1. 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

2. 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

3. Maximum Power Transfer The load receives the maximum amount of power if R L = R TH Why?

4. Maximum Power Transfer – Derivation Load voltage: Delivered power:

5. Maximizing power Set derivative of power to zero: Chain rule: Set numerator to zero:

6. Maximum Power Delivered Delivered power: Letting R L = R TH :

7. 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?

8. Example 1(a): Load Design

9. Example 1(b): Power delivered

10. Example 2 Determine the Norton equivalent of the circuit of example 1

11. 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!

12. 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

13. uA741 op-amp schematic Source: RFIC Technologies web site

14. Ideal Operational Amplifiers Typical circuit schematic symbol: Three-terminal device (2 inputs, 1 output) Operation characterized by: – Voltage difference between input terminals (  v in ) – Currents into the input terminals (i p and i n )

15. Ideal Operational Amplifier “Rules” More complete circuit symbol (Power supplies shown) Assumptions: i p = 0, i n = 0  v in = 0 V - < v out < V +

16. 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.  v in is multiplied by a large number to get v out 4.v out is limited by the external power supplies

17. Op-amp circuit – example 1 Find V out

18. Op-amp circuit – example 2 Find V out