DC, Superposition

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DC, Superposition

• DC is all straightforward
• Superposition Does 112?

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Admin

• Test this weekend
• Complete TODAY
• Parallel equivalents
• G vs. R
• Redrawing ugly circuits and simplifying
• Voltage divider

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Learning outcomes

• Analyze simple series and parallel circuits in
  terms of energy, charge, current, voltage
• Derive Kirchhoff's Voltage Law (KVL) from energy
  /charge considerations
• Analyze series and parallel DC circuits
• Explain implications for design and power
  consumption
• Analyze Combined Series/parallel circuits
• Explain Principle of Superposition
• Use Superposition to analyze ccts.

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Series/Parallel concepts

• Parallel circuits common voltage
• House/car examples
• Overload (Infinite current)
• Series Circuits
• Old-fashioned Christmas light example,
  LED/resistor example ? more complex circuits
• Combined series/parallel intro
• Discuss implications for design and for analysis

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Derive KVL

• Simple DC series circuit (fig 3.2, 3.7)
• Energy losses
• Constant charge flow constant current
• Zero-Sum voltages
• Voltage rises and drops
• Example multiple batteries in portable
  electronics
• Physics Conservation of Energy nothing lost

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KVL

• Summation form of KVL
• Sum(Vn) Sum(rises) - Sum(drops) 0
• EG (Measure voltages around a series cct.)

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Total Equiv Resistance

• Convert KVL to Ohms law form
• V I1R1 I2R2 I3R3 INRN
• I(R1 R2 R3 RN) for a series cct
• Total R sum of series R
• Can now completely analyze a series DC cct.
• (Discuss Analysis is necessary for design)

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EG complete analysis

• For a given series cct. find following (with
  units) (EG 3.2.3)
• Total R of all the loads
• Total I
• Current in each load (R )
• Voltage across each R
• Check with KVL
• Power (total) delivered to loads
• Power in each load
• Conservation of power
• Discuss relevance to a simple device or a room
  full of servers

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Voltage Divider Rule

• Voltage shared over two loads
• ITOT I1 I2
• VTOT/RTOT V1 / R1
• V1 R1 x VTOT/RTOT
• EG two resistors
• Put this equation into software easily
• Discuss Application to varying loads, sensors,
  microphones, etc.

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DC Parallel Circuits

• Simple circuit with three parallel resistances
  (fig 3.13)
• Energy is conserved therefore current is shared
• Voltage is constant, Current is shared

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Kirchhoff's Current Law

• Sum of currents entering and leaving a node 0
• Sum(In) IENTERING_NODE ILEAVING_NODE
• Sum(In) I1 I2 I3 IN
• Use for entering, - for leaving

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Derive Parallel Resistance Equation

• KCL ? ISOURCE I1 I2 I3 IN
• Use Ohms Law
• VT/RT V1 / R1 V2 / R2
• But VT V1 V2 VN V
• Take out common factor (V)
• V/RTV(1/ R1 1/ R2 1/ RN )
• Divide through by V
• 1/RT 1/ R1 1/ R2 1/ RN
• RT 1/(1/ R1 1/ R2 1/ RN )

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Example

• EG with varying resistances
• Battery plus 3 R in //e
• 12V battery, 3 O, 4 O, 6 O
• Use Ohms Law calc current in each
• Calc total current
• Therefore equiv R ?
• Calc equiv R with formula ? calc current

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Implications

• Equivalent R is LESS than smallest R
• Higher I through smaller R

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Two parallel R

• Use Equiv voltage Equation
• REQ 1/(1/ R1 1/R2 )
• REQ R1 xR2 /( R1 R2 )
• EG 10V batt R 5 O, 2 O
• Current in each by Ohms law
• Total Current by KCL
• Equiv R by eqn ? calc currents

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Conductance

• G 1/R (in mhos)
• Equivalent conductance of parallel cct.
• GT G1 G2 GN Sum (Gn)
• EG R 2.5 KO, 5 KO, 5 KO in //e
• Convert to Mhos
• Add
• Convert back

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Current Divider Rule

• Consider multiple branches
• What is current in any one branch?
• I1 IT x RTOT/R1

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Series/Parallel

• With the tools we have developed we can now
  handle very complex circuits by reducing them
  down to series and parallel resistances and using
  equivalent resistance
• RSER R1 R2
• 1/RPARA 1/R1 1/R2
• EG multiple Series and Parallel branches
• EG 3.4.1, Fig 3.20

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Measuring I, V, R

• Measure without interfering
• Measuring V
• Infinite impedance
• Measuring I
• Must break circuit
• Zero impedance
• Dont measure V with this set-up!
• Blows fuses (if youre lucky)
• Measuring R
• Must separate out parallel paths
• Provide a voltage

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Voltage and current sources

• Ideal V source zero resistance
• Real voltage sources are not perfect
• Battery has internal resistance (v. Low)
• Called Internal Impedance
• Other components have it too
• Even wire
• Current source has RINT in //e
• See 3.6 for more detail

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Superposition

• Complex circuits
• Multiple Sources
• How does this occur in real circuits?
• Induced voltages in communication channels
• Multiple power supplies interacting
• EG battery and charger
• Some devices provide energy
• Connecting two powered systems together
• Assumptions of Superposition

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Voltage Sources - Series

• Series voltage sources
• Add them
• If one is reversed current will flow wrong way
• Subtracts the voltage
• May damage battery chemically (recharging?)

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Sources - Parallel

• Parallel voltage sources
• Should be same voltage
• Increases current (energy) capability
• High currents will flow if not same voltage
• Consider 2 batts in parallel one reversed!
• Show batts with their internal impedance

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Superposition

• Replace all but one sources with their internal
  impedance
• Calculate all currents/voltages in all loads
• note direction of current flows
• Repeat for each source
• Add the results (this is Superposition)

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Example

• EG 3.5.2 p77
• T network w. 2 V sources
• Calculate power last (why?)