Engineering Science EAB_S_127 Electricity Chapter 3 & 4

Overview  Measurement of electrical resistance  The Wheatstone Bridge  Capacitance  Energy stored in a capacitor  Charging and Discharging through a resistor  Time constants

The Wheatstone Bridge  We use an “Ohmmeter” to measure an unknown resistance  The heart of the simplest Ohmmeter is a so-called “Wheatstone Bridge” circuit  If R 1 was a variable resistor, we can adjust it until V ab = 0

The Balanced Wheatstone Bridge  When V ab = 0, a special condition occurs: the bridge is said to be “balanced”, i.e. V a = V b  This implies that i g = 0, hence from KCL, i 4 = i 3 and i 2 = i 1  Further, from Ohm’s Law; i 4 R 4 = i 2 R 2 and i 3 R 3 = i 1 R 1

The Wheatstone Bridge continued  Hence

The Wheatstone Bridge: Example  Calculate R 1 in a Wheatstone bridge when it is balanced and when R 2 = 300 Ω, R 3 = 200 Ω, R 4 = 100 Ω.  Answer:

Capacitance  Capacitors are devices which store electrical charge  A capacitor consists of two plates separated by an insulator, as shown in Figure 4.1  The negative plate is connected to a low potential and the positive plate to a high potential Figure 4.1 Q V Positive plate Negative plate Insulator

Capacitance continued  The total amount of the charge stored, is denoted by Q and the voltage across the plates by V  The capacitance then is defined as  Where C is in Farads  1 Farad = 1 Coulomb per Volt Figure 4.1 Q V Positive plate Negative plate Insulator

Energy Stored in a Capacitor  When charged, a capacitor stores electrical energy  Recall the formula for electrical energy in a circuit, which is W = VQ  However, we need to be careful as the voltage between the plates in a capacitor varies from 0 to V  Hence, to be more accurate we should use the average voltage  Soand we know  Hence

Energy Stored in a Capacitor: Example  Question: A capacitor is supplied with 10 V in a circuit. If its capacitance is 150µF, what is the electrical energy stored in the capacitor?  Answer:

Charging and Discharging a Capacitor  Charging and discharging a capacitor from a DC (direct current) source is shown below  We assume that the voltage source, V, has no internal resistance  If the switch was held in position 2 for a long time, then the capacitor would be completely discharged, V c = 0V V

Charging a Capacitor  If the switch is then moved to position 1, current will start to flow through the resistor R, thereby charging the capacitor, C  The voltage across the plates of the capacitor will rise in time, until after a long time, the capacitor will have the same voltage as the supply, V V VCVC

Discharging a Capacitor  If the switch is then moved back to position 2, current will start to flow through the resistor R, thereby discharging the capacitor, C  The voltage across the plates of the capacitor will fall in time, until after a long time, the capacitor will have no charge at all and again, V c = 0V V VCVC

Time Constant of an RC Circuit  It can be shown mathematically, that the time for the voltage to fall to 37% of its original voltage,  = RC  The charging and discharging curves have an exponential nature  When discharging  When charging VCVC VCVC

RC Time Constant: Example  Question: If R = 1000  and C = 0.1  F, what is the time constant of the circuit?  Answer:  = RC = 1000x0.1x10 -6 = 0.1 x10 -3 = 100  s  Hence, when discharging, the following equation can be used to calculate the voltage  When charging

Summary  Learning Outcomes:  Wheatstone Bridge  Balanced Condition  Capacitors and capacitance  Energy stored in a capacitor  Charging a capacitor  Time constants  Exponential charging and discharging curves