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Lecture 12 (parts A & B) •Review: •Source transformations •Maximum power transfer •Derivation of maximum power transfer •Thévenin theorem examples •Operational Amplifiers •Related educational materials: –Chapters 4.5, 4.6, 5.1-5.4 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: VL VOC RL RL RTH • Delivered power: V V RL PL RL RL RL RTH 2 L 2 OC PL RL 2 Maximizing power • Set derivative of power to zero: PL 0 RL RL • Chain rule: 2 RL 0 VOC 2 RL RTH ( RL RTH )2 2 RL ( RTH RL ) V 0 4 ( RL RTH ) 2 OC • Set numerator to zero: RL RTH Maximum Power Delivered • Delivered power: 2 OC V PL RL RL RL RTH • Letting RL = RTH: 2 VOC PL 4 RTH 2 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 uA741 op-amp schematic • Source: RFIC Technologies web site Ideal Operational Amplifiers • Typical circuit schematic symbol: ip + vp - vn = vin in - • 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