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Circuit Theory 2

- Topics
- Impedance matching,
- Impedance transformation,
- Bandwidth and Noise.
- How you will use these topics
- Understanding the rangefinder circuits,
- Optimizing rangefinder performance,
- Understanding transducer models.

Impedance matching

- What is impedance matching?
- Any signal source has an associated source

impedance. - Any load circuit driven at a port exhibits an

associated impedance across the terminals of the

port. - Matching these impedances optimizes the transfer

of power from the source to the load.

- A source with a voltage divider

- A composite load circuit

Thevenins and Nortons Theorems

- Ideal sources
- An ideal voltage source has zero source

impedance. - An ideal current source has zero source

admittance. - Thevenins theorem any electrical signal source

is equivalent to an ideal voltage source in

series with a source impedance. - Nortons theorem any electrical signal source is

equivalent to an ideal current source in parallel

with a source admittance. - (Admittance is the complex reciprocal of

impedance)

Thevenin and Norton example

- Actual source
- Thevenin equivalent
- Norton equivalent

DC matching simple example and proof

- A battery with voltage Vg and internal resistance

Rg is connected to a load resistor Rl. - The power delivered to Rl is

- The maximum deliverable power occurs for

Impedance transformation

- Ideally is a lossless two-port circuit.
- Conservation of energy
- Reciprocal power can flow either way.
- Realizations
- Wideband transformer with n1, n2 turns,
- Resonant Pi network at resonance,

Magnetic transformers

- The magnetic field is

proportional to the total primary current linking

the magnetic path. - Magnetic induction is

the permeability of the magnetic path. - The voltage induced per turn is

, so the secondary voltage

, where . - Similarly, , so

. - Conservation of energy requires

, so - Impedance is transformed as

.

Pi net

(blue real, olive imaginary)

Bandwidth and Noise

- Note the Pi net bandwidth was about 3 kHz.
- The transformer was wideband.
- Why is bandwidth important?
- Response time is inversely proportional to

bandwidth. This limits the range resolution of

the rangefinder. - Noise is directly proportional to bandwidth.

This limits the maximum operational distance for

the rangefinder. - Therefore, a design tradeoff exists between range

resolution and maximum range.

Impedance matching homework problem

- Find the load impedance that accepts the

maximum deliverable power from a 5 V sinusoidal

source Vg at 40 kHz, having a source impedance

due to an internal series resistance of 50 Ohms

and inductance 500 uH. What components,

values, and topology will implement this load?

Sources homework problem

- Show the Thevenin and Norton equivalents of the

circuit shown.

- Extra credit
- Suppose you were given a sealed box 0.05m

x 0.05m x 0.02m, weighing 0.1kg with two

electrical terminals. It is totally opaque, even

to x-rays and ultrasound and cannot be

disassembled for inspection. When open

circuited, the voltage across the terminals is 1

V. When short circuited, the current between

the terminals is 1 A. You are told this is

either a Thevenin source or a Norton source. How

would you determine which? You may make any

measurements you like on the unit.

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