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NETWORK THEOREMS MODULE Superposition • Superposition Theorem – the response of a circuit to more than one source can be determined by analyzing the circuit’s response to each source (alone) and then combining the results Insert Figure 7.2 Superposition Insert Figure 7.3 Superposition • Analyze Separately, then Combine Results Use superposition to find the current ix. Current source is zero – open circuit as I = 0 and solve iXv Voltage source is zero – short circuit as V= 0 and solve iXv i X i Xv i Xc Use superposition to find the current ix. The controlled voltage source is included in all cases as it is controlled by the current ix. Voltage and Current Sources Insert Figure 7.7 Voltage and Current Sources Insert Figure 7.8 Voltage and Current Sources Insert Figure 7.9 Source Transformation VS Under what condition, the voltage and current of the load is the same when operating at the two practical sources? For voltage source iL RS + + _ VL RL _ For current source i S RP VL RL R P RL iL , + IS RP VL _ VS VL RL R S RL We have, RL VS iS RP R S RL RP RL VS RP RS , iS RS Voltage and Current Sources • Equivalent Voltage and Current Sources – for every voltage source, there exists an equivalent current source, and vice versa Thevenin’s Theorem • Thevenin’s Theorem – any resistive circuit or network, no matter how complex, can be represented as a voltage source in series with a source resistance Thevenin’s Theorem • Thevenin Voltage (VTH) – the voltage present at the output terminals of the circuit when the load is removed Insert Figure 7.18 Thevenin’s Theorem • Thevenin Resistance (RTH) – the resistance measured across the output terminals with the load removed Thévenin Equivalent Circuits Thévenin Equivalent Circuits Vt voc voc Rt isc Thévenin Equivalent Circuits Finding the Thévenin Resistance Directly When zeroing a voltage source, it becomes a short circuit. When zeroing a current source, it becomes an open circuit. We can find the Thévenin resistance by zeroing the sources in the original network and then computing the resistance between the terminals. Computation of Thévenin resistance Equivalence of open-circuit and Thévenin voltage A circuit and its Thévenin equivalent Superposition As the voltage source does not contribute any output voltage, Only the current source has the effect. Determine the Thévenin and Norton Equivalents of Network A in (a). Source transformation Find the Thévenin equivalent of the circuit shown in (a). v As i = -1, therefore, the controlled voltage source is -1.5V. Use nodal analysis at node v, v (1.5) v 1, v 0.6 3 2 Thus, Rth =v/I = 0.6/1 = 0.6 ohms Applications of Thevenin’s Theorem • Load Voltage Ranges – Thevenin’s theorem is most commonly used to predict the change in load voltage that will result from a change in load resistance Applications of Thevenin’s Theorem • Maximum Power Transfer – Maximum power transfer from a circuit to a variable load occurs when the load resistance equals the source resistance – For a series-parallel circuit, maximum power occurs when RL = RTH Applications of Thevenin’s Theorem • Multiload Circuits Insert Figure 7.30 Norton’s Theorem • Norton’s Theorem – any resistive circuit or network, no matter how complex, can be represented as a current source in parallel with a source resistance Norton’s Theorem • Norton Current (IN) – the current through the shorted load terminals Insert Figure 7.35 Computation of Norton current Norton’s Theorem • Norton Resistance (RN) – the resistance measured across the open load terminals (measured and calculated exactly like RTH) Norton’s Theorem • Norton-to-Thevenin and Thevenin-to-Norton Conversions Insert Figure 7.39 Step-by-step Thévenin/NortonEquivalent-Circuit Analysis 1. Perform two of these: a. Determine the open-circuit voltage Vt = voc. b. Determine the short-circuit current In = isc. c. Zero the sources and find the Thévenin resistance Rt looking back into the terminals. 2. Use the equation Vt = Rt In to compute the remaining value. 3. The Thévenin equivalent consists of a voltage source Vt in series with Rt . 4. The Norton equivalent consists of a current source In in parallel with Rt . Maximum Power Transfer The load resistance that absorbs the maximum power from a two-terminal circuit is equal to the Thévenin resistance. Power transfer between source and load Graphical representation of maximum power transfer