Circuit Theory

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Circuit Theory

  • What you will use this for
    • Power management
    • Signals between subsystems
    • Possible analog data types
  • How the knowledge will help you
    • Understanding power and energy requirements
    • Behavior of digital electric signals
    • Analog signal conditioning and limitations
    • Understanding associated technologies

Circuit theory Topics

  • Circuit Topology
  • Voltage, Current and Power
  • Kirchoff’s Laws
  • Circuit components
  • DC circuits
  • AC circuits
  • We will consistently use Systeme International d’Unites, or SI units here.
  • Basic units are Meters[m], Kilograms[kg], Seconds[s], and Amperes[A].

Circuit Topology

  • A circuit consists of a mesh of loops
  • Represented as branches and nodes in an undirected graph.
  • Circuit components reside in the branches
  • Connectivity resides in the nodes

Voltage, Current and Power (1)

  • The concept of charge
    • The Coulomb [C] – the SI unit of charge
    • An electron carries -1.6e-19 [C]
    • Conservation of charge
  • The concept of potential
    • Attraction/repulsion of charges
    • The electric field
    • The energy of moving a charge in a field

Voltage, Current and Power (2)

  • Voltage is a difference in electric potential
    • always taken between two points.
    • Absolute voltage is a nonsensical fiction.
    • The concept of ground is also a (useful) fiction.
  • It is a line integral of the force exerted by an electric field on a unit charge.
  • Customarily represented by v or V.
  • The SI unit is the Volt [V].

Voltage, Current and Power (3)

  • Current is a movement of charge.
  • It is the time derivative of charge passing through a circuit branch.
  • Customarily represented by i or I.
  • The SI unit is the Ampere [A].

Voltage, Current and Power (4)

  • Power is the product of voltage by current.
  • It is the time derivative of energy delivered to or extracted from a circuit branch.
  • Customarily represented by P or W.
  • The SI unit is the Watt [W].

Kirchoff’s Laws

  • These laws add up to nothing! Yet they completely characterize circuit behavior.
  • Kirchoff’s Voltage Law (KVL) - The sum of voltages taken around any loop is zero.
    • The start and end points are identical; consequently there is no potential difference between them.
  • Kirchoff’s Current Law (KCL) – The sum of currents entering any node is zero.
    • A consequence of the law of conservation of charge.

Circuit components

  • Active vs. Passive components
    • Active ones may generate electrical power.
    • Passive ones may store but not generate power.
  • Lumped vs. Distributed Constants
    • Distributed constant components account for propagation times through the circuit branches.
    • Lumped constant components ignore these propagation times. Appropriate for circuits small relative to signal wavelengths.
  • Linear, time invariant (LTI) components are those with constant component values.

Active circuit components

  • Conservation of energy: active components must get their power from somewhere!
  • From non-electrical sources
    • Batteries (chemical)
    • Dynamos (mechanical)
    • Transducers in general (light, sound, etc.)
  • From other electrical sources
    • Power supplies
    • Power transformers
    • Amplifiers

Passive lumped constants

  • Classical LTI
    • Resistors are AC/DC components.
    • Inductors are AC components (DC short circuit).
    • Capacitors are AC components (DC open circuit).
  • Other components
    • Rectifier diodes.
    • Three or more terminal devices, e.g. transistors.
    • Transformers.

DC circuits

  • The basic LTI component is the Resistor
    • Customarily represented by R.
    • The SI unit is the Ohm [].
  • Ohm’s Law: V = I R
  • Ohm’s and Kirchoff’s laws completely
  • prescribe the behavior of any DC circuit
  • comprising LTI components.

Example: voltage divider

  • Assume no current is drawn at the output
  • terminals in measuring Vout. Ohm’s Law
  • requires that VR1 = IR1 R1 and VR2 = IR2 R2,
  • which is also Vout. KCL says the current
  • leaving resistor R1 must equal the current
  • entering R2, or IR1 = IR2, so we can write
  • Vout = IR1 R2. KVL says the voltage around the loop including the battery
  • and both resistors is 0, therefore Vin = VR1 + Vout, or Vin = IR1 R1 + IR1 R2.
  • Thus, IR1 = Vin / (R1 + R2), and
  • Vout = Vin R2 / (R1 + R2).

AC circuits -- Components

  • Basic LTI components
    • Resistor, R, [] (Ohms)
    • Inductor, L, [H] (Henrys)
    • Capacitor, C, [F] (Farads)
  • Frequency
    • Repetition rate, f, [Hz] (Hertz)
    • Angular,  = 2f, [1/s] (radians/sec)

AC Components: Inductors

  • Current in an inductor generates a magnetic field,
  • B = K1 I
  • Changes in the field induce an inductive voltage.
  • V = K2 (dB/dt)
  • The instantaneous voltage is
  • V = L(dI/dt),
  • where L = K1K2.
  • This is the time domain behavior of an inductor.

AC Components: Capacitors

  • Charge in a capacitor produces an electric field E, and thus a proportional voltage,
  • Q = C V,
  • Where C is the capacitance.
  • The charge on the capacitor changes according to
  • I = (dQ/dt).
  • The instantaneous current is therefore
  • I = C(dV/dt).
  • This is the time domain behavior of a capacitor.

AC Circuits – Laplace Transform

  • Transforms differential equations in time to algebraic equations in frequency (s domain).
  • where the frequency variable s =  + j.
  • For sinusoidal waves,  = 0, and s = j.
  • Resistor behavior in s domain: v= iR.
  • Inductor behavior in s domain: v= i (jL).
  • Capacitor behavior in s domain: i= v (jC).

AC circuits -- Impedance

  • Impedance and Ohm’s Law for AC:
    • Impedance is Z = R + jX,
    • where j = -1, and X is the reactance in [].
    • Ohm’s AC Law in s domain: v = i Z
  • Resistance R dissipates power as heat.
  • Reactance X stores and returns power.
    • Inductors have positive reactance Xl=L
    • Capacitors have negative reactance Xc=-1/C

Impedance shortcuts

  • The impedance of components connected in parallel is the reciprocal of the complex sum of their reciprocal impedances.
  • The impedance of components connected in series is the complex sum of their impedances.

Example: low pass filter

  • Magnitude and phase plots of A, where RC=1. The magnitude plot is log/log, while the phase plot is linear radians vs. log freq.

Homework problem

  • Derive the filter gain of the pictured circuit.
  • Plot the magnitude and phase of the filter for
  • L = 6.3e-6 [H], R = 16 [], and C = 1.0e-7 [F].
  • For extra credit, also plot for R = 7 [] and 50 [].

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