
Circuit Theory

Date  29.11.2016  Size  10.06 Kb. 
 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 1.6e19 [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.
 Conservation of energy: active components must get their power from somewhere!
 From nonelectrical 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, = 2f, [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.
 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.
 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 = 6.3e6 [H], R = 16 [], and C = 1.0e7 [F].
 For extra credit, also plot for R = 7 [] and 50 [].
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