Circuit Theory 2

1. 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.

2. 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:

3. Thevenin’s and Norton’s Theorems • Ideal sources • An ideal voltage source has zero source impedance. • An ideal current source has zero source admittance.* • Thevenin’s theorem: any electrical signal source is equivalent to an ideal voltage source in series with a source impedance. • Norton’s 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)

4. Thevenin and Norton example • Actual source • Thevenin equivalent • Norton equivalent

5. 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

6. 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,

7. 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 .

8. Pi net (blue = real, olive = imaginary)

9. 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.

10. 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?

11. Sources homework problem • Show the Thevenin and Norton equivalents of the circuit shown. • Extra credit: Suppose you were given a sealed box 0.05[m] x 0.05[m] x 0.02[m], weighing 0.1[kg] 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.