Basic DC Circuits Review

Prefixes Prefixes come in handy when trying to express high or low numbers. Prefixes Symbol Value atto a 10-18 femto f 10-15 pico p 10-12 nano n 10-9 micro µ 10-6 milli m 10-3 kilo k 103 mega M 106 giga G 109 tera T 1012 EXAMPLES: 1,000 1 k 5.68×10-3 5.68 m 0.000056 56 µ 1,212,000,000 1.212 G 0.000000000005 5 p 2.5×10-10 250 p

Circuit Conventions There will be some circuit symbols used in illustrations before they are properly discussed in this presentation. In order to prevent confusion, these symbols will be briefly introduced in this slide and later discussed in more detail. Representations of voltage supplies used in this tutorial. Independent Voltage Source Batteries Dependent Ground Ground is used as a reference and is typically at a 0 volt potential. Representations of current supplies used in this tutorial. Independent Current Source Resistor Dependent

(Independent Voltage Source) Voltage (V) Voltage is an electrical pressure which causes current to flow through a resistance. It is measured in volts (V). Two common DC voltage supplies are shown below:  (Batteries) (Independent Voltage Source) The “long side” or + terminal of a battery is called the anode. The “short side” or – terminal of a battery is called the cathode.

Voltage Voltage can be compared to the pressure of water in a tank. As the height of water in a tank increases, so does the water pressure. This increase in pressure causes more water to flow out of an opening in the bottom of a tank, much like how a higher voltage (higher electrical pressure) produces more current through a resistance. 2 A 1 A 3 A 3 Ω 9 V 3 V 6 V

Voltage Polarity Vo = 9.15 V Vo = -9.15 V Voltage polarity is denoted by a + and – symbol. When connecting the positive (red) lead of a multimeter to the positive terminal of the battery with the negative (black) lead to the negative terminal of the battery, a positive value of voltage will be displayed. However, if you were to connect the red lead to the negative terminal and the black lead to the positive terminal, a negative voltage would be displayed. Here we get -9.15 V, since the red lead is hooked to the negative terminal of the battery and the black is connected to the positive terminal. Here we get 9.15 V, since the red lead is hooked to the positive terminal of the battery and the black is connected to the negative terminal. Vo = 9.15 V Vo = -9.15 V - + 9.15V -9.15 V The multimeter is essentially an open circuit when measuring voltage. 9.15 V Vo Digital Multimeter - + - + Let’s look at the following source: Now let’s see what happens when polarity is reversed.

The arrow-head denotes the positive side. Voltage Polarity There are two basic ways that we can represent voltage polarity. The first is a plus/minus representation (shown on the left) and the second is an arrow. Each of these representations show equivalent polarities. In the example below we are showing the voltage polarity across a resistor. + 10 V 10 V 10 V 10 V - The arrow-head denotes the positive side.

Voltage Addition & Subtraction Notice that the black lead is connected to the positive terminal of the 9.08 V source, and the red lead is connected to the negative terminal of the 8.80 V source. We can find Vo by adding these sources in series, realizing that the configuration sums two negative voltages (remember the - to +, - to + relationship). Therefore: Vo = V1 + V2 = (-8.80 V) + (-9.08 V) = -17.88 V When appearing in series, multiple voltage sources can be added to or subtracted from one another depending on their terminal connections. In the following example we will be working with two batteries which have additive polarities, meaning that with respect to ground both batteries have the same terminal configurations. In regards to the following example this means that the positive terminal of one battery is connected to the negative terminal of the other battery, giving (from ground) a negative to positive, negative to positive configuration. Notice that the black lead is connected to the negative terminal of the 8.80 V source, and the red lead is connected to the positive terminal of the 9.08 V source. This allows us to find Vo by the following equation due to the - to +, - to + terminal connections. Therefore: Vo = V1 + V2 = 8.80 V + 9.08 V = 17.88 V Note: With this polarity of voltage, the value for V2 is 9.08 V. Note: With this polarity of voltage, the value for V1 is 8.80 V. Note: With this polarity of voltage, the value for V2 is -9.08 V. Note: With this polarity of voltage, the value for V1 is -8.80 V. V1 = 8.80 V V1 = -8.80 V V2 = -9.08 V V2 = 9.08 V - + + - V2 9.08 V -9.08V -17.88V 8.80V 9.08V 17.88V -8.80V + - Vo + - Digital Multimeter - + V1 8.80 V + - + - Vo = -17.88 V Vo = 17.88 V

Voltage Addition & Subtraction For this opposing configuration of V1 and V2 (remember the – to +, + to – relationship with respect to ground discussed earlier), we find the value of Vo by the following equation: Vo = V1 + V2 = 9.15 V + (-8.85 V) = 0.3 V For polarity of the leads switched and remembering the – to +, + to – relationship with respect to ground discussed earlier, Vo can be found using the following equation. Vo = V1 + V2 = (-9.15 V) + 8.85 V = -0.3 V Now, with a slightly different configuration of batteries we see what happens when the polarities oppose one another. When two voltage sources oppose each other it means that their terminal connections are opposite with respect to ground. In our next example we have a negative to positive configuration with respect to ground connected to a positive to negative configuration. With voltage polarities in this manner, the sources will work to cancel each other out. Note: With this polarity of voltage, the value for V1 is -9.15 V. Note: With this polarity of voltage, the value for V2 is -8.85 V. Note: With this polarity of voltage, the value for V2 is 8.85 V. Note: With this polarity of voltage, the value for V1 is 9.15 V. Notice that the V2 measurement polarity is opposing the polarity of the battery, therefore the value for V2 is -8.85 V in this case. - - + + V2 8.85 V + - 9.15V -8.85V -9.15V -0.3V 0.3V 8.85V Vo - + Digital Multimeter - V1 + 9.15 V + - + - Here we have the measurement polarity of V1 opposing that of the 9.15 V battery. This gives a value for V1 of -9.15 V. Vo = -0.3 V V2 = 8.85 V V1 = 9.15 V V1 = -9.15 V Vo = 0.3 V V2 = -8.85V

Voltage Addition & Subtraction Let’s look at an example with more than two batteries: Since the polarity of V3 is opposite of the 10 V source, the value for V3 is -10 V. Since we now know the values for V1, V2, V3, and V4 we can calculate the value of Vout. Since V1, V2, and V4 all have polarities matching those of their corresponding voltage sources, the values for V1, V2, and V4 are 10 V. This concept was shown previously with the digital multimeter. For simplicity, we will assume that all the batteries are 10 volts. 10V V1 10V V2 Vout Vout = V1 + V2 + V3 + V4 10V V3 10 V 10 V -10 V 10 V Vout = 20 V 10V V4 Note: We can add V1, V2, and V4 due to their direction. However, the difference in direction in V3 means that its value must be subtracted.

Current (I) Current is the movement of electrons through a conductor, which is the time derivative of charge (dq/dt). It is established by a potential difference (or voltage) across a resistance and is measured in the quantity amperes or amps (A). The common DC current source is shown below: (Independent Current Sources) The arrow of the independent current source represents the direction of current flow.

Current Current is not across two points as is voltage, but flows through a circuit element. Note: Multimeter is set to measure current here. It essentially acts as a short circuit to take this measurement. Let’s consider the following circuit: 1 kΩ I -9 mA 9 mA 9 mA Digital Multimeter - + 1 kΩ - Current is denoted positive when entering the red (positive) lead. Current is denoted negative when entering the black (negative) lead. + I 9 mA - + Click to see what happens when leads are switched.

Current Current can only flow through a closed loop. It must travel where there is a defined path. This concept is pictured below with current depicted in red. R1 R3 Notice, there is zero current flow through R3, since there is no closed path for current to flow. R2

Current During analysis, the sign of current flowing from a positive terminal of a voltage supply to a negative terminal is considered positive by historical convention (shown below in red). However, experience tells us that negatively charged electrons flow in the opposite direction (shown below in purple). R1 +I -I R2

Nodes A node is a connection between one or more elements in a circuit. Here, the nodes of each circuit are circled in red. Notice that the wires composing each node have no resistance, thus there is no voltage drop within the red areas. Note: When taking measurements with a digital multimeter the negative lead is connected to ground (node 1).

Nodes Notice that this voltage reading is also 9 V and that the voltage dropped across R2 is equal to the voltage across R1. The reason is because both of these resistors share the same two nodes. From the previous slides, you might have known that this configuration would result in a read out of 9 V. The voltage across R3 and R4 (taken at the outer nodes) is also 9 V because these are the same two nodes as those shared across R2 and R1. For the same reason we also measure the same 9 V across R5. However, you might not have known that the voltage dropped across R1 was also 9 V. Next, we will use the digital multimeter to measure node voltages in a circuit containing two nodes. 9 V Digital Multimeter - + R1 R2 R3 R5 9 V R4

Nodes What result would we get when measuring the top (red) node in reference to the orange node located between R3 and R4? Now we are going to take a closer look at the previous example to see the effects of choosing a reference point when measuring node voltage. The measurement would be a value of voltage less than 9 V but greater than 0 V. Previously, when measuring the voltage across R3 and R4, we determined the voltage to be 9 V. This is because we are measuring the top (red) node in reference to ground (the purple node). Now, Can you guess what the multimeter would read when measuring the voltage of the top (red) node with respect to itself? If your answer was 0 V, then you were correct! 9 V 0 V < 9 V ??? R1 R2 R3 R5 Digital Multimeter 9 V - + R4

Branches A branch is a part of a circuit that contains one or more circuit elements in series with a separate node at each end. Notice that, the current flowing through a branch is equal for every element contained in the branch network. For example, IS I3 I1 I2 R1 Note: The current I1 flows through R1, R2, and R3. The value of I1 does not change through the branch. The same is true of I2 and I3. R5 R4 R2 V R6 R3

actual representation Resistance (R) Resistance is a hindrance/opposition to the passage of an electrical current. Resistance in a circuit is represented by a resistor. The unit of resistance is the ohm (Ω). The symbol used to represent a resistor is schematic capture actual representation Materials such as metal (conductors) have a small resistance, where materials such as rubber (insulators) have a large resistance.

actual representation Capacitance (C) Capacitance is the ratio of charge to voltage across two conductive elements (or plates). Capacitance is represented by a capacitor in circuits and measured in farads (F). A farad is a very large value of capacitance. A more likely value of capacitance would be 0.01 µF (1x10-8 F). The symbol used to represent a capacitor is + - schematic capture actual representation Some capacitors, especially electrolytic capacitors, are polarized.

Capacitance Continued When analyzing a steady-state DC circuit, capacitors act as open circuits—meaning there is no steady-state DC current flowing through them (infinite resistance). C R = ∞ Ω R1 C R1 V R2 R3 V R2 R3

actual representation Inductance (L) Inductance is the property of an electric circuit by which an electromotive force is induced as the result of a changing magnetic flux. This is represented in a circuit by an inductor. The unit of inductance is the Henry (H). The symbol used to represent an inductor is schematic capture actual representation

Inductance When analyzing a steady-state DC circuit, inductors act as short circuits— meaning that steady-state current is passed directly through them (zero resistance). L R = 0 Ω R1 L R1 V R2 R3 R2 R3 V

Material Properties Resistivity (ρ)– Resistivity is the intrinsic property that accounts for the nature of a material. It is defined as the ability of a material to resist electrical conduction, with units ohm-meter (Ωm).  The resistance of a material is related to its resistivity such that: R = ρ (L/A) where, L ρ = resistivity of material L = length of conductor which current flows along A = cross-sectional area of conductor that current flows through Some Material I h w

Material Properties Conductivity (σ) – Conductivity is the inverse of resistivity. It is defined as the ability of a material to conduct electricity, with units inversed ohm-meter (Ωm-1). The conductance of a material is related to its conductivity by: G = σ (A/L) = 1/R, where L σ = conductivity of material L = length of conductor which current flows through A = cross-sectional area of conductor that current flows through Some Material I h w