Lecture 12 Review Source transformations Maximum power transfer

Lecture 12 • Review: • Source transformations • Maximum power transfer • Derivation of maximum power transfer • Thévenin theorem examples • Operational Amplifiers • Related educational modules: –Sections 1. 7. 5, 1. 8. 0, 1. 8. 1

Using Source Transformations in Circuit Analysis • Any voltage source in series with a resistance can be modeled as a current source in parallel with the same resistance and vice-versa

Maximum Power Transfer • The load receives the maximum amount of power if RL = RTH • Why?

Maximum Power Transfer – Derivation • Load voltage: • Delivered power:

Maximizing power • Set derivative of power to zero: • Chain rule: • Set numerator to zero:

Maximum Power Delivered • Delivered power: • Letting RL = RTH:

Example 1: Maximum power transfer (a) Determine the load resistance, R, which absorbs the maximum power from the circuit. (b) What is the maximum power delivered to the load?

Example 1(a): Load Design

Example 1(b): Power delivered

Example 2 • Determine the Norton equivalent of the circuit of example 1

Operational Amplifiers • So far, with the exception of our ideal power sources, all the circuit elements we have examined have been passive – Total energy delivered by the circuit to the element is non -negative • We now introduce another class of active devices – Operational Amplifiers (op-amps) – Note: These require an external power supply!

Operational Amplifiers – overview • We will analyze op-amps as a “device” or “black box”, without worrying about their internal circuitry – This may make it appear as if KVL, KCL do not apply to the operational amplifier – Our analysis is based on “rules” for the overall op-amp operation, and not performing a detailed analysis of the internal circuitry • We want to use op-amps to perform operations, not design and build the op-amps themselves

u. A 741 op-amp schematic • Source: RFIC Technologies web site

Ideal Operational Amplifiers • Typical circuit schematic symbol: • Three-terminal device (2 inputs, 1 output) • Operation characterized by: – Voltage difference between input terminals ( vin) – Currents into the input terminals (ip and in)

Ideal Operational Amplifier “Rules” • More complete circuit symbol • (Power supplies shown) • Assumptions: • ip = 0, in = 0 • vin = 0 • V - < vout < V +

Notes on op-amp operation 1. Output current is generally not known (it is provided by external power supplies) 2. KCL at input nodes is generally a good starting point in op-amp circuit analysis 3. vin is multiplied by a large number to get vout 4. vout is limited by the external power supplies

Op-amp circuit – example 1 • Find Vout

Op-amp circuit – example 2 • Find Vout