1. Lecture 4 Kirchoff: The Man, The Law, The Field EC 2 Polikar

2. Ohm: The law The amount of current passing through a conductor is directly proportional to the voltage across the conductor and inversely proportional to the resistance of the conductor. + _ V R I V: The voltage difference between two locations, aka potential difference. It is analogous to pressure. I: The amount of current flowing between these two points. It is analogous to flow. R: Resistance between the two points. So how did Ohm come with this law…?

3. Power A conductor’s resistance to electric current produces heat. The greater the current passing through the conductor, the greater the heat. Also, the greater the resistance, the greater the heat. A current of I amps passing through a resistance of R ohms for t seconds generates an amount of heat equal to joules (a joule is a unit of energy equal to 0.239 calorie). Energy is required to drive an electric current through a resistance. This energy is supplied by the source of the current, such as a battery or an electric generator. The rate at which energy is supplied to a device is called power, that is power is energy supplied per unit time. Power is often measured in units called watts. The power P supplied by a current of I amp passing through a resistance of R ohms is given by

4. Gustav Robert Kirchoff Born: 12 March 1824 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 17 Oct 1887 in Berlin, Germany

6. Kirchoff’s Laws  Two sets of rules define circuit analysis:  Kirchoff’s current laws  Kirchoff’s voltage laws  KCL: The algebraic sum of all currents in any given node is zero  KVL: The algebraic sum of all voltages in any given loop is zero  What is a node?  What is a loop?  What is an algebraic sum…?

7. Assumptions & Definitions  Wires: We connect the components to each other by wires.  Wires, just like any other component in a circuit, has an internal resistance. However, since this resistance is so small, it is usually ignored. We will “model” the wire as a zero-resistance device, just like we model the ideal current meter as a zero-resistance device.  Nodes: A node is defined as the point where two or more components are connected to each other. You must show care, however, in locating and counting the nodes. + - v A R C R D R F R E R B How many nodes are there in this circuit?

8. Definitions  Loops: Any trace that starts and ends at the same point is a “closed loop”.  A loop typically follows components, but it can, jump across “space”.  We need to understand how to locate, identify and count loops.  Short / open circuit: A short circuit is simply connection of wires with no other components in between. Open circuit is a lack of connection between two nodes. How many loops are there in this circuit? + - v 1 R 1 R 2 R 4 R 3 R 5 + vava Can you identify short and open circuits ? -

9. Kirchoff’s Current Law (KCL)  Algebraic sum of all currents at any given node must be zero.  This is equivalent to saying sum of all currents entering a node (let’s call these negative currents) and the sum of all current leaving a node (and let’s call these positive currents), must be equal.  In other words, no current (or no charge) may accumulate at a node. Current must flow! A B + V _ i1i1 i3i3 i5i5 i7i7 i2i2 i4i4 i6i6 i 11 i9i9 i8i8 i 10 For node A: -i 1 +i 2 +i 3 =0  i 1 =i 2 +i 3 How about other nodes?

10. Kirchoff’s Voltage Law (KVL)  The algebraic sum of all voltages in a loop must sum to zero!  The law states that energy must be conserved.  This is where things may get little tricky, since you must follow a norm for defining polarities:  You may follow this simple rule: A voltage source produces energy (power), so there is an increase in potential energy when you move from its – to + terminal.  Resistors consume energy / dissipate power, so there is a potential drop across the terminals of a resistor, equivalent to the value of the resistor times the current passing through that resistor (by Ohm’s law). + _ V R1R1 I R2R2 For the loop on the left: +V- I.R 2 – I.R 1 = 0, or V=I(R 1 +R 2 )

11. More on KVL  How about this circuit?  A good practice is always to choose your current directions first, and then follow all potential drops and rises. Do not be concerned about choosing the right direction. If at the end you find out that the results for a current is negative, that means you picked the polarity wrong. A B + V _ i1i1 i3i3 i5i5 i7i7 i2i2 i4i4 i6i6 i 11 i9i9 i8i8 i 10

12. Open & Short Circuits  Remember:  No current passes through an open circuit, however, there can, and usually is a potential difference (voltage) across the terminals of the open circuit.  The potential difference between two points on a short circuit is, be definition, zero; however, there can, and usually is a nonzero current flowing through a short circuit. A B + V _ i CD =0 + v CD - i BE ≠0 v BE =0 C D E