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:

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)

Thevenin and Norton example  Actual source  Thevenin equivalent  Norton equivalent

DC matching: simple example and proof  A battery with voltage V g and internal resistance R g is connected to a load resistor R l.  The power delivered to R l 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 n 1, n 2 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 V g 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 0.05[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.