General/Chapter 4 Study Guide

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