4.3 Thevenin TheoremExample 4.14.5 Maximum Signal TransferThevenin and Norton Theorems Prof. Bogdan Adamczyk Grand Valley State University Page 1 of 24 Padnos School of Engineering 2007 4.3 Thevenin Theorem In practice, we often interface or connect two circuits. Circuit interface occurs frequently in electrical and electronic systems, so special analysis methods are used to handle them. For the two-terminal interface shown in Figure 4.3-1 we normally think of one circuit as the source S and the other as the load L. Load Source = linear two-terminal circuit a b Figure 4.3-1 A two-terminal interface It often occurs in practice that a particular element in a circuit is variable (usually called the load) while other elements are fixed. Each time the load circuit changes, the entire system (source circuit and the load circuit) has to be analyzed again. Thevenin theorem provides a technique by which the linear source circuit is replaced by an equivalent circuit containing one voltage source and one resistor in series with it, as shown in Figure 4.3-2. RT VT + − Load a b Source Figure 4.3-2 Thevenin equivalent circuitThevenin and Norton Theorems Prof. Bogdan Adamczyk Grand Valley State University Page 2 of 24 Padnos School of Engineering 2007 The circuit to the left of terminals a-b in Figure 4.3-2 is known as the Thevenin equivalent circuit. The Thevenin equivalent circuit consists of a voltage source VT in series with a resistance RT. Note that the equivalence means that the i-v characteristics at the terminals a-b are unchanged when we replace the original circuit with the Thevenin equivalent circuit. We need to define the Thevenin voltage VT and the Thevenin resistance RT. The Thevenin voltage VT is the open-circuit voltage, that is the voltage seen across the terminals of the source circuit, with no load connected across them as shown in Figure 4.3-3. a Source = linear two-terminal circuit + voc = VT − b Figure 4.3-3 Thevenin voltage VT The Thevenin resistance RT is the resistance seen between the terminals a-b when all independent sources are turned off (voltage sources replaced by short circuits, current sources replaced by open circuits) as shown in Figure 4.3-4. a Linear circuit w/ independent sources set equal to zero Rin = RT b Figure 4.3-4 Thevenin resistance RTThevenin and Norton Theorems Prof. Bogdan Adamczyk Grand Valley State University Page 3 of 24 Padnos School of Engineering 2007 Thevenin principle is particularly useful when we wish to find the resulting current, voltage, or power delivered to a load. Consider a linear circuit terminated by a load RL as shown in Figure 4.3-5. RL a Linear Circuit IL b Figure4.3-5 Linear circuit terminated by a load The circuit to the left of the load can now be replaced by its Thevenin equivalent circuit, as shown in Figure 4.3-6. RL a VT + − RT IL b Figure4.3-6 Thevenin equivalent circuit The current through the load and the voltage across the load may now be easily determined as: LTTLRRVI+= (4.3.1) TTLLLLLVRRRIRV+== (4.3.2)Thevenin and Norton Theorems Prof. Bogdan Adamczyk Grand Valley State University Page 4 of 24 Padnos School of Engineering 2007 The power delivered to the load can now be easily computed as LLIVP = (4.3.3) Alternatively, we could choose a load that satisfies imposed constraints on either voltage, current or power delivered to it by the source circuit as illustrated in the following example. Example 4.1 Consider the circuit shown in Figure 4.3-6. Select a load resistance so that the source circuit delivers VL voltage to the load. VL − + R2 R1 R3a + − VS RL b Figure 4.3-6 Application of Thevenin theorem This task is easily handled once we have the Thevenin equivalent for the source circuit. The Thevenin voltage is found by disconnecting the load at the terminals a and b, as shown in Figure 4.3-7, and determining the open circuit voltage between them.Thevenin and Norton Theorems Prof. Bogdan Adamczyk Grand Valley State University Page 5 of 24 Padnos School of Engineering 2007 R1 R3a + − + VS R2 VOC = VT − b Figure 4.3-7 Determining VT Using voltage divider, we obtain STVRRRV212+= (4.3.4) Next, we find Thevenin resistance by suppressing all sources and determining a resistance between the nodes a and b, as shown in Figure 4.3-8. R3R2 R1 a RT b Figure 4.3-8 Determining RTThevenin and Norton Theorems Prof. Bogdan Adamczyk Grand Valley State University Page 6 of 24 Padnos School of Engineering 2007 We note that R3 is in series with a parallel combination of R1 and R2, thus 32321RRRRRRT++= (4.3.5) The resulting Thevenin equivalent circuit is shown in Figure 4.3-9. + a RT VL − + − VT RL b Figure 4.3-9 Equivalent Thevenin circuit Using voltage divider, we may write TLTLLVRRRV+= (4.3.6) Solving the above equation for RL produces ()TLLTLLTLLTLRVVVRRVRVRV=−=+ or LTTLLVVRVR−= (4.3.7) where VT and RT are given by Eqs. (4.3.4) and (4.3.5), respectively, and VL is specified voltage across the load.Thevenin and Norton Theorems Prof. Bogdan Adamczyk Grand Valley State University Page 7 of 24 Padnos School of Engineering 2007 4.4 Norton Equivalent Circuit In 1926, an American engineer Edward L. Norton proposed an alternative solution to replacing a linear two-terminal circuit by an equivalent circuit. Norton theorem provides a technique by which the linear source circuit is replaced by an equivalent circuit containing one current source and one resistor in parallel with it, as shown in Figure 4.4-1. RN Load IN Source a b Figure 4.4-1 Norton equivalent circuit The circuit to the left of terminals a-b in Figure 4.4-1 is known as the Norton equivalent circuit. We need to define the Norton current IN and the Norton resistance RN. We know, from the source transformation that TNRR = (4.4.1) Thus, we find RN in the same way we find RT. That is, the Norton resistance RN is the resistance seen between the terminals a-b when all independent sources are turned off (voltage sources replaced by short circuits, current sources replaced by open circuits). To find the Norton current IN, we need to determine he short-circuit current isc flowing from terminal a to terminal b as shown in Figure 4.4-2.Thevenin and Norton Theorems Prof.