EE301 AC Source Transformation and Nodal Analysis

Transcription

1 EE0 AC Source Transformaton and Nodal Analyss Learnng Ojectves. Construct equvalent crcuts y convertng an AC voltage source and a resstor to an AC current source and a resstor. Apply Nodal Analyss to an AC Crcut DC Source transformaton Source transformaton s the process of replacng a voltage source vs n seres wth a resstor R y a current source s n parallel wth a resstor R, or vce versa. v R or s s s vs R AC Source transformaton A voltage source wth mpedance Z n seres s the same as a current source wth an mpedance Z n parallel. Example: Convert the voltage source to a current source Soluton: Example: Usng source transformatons, determne the voltage drop VR across the ohm resstor. Soluton:

2 EE0 AC Source Transformaton and Nodal Analyss Example: Usng source transformatons, determne the voltage drop VR across the 0 ohm resstor. Soluton: AC Nodal Analyss AC Nodal Analyss s exactly the same procedurally as DC Nodal Analyss. The only dfference s that the numers are now complex. Gven a crcut wth n nodes: Step. Select a reference node. Step. Assgn voltages Va, V, Vc. to the remanng nodes, dentfyng any known voltages. Step. Assume a drecton for the current passng through each resstor adjacent to a node wth an unknown voltage and express the ranch currents n terms of node voltages. Step 4. Apply KCL to each node. Step 5. Solve the resultng smultaneous equatons to otan the unknown node voltages. So, let s take t slow y consderng the crcut elow: 0 0V 0 Ω 7 0V

3 EE0 AC Source Transformaton and Nodal Analyss We frst select a reference node, and assgn laels to the remanng nodes. The crcut aove has four nodes. Let s select the reference node (ground) to e the ottom node (Step complete!), and let s lael the remanng three nodes as a,, and c, as shown elow: a c 0 0V 0 Ω 7 0V Now, each of the laeled nodes, a,, and c, wll have a voltage assocated wth t: Va, V, V c. Now we are not qute done wth Step ; we have to dentfy any know voltages. Look at the fgure aove: Are any of the quanttes Va, V, V c known? V 00, V UNK, V 7 0 Step complete! a c Next, for Step, we assume a drecton for the current passng through each resstor adjacent to a node wth an unknown voltage (node ). Stated another way, we are gong to wrte the ranch currents. We have chosen to lael the currents as,, n the drecton shown elow. a c 0 0V 0 Ω 7 0V But we re not done wth Step : we now have to express each of the ranch currents n terms of node voltages. Ths s done usng Ohm s Law, ut you have to e careful aout the polartes. The way I have laeled the drecton of assumes that the voltage drop across the 4 resstor has t s postve polarty at node. Stated another way, I am assumng that V s greater than V a. So, expressng the currents as,, n terms of the node voltages, we have: V V V 00V a 6 j8 6 j8 V 0 V j0 j0 V Vc V 70

4 EE0 AC Source Transformaton and Nodal Analyss Step complete! Next, we apply KCL to node. KCL KCL : 0 V 00V V V 70 : 0 6 j8 j0 Step 4 complete! Ths s one equaton wth one unknown. At ths pont the Nodal Analyss s essentally fnshed and the algera egns. You can solve ths wth some smple algera. It s left to the student to compute the value of V. Now that we know V we can determne any other quantty n the crcut (such as the currents,, ). That s t!! That s all there s conceptually to AC Nodal Analyss!! Example: Use Nodal Analyss to determne the voltage at node n the crcut elow. Soluton: 0 Ω 5 5V 5 Ω 0A Step &: Select a reference node and assgn voltages Va, V, Vc to the remanng nodes, dentfyng any known voltages. a c 0 Ω 5 5V 5 Ω 0A V 5 5, V UNK, V UNK a c 4

5 EE0 AC Source Transformaton and Nodal Analyss Note that, strctly speakng we don t know the voltage at node c however a KCL equaton at ths node would e trval and s not necessary. So we only need one KCL equaton and node : KCL : 0 Step s complete. In ths case, snce s a current source and constant, we can smply use the constant and there s no need to express t n terms of voltage and resstance. Note the polarty of s opposte the red arrow so t s negatve. V V V 55V a 0 0 V 0 V j5 j5 0A Now express the KCL equaton n terms of voltage and resstance: KCL V 00V V : 0A 0 0 j5 Step 4 s complete and ths s the end of the Nodal Analyss. From here on the prolem s strctly algera. Example: In lght of the prevous example, determne I UNK n the crcut aove, assumng I UNK ponts from node c to node. Soluton: 5