Kirchhoff's Voltage Law (KVL)
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1 Kirchhoff's Voltage Law (KVL)
2 Voltage conservation law Analogy: Energy Balance Water Distribution Network Typical network includes the energy sources and loads Energy source Load Tap water Pond/Lake/iver/Sea 3 rd floor nd floor st floor Potential Energy Position in the network Water potential energy increases from zero to high level by the pump and then gradually goes back to zero as the water passes through the load
3 Energy Balance in Electric Networks 0 9 V ϕ ϕ ϕ3 ϕ4 ϕ5 ϕ6 Electric Potential (same 9 as ) Position in the network
4 Voltage Change along the loop V = ϕ ϕ 9 V ϕ ϕ3 ϕ4 ϕ5 V3 = ϕ3 ϕ V43 = ϕ4 ϕ3 ϕ ϕ6 V54 = ϕ5 ϕ4 V54 = ϕ5 ϕ4 V65 = ϕ6 ϕ5 V6 = ϕ ϕ6 V =0
5 Kirchhoff's Voltage Law (KVL) "The algebraic sum of all voltages in a loop must equal zero discovered in 847 by Gustav. Kirchhoff, a German scientist
6 KVL for multi-mesh multi-node circuits Example: potential distribution along the mesh b b 0. Assume the current directions, i b (through b ), i 4 ( 4 ) and i 3 ( 3 ). Enumerate all the different nodes along the mesh path 3. Move along the path and account for the potential changes (i.e. the voltages).
7 KVL for multi-mesh circuits Let us analyze the potential distribution along the mesh b Node Voltage Potential b 3 0 +E b -i b * b -i 4 * 4 -i 3 * 3 +E b +E b -i b * b +E b -i b * b -i 4 * 4 +E b -i b * b -i 4 * 4 - i b * 3 0 (!) When passing the resistor along the current direction, the potential decreases (the voltage is negative); When passing the resistor against the current direction, the potential increases (the voltage is positive); The total potential change along the close path is equal to zero (the starting point and the end point are the same!). Therefore, the KVL for the mesh b is +E b -i b * b i 4 * 4 i 3 * 3 = 0
8 Series electric circuits Current flow Electrons flow Three resistors (labeled,, and 3), connected in a chain from one terminal of the battery to the other. In a series circuit (or a sub-circuit), there is only one path for current to flow.
9 Series electric circuits ϕ ϕ Current flow V B Electrons flow ϕ4 = 0 ϕ3 Let us assign the potentials to all the nodes. Let us find the voltages across the battery and all the resistors: V B = ϕ ϕ4 V = ϕ ϕ V 3 = ϕ3 ϕ V 43 = ϕ4 ϕ3 V B + V +V 3 +V 43 = (ϕ ϕ4)+ (ϕ ϕ)+ (ϕ3 ϕ)+ (ϕ4 ϕ3)=0
10 Series electric circuits ϕ ϕ Current flow V B Electrons flow ϕ4 = 0 ϕ3 V B + V +V 3 +V 43 = 0 this is the KVL From the Ohm s law: V = - I V 3 = - I V 43 = - I 3 V B = I + I + I 3 V B I = + + 3
11 Simple series circuits V = I ; + - E = V =V V 3 =V 34 I = E / ( ); I = 9V/8kΩ = 0.5 ma; V = 0.5 ma 3 kω =.5 V V = I = 5 V; V 3 = I 3 =.5 V; V =V Σ(V ij ) = = 9V = E +
12 Series circuit rule: V + - E = + - V Circuit Current = I - + Input voltage Sumof all the resistances = V 3 E The voltage across any resistor can be found from the Ohm s law: For instance, V = V = I...
13 Equivalent resistance eq Can we replace these three resistors with just one to simplify the circuit? What replacement do we call equivalent? Single resistor is equivalent to several resistors connected in series if the current in the circuit remains the same. In the actual circuit: I = E / ( ); In equivalent circuit: I = E / eq ; From these: eq = ; For the series connection, the equivalent resistance = sum (all the series resistances)
14 Equivalent resistance of the series circuit eq If some resistors in the network or a part of it, are connected in series, then the equivalent resistance is: eqs = ;
15 Equivalent resistance of the series circuit Any series connection of two or more resistors can be replaced by the equivalent resistor, not necessarily the whole circuit eq67 = = 4 kohm eq34 = = 4 kohm
16 Series circuit as a voltage divider + - E = + - V OUT = V - + Using the series circuit rule, I = E + V = I = E + The output voltage V is a fraction of the input voltage E, defined by the ratio of the output resistance over the total circuit resistance.
17 Voltage distribution in a series circuit E I V + = = E = V V E I V + = = V V =
18 Series and parallel circuit summary Series Parallel Same current flows through all the series components. I = I = I 3 =. Total voltage drop is equal to the sum of voltage drops across the series components: V T = V + V + V 3 + The equivalent resistance is equal to the sum of all the series resistances: EQ = Voltage division (for -element circuit): V = V Same voltage is applied across all the parallel components. V = V = V 3 =. Total current is equal to the sum of the currents through all the parallel components: I T = I + I + I 3 + The equivalent conductance is equal to the sum of all the parallel conductances: G EQ = G + G + G 3 + or / EQ = / + / +/ 3 + Current division (for -element circuit): I G = = I G
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