Ohm s Law, Kirchhoff s Laws and AC Circuits

Transcription

1 Appendix C Ohm s Law, Kirchhoff s Laws and AC Circuits C.1 Introduction This write up deals with the behaviour of circuits consisting of resistances, capacitances and inductances, and sinusoidal voltage sources. We assume that the source has been turned on for a long time (compared to any characteristic time constants of the circuit), and that we are dealing with the steady state circuit operation. (The transient operation of such circuits are discussed in write ups for individual experiments.) C.1.1 Reactance and Impedance A sinusoidal voltage V (t) can be written as: V (t) = V 0 cos (ωt + φ) (C.1) where V 0 is the amplitude, ω is the angular frequency in rads/sec, and φ is the phase angle in radians. The phase angle can be determined from V (t=0) : V (t=0) = V 0 cos φ (C.) When such a source is used to drive an RLC circuit (one containing resistors, capacitors and inductors), the current and voltage associated with any branch will also vary sinusoidally at the same frequency, but with a different amplitude and possibly a different phase angle.

2 C- Ohm s Law, Kirchhoff s Laws and AC Circuits Resistors For example, if Equation C.1 refers to the voltage applied to a simple resistor R, Ohm s Law tells us that: V (t) = I(t)R (C.3) and thus the current is given by: I(t) = V (t) R = V 0 R cos (ωt) (C.4) (C.5) where we have taken the phase angle φ to be zero for convenience. We see that the current through the resistor varies sinusoidally, and that it is in phase with the applied voltage. Capacitors Let us analyze the situation when a voltage (Equation C.1) is applied to a capacitor. The current flowing through the capacitor is, of course, the time derivative of the charge stored in the capacitor: and, since we have for the current: I(t) = dq dt Q(t) = CV (t) = CV 0 cos (ωt) (C.6) (C.7) I(t) dv (t) = C dt (C.8) = ωcv 0 sin (ωt) (C.9) Using the fact that ( sin θ = cos θ + π ) we can also write the current as: I(t) = ωcv 0 cos ( ωt + π ) (C.10) which shows that the current through the capacitor is also sinusoidal, but we see that there is a phase difference of π/ between the voltage and the

3 C.1 Introduction C-3 current. Thus the current leads the voltage, or more commonly, the voltage lags the current. Note that the phase difference of π/ means that the voltage across the capacitor is zero when the current is at its peak value. We can also write Equation C.9 in a form which looks like Ohm s Law: ( V 0 cos ωt + π ) = 1 ωc I(t) (C.11) = X C I(t) (C.1) where X C = 1 ωc and X C is known as the Capacitive Reactance and is somewhat analogous to resistance in DC circuits; ie. it is the ratio between the maximum voltage across the device and the maximum current through the device. Note that X C is frequency dependent. Inductors Lastly, consider what happens when a sinusoidal current flows through an inductor. The voltage across an inductance L is given by: V (t) = L di dt (C.13) and if the current is given by: I(t) = I 0 cos (ωt) (C.14) the voltage is then: V (t) = ωli 0 sin (ωt) (C.15) = ωli 0 cos ( ) ωt + π (C.16) Thus the voltage across the inductor leads the current by a phase angle of π/. Analogous to the case above, X L = ωl and now the Inductive Reactance is X L.

4 C-4 Ohm s Law, Kirchhoff s Laws and AC Circuits C.1. Ohm s Law and AC Circuits The total of resistances and reactances in a circuit or a branch is called the impedance Z, where V = IZ (C.17) This is the AC analog of Ohm s Law for DC circuits. It looks similar to Ohm s Law for DC circuits, but now the phases of I, V, and Z must be taken into account. Since they have both a magnitude and a phase, then it is clear that V, I, and Z are all vectors. In addition, I and V can be time dependent (in general Z is not) and so V and I may be better represented as V and I, and Equation C.17 is better written as V = IZ (C.18) C.1.3 Phasors To determine the impedance of a circuit, (i.e. its resistance including both magnitude and phase information), and also the voltages and currents, it is very convenient to introduce the use of complex algebra. That is, we represent voltages, currents and impedances by complex quantities, with the implicit understanding that we take the real parts of final answers to compare with measured values. This is an eminently sensible thing to do, since our instruments are not capable of measuring anything but real quantities! With this in mind, we can make use of the extremely important result: e ±jθ = cos θ ± j sin θ (C.19) where j = 1 (Here j replaces i to avoid confusion between i and I.) Let us use Equation C.19 to represent a sinusoidal voltage: V (t) = V 0 cos ωt (C.0) = R (V 0 e jωt ) (C.1) = R V (C.) where the notation R() means to extract the real part of the complex expression. What do you think I() means when applied to a complex expression?

5 C.1 Introduction C-5 We can rewrite Equation C. as V = V 0 e jωt (C.3) Let us examine the capacitor again, this time using our complex notation. With the applied voltage given by Equation C.3, the current through the capacitor is: I = C d V dt = C d dt (V 0e jωt ) = jωc (V 0 e jωt ) = jωc V Rearranging gives V = I ( 1 ) jωc Notice the enormous simplification. If we define the capacitive impedance as Z C = 1 jωc = j ωc = j 1 ωc (C.4) (C.5) (C.6) then the above equation becomes V = IZ which is analogous to Ohm s Law for DC circuits. Since then we have Next, notice that so we can write X C = 1 ωc Z C = jx C j = e (jπ/) I = ωcv 0 e j(ωt+π/) (C.7) (C.8) (C.9)

6 C-6 Ohm s Law, Kirchhoff s Laws and AC Circuits and after extracting the real part of the current: ( I(t) = ωcv 0 cos ωt + π ) This is identical to our earlier result in Equation C.10. current flowing through an inductance L is represented as (C.30) Similarly, if the I = I 0 e jωt the voltage across the inductor will be given by V = jωl ( I 0 e jωt) = I (jωl) (C.31) and if we define the inductive impedance as Z L = jωl (C.3) where we have X L = ωl Z L = jx L (C.33) (C.34) Notice that the capacitive and inductive impedances are now imaginary. This means that circuit impedances will be complex quantities (in general) and for both capacitors and inductors, Z = X. C.1.4 Kirchhoff s Laws and AC Circuits Using complex numbers to reflect the vector nature of circuit parameters results in the following formulation of Kirchhoff s laws for AC circuits I = 0 where now it is a vector sum V = IZ and this is a vector sum also. We have gained an enormous mathematical advantage the total impedance of an AC circuit can be determined in the same way as resistors are combined

7 C.1 Introduction C-7 Figure C.1: Sample Phasor Diagram in DC circuits with the understanding that now all quantities are vectors, and specifically Z R = R = R 0 Z C = jx C = j ωc = 1 ωc 90 Z L = jx L = jωl = ωl 90 As an example of how this works, consider a simple RL circuit. impedance Z of this circuit is given by: The Z = R + Z L (C.35) = R + jx L (C.36) = R + jωl (C.37) Figure C.1 shows a Phasor Diagram representing the impedance of the circuit. From the diagram, it should be clear to you that a phasor is nothing more than a vector in the complex plane. As such, we are interested in two quantities: the length of the phasor, which is the magnitude of Z, denoted by Z, and the angle φ that the phasor makes with the Real axis. Clearly, the magnitude of Z is just the square root of the real part squared plus the

8 C-8 Ohm s Law, Kirchhoff s Laws and AC Circuits imaginary part squared, which is obtained from: or Z = ZZ (C.38) = (R + jωl) (R jωl) (C.39) = R + ω L (C.40) Z = ( R + ω L ) 1/ (C.41) where the notation Z denotes complex conjugation (multiply the imaginary part by 1). The phase angle φ is just φ = arctan I(Z) R(Z) (C.4) = arctan ωl R (C.43) This gives us yet another way to represent the complex impedance, namely Z = Z e jφ (C.44) Now we can find the current in the circuit very easily: so I = V Z I(t) = V (t) Z = V 0e jωt R+jωL = V 0e jωt Z e jφ = V 0 Z ej(ωt φ) (C.45) (C.46) (C.47) (C.48) Figure C. shows a phasor diagram representing the input voltage and the circuit current. Both phasors rotate counterclockwise as ωt increases. Notice that the voltage leads the current, as expected for an inductive circuit, and that the angle between the voltage and current phasors is the phase angle φ determined from the complex impedance. A final point: the projection of the voltage phasor on the real axis is what you actually measure with an AC voltmeter; similarly for the current phasor (this is the correspondence between a physical measurement and the mathematical operation of extracting the real part of a complex quantity).

9 C.1 Introduction C-9 Figure C.: Phasors Changing Over Time Once the circuit current has been determined, the voltages across the individual components are easily determined: V R (t) = I(t) R V L (t) = I(t) X L