NETWORK THEOREMS

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Superposition Theorem: Two Loop Problem To apply the superposition theorem to calculate the current through resistor R1 in the two loop circuit shown, the individual current supplied by each battery is calculated with the other battery replaced by a short circuit. For R1 =Ω, R2 = Ω, R3 = Ω, and voltages V1 = V and V2 = V, the calculated currents are = A, = A with a resultant current in R1 of = A. Note: To avoid dealing with so many short circuits, any resistor with value zero will default to 1 when a voltage is changed. It can be changed back to a zero value if you wish to explore the effects of short circuits. Ohms and amperes are the default units, but if you put in resistor values in kilohms, then the currents will be milliamperes. Other approaches to two-loop circuits

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Superposition Theorem The total current in any part of a linear circuit equals the algebraic sum of the currents produced by each source separately. To evaluate the separate currents to be combined, replace all other voltage sources by short circuits and all other current sources by open circuits. Application to two-loop problem	Index DC Circuits
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Thevenin's Theorem Any combination of batteries and resistances with two terminals can be replaced by a single voltage source e and a single series resistor r. The value of e is the open circuit voltage at the terminals, and the value of r is e divided by the current with the terminals short circuited. Thevenin voltage Thevenin resistance Numerical example Norton equivalent to AC Version Application in Digital to Analog Converter

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Thevenin Voltage The Thevenin voltage e used in Thevenin's Theorem is an ideal voltage source equal to the open circuit voltage at the terminals. In the example below, the resistance R2 does not affect this voltage and the resistances R1 and R3 form a voltage divider, giving Thevenin resistance Numerical example Norton equivalent	Index DC Circuits
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Thevenin/Norton Resistance The Thevenin resistance r used in Thevenin's Theorem is the resistance measured at terminals AB with all voltage sources replaced by short circuits and all current sources replaced by open circuits. It can also be calculated by dividing the open circuit voltage by the short circuit current at AB, but the previous method is usually preferable and gives The same resistance is used in the Norton equivalent. Thevenin voltage Numerical example Norton equivalent

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Thevenin Example Replacing a network by its Thevenin equivalent can simplify the analysis of a complex circuit. In this example, the Thevenin voltage is just the output of the voltage divider formed by R1 and R3. The Thevenin resistance is the resistance looking back from AB with V1 replaced by a short circuit. For R1 =Ω, R2 = Ω, R3 =Ω, and voltage V1 = V the Thevenin voltage is V since R1 and R3 form a simple voltage divider. The Thevenin resistance is = Ω. Apply to two-loop problem AC example Norton equivalent

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Norton's Theorem Any collection of batteries and resistances with two terminals is electrically equivalent to an ideal current source i in parallel with a single resistor r. The value of r is the same as that in the Thevenin equivalent and the current i can be found by dividing the open circuit voltage by r. Norton current Norton resistance Numerical example Thevenin equivalent	Index DC Circuits
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Norton Current The value i for the current used in Norton's Theorem is found by determining the open circuit voltage at the terminals AB and dividing it by the Norton resistance r. Norton resistance Numerical example Thevenin equivalent	Index DC Circuits
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Norton Example Replacing a network by its Norton equivalent can simplify the analysis of a complex circuit. In this example, the Norton current is obtained from the open circuit voltage (the Thevenin voltage) divided by the resistance r. This resistance is the same as the Thevenin resistance, the resistance looking back from AB with V1 replaced by a short circuit. For R1 = Ω, R2 = Ω, R3 = Ω, and voltage V1 = V the open circuit voltage is = V since R1 and R3 form a simple voltage divider. The Norton resistance is = Ω. and the resulting Norton current is = A Apply to two-loop problem Thevenin equivalent

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