# 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
• 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)
• 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 [].