the formula dB = 10 \log(PR), which rearranges to PR = 10^{-\frac{1}{10}} = .794 so the power is 79.4% of what it was. This corresponds to a loss of 100 - 79.4 = 20.6 or 20.6%

Section 4.2: AC Power

  • oscilloscope measures 200 V peak-to-peak across 50 Ohm dummy load. What is PEP output? PEP = \dfrac{E_{RMS}^2}{R} = \dfrac{(0.707 V_{PK})^2}{R} = \dfrac{(0.707 \frac{V_{PK-TO-PK}}{2})^2}{R} = 245 V
  • AC signal producing same power dissipation in a resistor as a DC signal of the same voltage is the AC signal with an RMS voltage equal to the DC voltage: V_{RMS,AC} = V_{DC}
  • A sine wave with a peak voltage V_{PK} = 17 \text{V} has an RMS voltage of V_{RMS} = 0.707 V_{PK} = 12 \text{V}
  • For an unmodulated carrier, PEP = average power
  • The RMS voltage across a 50 Ohm dummy load dissipating 1200 W is  P = \frac{V_{RMS}^2}{R} so E_{RMS} = \sqrt{PR} = 245 V
  • If average power is measured as 1060 watts for an unmodulated carrier, its PEP output is 1060 watts. For unmodulated signals, PEP = average power
  • If oscilloscope measures 500 V peak-to-peak across 50 ohm load, PEP is PEP = \dfrac{E_{RMS}^2}{R} = \dfrac{(0.707 \frac{V_{PK-TO-PK}}{2})^2}{R} = 625 W

Section 4.3: Basic Components

(Fill in)

Resistors

  • The change in resistance is a function of the resistor's temperature coefficient
  • Inductive resistors can affect RF circuits and change signals (contain metal winding)
  • Use non-inductive resistors in RF circuits

Inductors:

  • Double lines in symbol mean metal core
  • Inductors store an amount of magnetic energy, from the current flowing through it
  • Higher inductance means more magnetic energy stored
  • Higher permeability of core increases inductance
  • Mutual inductance - current generated from a shared magnetic core
  • To avoid mutual inductance, use torroidal inductors, or place inductors at right angles
  • Inductor material can be optimized for particular frequencies

Capacitors:

  • Basic structure: two conductors separated by a dielectric, which stores electrical energy while preventing DC current flow
  • The closer the surfaces, the larger the SA, the larger the dielectric energy storage, the higher the capacitance
  • Rolled up capacitors have significant parasitic inductance
  • Ceramic capacitors are more common at higher frequencies
  • Electrolytic capacitors use electrolyte gel/paste, pack higher capacitance into smaller volume
  • Polarized capacitors - current can only flow in 1 direction
  • Voltage rating of capacitors is the voltage above which the dielectric insulation will break down
  • Blocking capacitors l- block DC signals, but not AC signals
  • Bypass capacitors - low impedance path across high impedance circuit
  • Filter capacitors - smooth out rectified AC into DC power
  • Suppressor capacitors - absorb transient voltage spikes
  • Tuning capacitors - varying resonant circuit frequencies

Components in series/parallel:

  • series resistance is additive: ----R1----R2----R3---- R1+R2+R3
  • series inductance is additive: L1+L2+L3
  • series capacitance is reciprocal of reciprocals 1 / ( 1/C1 + 1/C2 + 1/C3 )
  • parallel resistances are reciprocal of reciprocals
  • parallel inductors are reciprocal of reciprocal
  • parallel capacitors are additive

Transformers:

  • Transformers utilize mutual inductance (shared magnetic core)
  • Inductors are called windings
  • Power applied to primary winding
  • Power extracted from secondary winding
  • Changing number of windings changes current (power is conserved)
  • significant changes between primary/secondary voltages requires changes in wire size
  • Step-up transformer: primary winding has higher current, so wound with larger diameter wire
  • Relation between voltage and number of windings:


\frac{E_s}{E_p} = \frac{N_s}{N_p}

Section 4.4: Reactance and Impedance

Reactance:

  • capacitors and inductors respond differently to AC and DC
  • resistance to AC is called reactance X (measured in ohms)
  • Reactance occurs because capacitors and inductors store energy

Capacitive reactance:

  • When DC applied to capacitor:
  • Current rushes in
  • Capacitor begins to store energy
  • Voltage in capacitor rises
  • Decrease in voltage leads to decrease in delta V, driving force of current
  • the more energy stored in capacitor, the lower the current that flows
  • eventually, current stops
  • Capacitor in DC circuit:
    • Capacitor initially looks like a short circuit (closed circuit)
    • After capacitor is charged, looks like an open circuit
    • Capacitors block DC current
  • When AC applied to capacitor:
    • At low frequencies, AC behaves like DC
    • Capacitor has enough time to charge, stop current
    • If AC voltage is higher frequency, capacitor never fully charges to reduce current very much
    • Capacitors block DC current, resist low frequency AC and pass high frequency AC
  • Opposition to AC current from stored energy is called capacitive reactance X_c and changes with frequency


X_c = \dfrac{1}{2 \pi f C}

Inductive reactance:

  • Inductors resist current in a complementary way to capacitors
  • When DC voltage applied to inductor:
    • Current rushes through coil and magnetic energy begins to fill the core
    • THe change in the magnetic field resists current initially, gradually lets more through
    • When inductor dielectric material is "fully charged," current can pass through it
    • WHen voltage first applied, inductor looks like an open circuit
    • AFfter time, inductor looks like closed circuit
  • Inductor treats DC in an opposite way from capacitor
  • If AC voltage applied to inductor:
    • Magnetic field perpetually changing
    • Current always opposed
    • If low-frequency AC, inductor's magnetic core has time to change nad let current pass through
    • An inductor blocks high-frequency AC, passes low-frequency AC currents, and acts as a short circuit for DC currents
  • Inductive reactance is opposition to AC current flow from stored energy and is denoted X_L


X_L = 2 \pi f L

In summary:

Capacitors oppose changes in voltage.

Inductors oppose changes in current.

Impedance:

  • General term for the opposition to current flow in an AC circuit, caused by reactance, resistance, or any combination
  • Impedance denoted Z (ohms)
  • Impedance is the ratio of voltage to current
  • Resistance is independent of frequency
  • Reactance is a function of frequency

Resonance:

  • Condition in which match between (frequency at which circuit or antenna naturally responds) and the (frequency of applied signal)
  • In a circuit with inductive/capacitive reactances, resonance means effects of inductor/capacitor on AC current cancel out
  • Resonant circuit: inductive reactance of L cancels with capacitive reactance of C, creating a short circuit and leaving the remaining resistance (load) as the only circuit impedance

Impedance transformation:

  • Transformers change voltages and current
  • Ratio of voltage to current is impedance
  • Impedance of transformer also changed (car transmission)


\frac{Z_s}{Z_p} = \left( \frac{N_{s}}{N_{p}} \right)^2

Impedance matching:

  • Interval impedance of components allows limits on power delivery
  • Example: hearing aid battery (high impedance) and D-cell (low impedance) have same E = 1.5 V
  • Maximum Power Transfer Theorem - maximum power transfer occurs when source and load output impedances are equal and purely resistive (no reactance)
    • Hence, adding inductors and capacitors to lengthen and shorten antennas, and make it resonant.
  • Maximum power happens at resonant frequency of the circuit
  • Amateur equipment: source impedance at output should be 50 ohms (for coax)
  • Antennas often designed with feed point impedance of 50 ohms (changes with frequency)
  • If impedance difference between transmitter output impedance and load impedance are too great, it can reflect power back and damange transmitter
  • To match impedance at transmitter output wtih impedance of antenna, use impedance-matching circuit
  • LC circuits (capacitors and inductors)
    • Pi network: two capacitors on either side of an inductor; the capacitors are connected to ground
    • T network: two capacitors on either side of an inductor; the inductor is connected to ground
  • Another way to match impedances is using transformers
  • Impedance transformers equalize impedances of source and load to maximize transfer of power
  • Stress caused by lots of power and high transformation ratios
  • High power can lead to core saturation and harmonic distortion

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