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ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS Independent voltage source Ideal source: § Across its terminals maintains voltage with prescribed waveform § It is able to deliver any current, including infinite, depending on load, to preserve given voltage § It is able to deliver infinite power Symbol Example of the voltage time function Loading characteristics at time instant tk Actual voltage source: § Deliverable power is finite § Maximum value of the delivered current is limited Symbol: Load characteristic: u(t) = ui(t) ¡ f [i(t)] u(t) = ui (t) ¡ Ri i(t) ui ik open-circuited (internal) voltage short-circuit current Loading characteristic of linear (thick black line) and example of non-linear (thin blue curve) voltage source Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS -1- Power, delivered to the circuit by voltage source: Pu = U ¢ Iu where Iu is current passing the voltage source (positive sign has current outgoing positive terminal of the voltage source) º can be negative (consuming power, e.g. accumulator in charger) Independent current source Ideal source: § Across its terminals maintains current with prescribed waveform § It is able to reach infinite voltage, depending on load, to keep delivering given current § It is able to deliver infinite power Loading characteristics at time instant tk Symbol: Example of the current time function Actual current source: § Deliverable power is finite § Maximum value of the terminal voltage is limited Symbol: Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS Load characteristic: i(t) = ii(t) ¡ g[u(t)] i(t) = ii (t) ¡ Giu(t) -2- ui ik open-circuited (internal) voltage short-circuit current Loading characteristic of linear (thick black line) and example of nonlinear (thin blue curve) current source !!! Warning !!! – These two connections from the point of view of output terminals acts like an ideal sources Ri Ii Ui Ri Power, delivered to the circuit by the current source: Pi = Ui ¢ I where Ui is voltage across current source (positive orientation of voltage is from the terminal, from which the current is flowing out) º can be negative (consuming power) Source compatibility If both loading characteristics are the same, we cannot distinguish from the point of view of output terminals (by measurement), if the source is actual voltage source, or current source Æactual sources can be replaced by compatible sources 150W i i u 75V u up = ui ; ik = any loading circuit ui Ri 0.5A 150W up = Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS ii = iiRi; Gi any loading circuit ik = ii -3- It is not possible to substitute following circuits – sources are still ideal! Example: I = 1A, R1 = 100 W, R2 = 200 W, R3 = 300 W Ui = IR2 = 200 V Ri = R2 + R3 = 500 Ð Thevenin’s theorem § Any linear active two-terminal („black box“, containing any number of circuit elements – sources, resistors, inductors, capacitors, …) can be replaced by the series connection of a voltage source and a passive two-terminal (resistor, or connection of resistor, capacitor or inductor, resulting on impedance) § The value of a passive two-terminal resistance (impedance) is total resistance (impedance) of whole active two-terminal after removing of all sources, from the point of view of terminals § Removing of voltage source: source is short-circuited (ideal voltage source has zero internal resistivity) § Removing of current source: source is opened (ideal current source has infinite internal resistivity /and not loaded has infinite voltage across its terminals/) § It is not possible to remove controlled source from the circuit! § Total resistivity (impedance) is tangent of loading characteristics, event the circuit contains controlled sources Æ the only method, if the circuit contains controlled sources Up Ri = Ik § Not valid for evaluation of total power of the circuit (different currents and voltages) – power is not linear function! Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS -4- Norton’s theorem § Any linear active two-terminal can be replaced by the parallel connection of a current source and a passive two-terminal § The value of a passive two-terminal resistance (impedance) is total resistance (impedance) of whole active two-terminal after removing of all sources, from the point of view of terminals Controlled sources § the value of voltage (current) is controlled by another circuit variable (voltage or current) § practical examples of devices containing controlled sources – transistor, operational amplifier voltage controlled voltage source current controlled voltage source uv = Kur uv = Rir voltage controlled current source current controlled current source iv = Hir iv = Gur Equivalence of active two-terminals Series connection of voltage sources n X u(t) = uk (t) k=1 Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS -5- Parallel connection of current sources n X i(t) = ik (t) k=1 ≈ ≈ Voltage source has 0 internal resistivity, it is short circuit with respect to current source and current supplied by current source passes voltage source freely; current source has ∞ resistivity with respect to voltage source, thus voltage source is disconnected Voltage source has 0 internal resistivity, it is short circuit with respect to current source and current supplied by current source passes voltage source; no current could be delivered to the connected circuit; current source has ∞ resistivity, it has no effect on voltage source Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS -6- Voltage source replacement: Current source replacement: The left terminal of circuit elements D1, D2 has voltage u(t); in the branch A the voltage source is disturbed by another voltage source connected in opposite way; to maintan voltage u(t) in branches B, C it is necessary connect another two voltage sources u(t) By connecting of second current source in series it is possible to add shorting strap to the C terminal; current passing strap is 0 (whole current pass only current sources) so the conditions are same like without strap Dividing of the circuit: If the circuit has two parts, connected together by two (or more) wires (these are two or more terminals), it is possible to divide the circuit on two independent parts, if the voltage conditions keep unchanged (we will connect two voltage sources maintaining the same value of voltage) Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS -7- Example: R1 R3 U1 R2 R4 → R1 + U1 R2 R3 U1 R4 Fundamental circuits – voltage divider · 2 or more circuit elements connected in series · common circuit variable – current U1 I= R1 + R2 U1 U2 = R2I = R2 R1 + R2 R1 I U1 R2 U2 U 2 = U1 Easy extendable to N resistors: R2 R1 + R2 Uj = U1 – current divider Rj N X Ri i=1 · 2 or more circuit elements connected in parallel · common circuit variable – voltage U = RI = I R1 I2 = R2 U R1 R2 I R 1 + R2 U R1 R2 1 =I R2 R1 + R2 R2 I2 = I R1 R1 + R2 Extension to N resistors is more complicated – e.g. for 3 resistors R1 R2 R3 R = ( R11 + R12 + R13 )¡1 = R1 R2 +R 1 R 3 +R2 R3 Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS -8- Step by step simplification method · the aim is to simplify circuit by searching of series and parallel connections of circuit elements starting from the output variable · this procedure is repeated step by step until reaching some fundamental circuit where it is easy to find voltage and passing current · then, we will return back to the original circuit, step by step, dividing voltages and currents example: Analyzed circuit · The aim is compute voltage U2 R1 U1 R2 R3 R4 Step 1 We can combine series connected resistors R3 and R4 together R1 U1 R2 R34 R1 U1 R234 U234 R1 U1 R3 R2 U234 U2 R4 U2 R34 = R3 + R4 Step 2 Now we will combine parallel connected resistors R1 and R34; Finally, resistors R1 a R234 form voltage divider, so that we can find voltage across resistor R234 R2 ¢ R34 R234 R234 = ; U234 = U1 R2 + R34 R1 + R234 Step 3 – back to the original circuit Now I know total voltage across resistors R3 a R4, so that I can use voltage divider rule to find voltage across resistor R4 – requested voltage U2 R4 U2 = U234 R3 + R4 Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS -9- Superposition theorem R1 I Ux R1 R3 R2 U I Ux1 R3 R1 R2 Ux2 = R3 R2 U + § If the circuit is supplied by N independent voltage or current sources, it is possible to remove N -1 sources, so that the circuit will be supplied just by one source § Repeat previous step for each source in the circuit, so we obtain N distinct results § Wanted circuit variable is the sum of N contributions from distinct sources § NOT APPLICABLE if the circuit is non-linear! § NOT APPLICABLE on controlled sources! § NOT APPLICABLE on power contributions from distinct sources to total power on resistors, total current passing resistor (or voltage across it) must be calculated first (using superposition theorem) and then power, P = RI 2 (power is not linear function) Example – the circuit above, R1 = 100 W, R2 = 300 W, R3 = 200 W, U = 250 V, I =1A Ux = Ux1 + Ux2 = I R2 R2R3 +U = 120 + 150 = 270 V R2 + R3 R2 + R3 , P R2 2702 ! = = 243 W 300 2 PU = U IU = U ( R2 U+R3 ¡ I R2R+R )= 3 PI 300 = 250 ¢ ( 250 500 ¡ 1 ¢ 500 ) = 250 ¢ (¡0:1) = ¡25 W h i R2 R3 = UI ¢ I = I ¢ (R1 + R2+R3 ) + Ux2 I = = [1 ¢ (100 + 120) + 150] ¢ 1 = 370 W Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS - 10 - Delta-star transformation 1 1 R01 R03 R12 R31 R02 2 3 3 R23 2 § The properties of all terminal pairs must be the same in both delta (∆) and star (wye, or Y) connections § In fact, total resistivity between distinct terminal pairs must be the same in both connections: R01 + R02 = R12(R23 + R31) R12 + R23 + R31 R02 + R03 = R23(R31 + R12) R12 + R23 + R31 R01 + R03 = R31(R12 + R23) R12 + R23 + R31 § Then, ∆-Y transformation (replacement of ∆ connection by equivalent Y connection) is R01 = R12R31 R12 + R23 + R31 R02 = R12R23 R12 + R23 + R31 R01 = R23R31 R12 + R23 + R31 Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS - 11 - § Y-∆ transformation (replacement of Y connection by equivalent ∆ connection) a) Total conductance between distinct terminal pairs must be the same in both connections: G03 (G01 + G02) G31 + G23 = G01 + G02 + G03 G12 + G23 = G02 (G01 + G03) G01 + G02 + G03 G12 + G31 = G01 (G02 + G03) G01 + G02 + G03 b) Then, G12 = G01G02 G01 + G02 + G03 G31 = G01G03 G01 + G02 + G03 G23 = G02G03 G01 + G02 + G03 c) Finally, R12 = R01 + R02 + R01R02 R03 R23 = R02 + R03 + R02R03 R01 R31 = R03 + R01 + R03R01 R02 Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS - 12 - Example: R14 R4 R1 R3 R2 Ux = U R14 = R34 R13 Ux In this circuit, resistors R1, R3 and R4 form ∆ connection. Computation of voltage Ux is laborious using step by step simplification or other methods. R4 R3 R1 R2 Ux Here, the resistors R1, R3 and R4 were transformed from ∆ connection to Y connection. Now, it is just simple voltage divider. R2 + R13 R14 + R13 + R2 R1R4 , R1 + R3 + R4 R13 = R1R3 , R1 + R3 + R4 R34 = R3R4 R1 + R3 + R4 Mnemonic: if in distinct node in ∆ connection are connected resistors Rx and Ry, then is convenient denote resistor connected into same node in Y connection Rxy. Then in nominator is product RxRy, in denominator sum of all three resistors in ∆ connection. Opposite procedure: R14R13 R1 = R14 + R13 + , R34 R4 = R14 + R34 + R3 = R13 + R34 + R13R34 , R14 R14R34 R13 Mnemonic: from Y to ∆ connection – if between two nodes are connected resistors Rx andR Ry in Y connection, then the resistivity of resistor, connected R R between same nodes is Rx + Ry + Rx z y . Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS - 13 - Coupled and not coupled inductors Sometimes it could be useful replace mutual coupled inductors by ordinary uncoupled inductors. If coupled inductors are connected into same node, then the replacement is: Pavel Máša – ELECTRICAL CIRCUITS 1 – LECTURE 3: ELEMENTARY ANALYSIS OF RESISTIVE CIRCUITS - 14 -