Reactance and Impedance in RC and RL Circuits Consider the RC circuit shown which is connected to an alternating (AC) voltage source V(t). The circuit current I C, which is also the capacitive current, can be derived as Q = C x V C take the derivative of both sides so dq/dt = C dv C /dt but dq/dt = I C so I C = C dv C /dt The waveforms for the capacitive voltage and current are shown 1
From the equation I C = C dv C /dt it can be seen that the current I C is a maximum when the expression dv C /dt is a maximum. Identify a point on the waveforms that represents this situation. It can also be seen that the current I C is zero when the expression dv C /dt is zero. Identify a point on the waveforms that represents this situation. The result that can be derived from these two statements is that there is a 90 phase shift between the capacitive voltage and current. In fact it can be said that The capacitive voltage lags the capacitive current by 90 Capacitive Reactance When an AC voltage source is connected to a capacitor, a non-zero, finite current flows and can be described as having an RMS, Peak or Peak-to-Peak value. This indicates that the capacitor is exhibiting an opposition to current flow similar to that of a resistor. This opposition to current flow is called Capacitive Reactance. The symbol for capacitive reactance is X C and the units of capacitive reactance are ohms (Ω). The capacitive reactance X C of a capacitor can be defined from Ohms Law as X C = V C /I C Factors Affecting Capacitive Reactance Capacitive reactance is inversely proportional to both the frequency f and the size of the capacitive C. The capacitive reactance X C of a capacitor can be calculated using these factors as: Examples Find the capacitive reactance X C of X C = 1/2πfC ohms 1. C = 0.1 µf at a frequency of 10 khz (159 Ω) 2. C = 0.022 µf at a frequency of 1.0 khz (7234 Ω) 3. C = 47 pf at a frequency of 10 khz (339 kω) Series RC Circuit This circuit has a resistor and capacitor in series so a result is then that the resistor and capacitor currents are the same. 2
In the capacitor the voltage lags the current by 90 and the resistor and capacitor currents are the same (in phase also). In the resistor the voltage and current are in phase so the resistor voltage and capacitor voltage have a 90 phase shift. In fact the phase relation can be expressed as: The capacitor voltage lags the resistor voltage by 90 Voltage Vector Diagram The voltages in the expression above can be represented in a vector diagram. The vector V R that is in phase with the circuit current is always plotted in the zero degree direction. The vector V C lags the resistor voltage by 90. The vector sum of V R and V C is the applied voltage V and is shown as the diagonal of the rectangle formed by V R and V C. This vector sum V can be shown as: V = V R + V C or as a complex number V = V R - j V C The applied voltage V is represented mathematically as a complex number with real and imaginary components. The real component of the applied voltage (V) is the resistor voltage, V R and the imaginary component of the applied voltage is the capacitor voltage, V C. Rectangular and Polar Forms of V The complex number expression for the applied voltage V (V = V R - j V C ) is given in rectangular form. Complex numbers can also be expressed in Polar Form. In Polar Form a complex number has a magnitude and phase angle. The Magnitude of the applied voltage V is found using the Pythagorean Theorem as 3
The Phase Angle of the applied voltage V is found using a Trigonometry ratio as φ = - tan -1 (V C /V R ) To convert from Polar to Rectangular form use: V R = V cos φ V C = V sin φ Examples 1. An RC circuit has V R = 50 V and V C = 50 V. What is the applied voltage? V = sqrt(50 2 + 50 2 ) = 70.7 V φ = - tan -1 (V C /V R ) = - tan -1 (50/50) = - tan -1 1 = -45 2. An RC circuit has an applied voltage of 10 V at a phase angle of -60. What are the values of V R and V C. V R = V cos φ = 10 cos 60 = 10 x 0.5 = 5 V V C = V sin φ = 10 sin 60 = 10 x 0.866 = 8.66 V Impedance of a Series RC Circuit. It is now apparent that both components, R and C, exhibit an opposition to AC current flow. These two oppositions can be combined into a total opposition to AC current flow called Impedance represented by the letter Z. The units of impedance are ohms (Ω). Impedance is represented mathematically as a complex number with real and imaginary components. The real component of impedance (Z) is resistance R and the imaginary component of impedance is capacitive reactance. In vector notation the impedance is represented as: Z = R + X C or as a complex number Z = R - j X C Impedance Vector Diagram A diagram of the vectors in the impedance sum notation can be represented as: 4
The Impedance Z is represented mathematically as a complex number with real and imaginary components. The real component of the impedance is the resistance and the imaginary component of the impedance is the capacitive reactance, X C. Rectangular and Polar Forms of Z The complex number expression for the applied voltage Z (Z = R - j X C ) is given in rectangular form. Complex numbers can also be expressed in Polar Form. In Polar Form a complex number has a magnitude and phase angle. The Magnitude of the Impedance Z is found using the Pythagorean Theorem as The Phase Angle of the Impedance is found using a Trigonometry ratio as φ = - tan -1 (X C /R) To convert from Polar to Rectangular Form use: R = Z cos φ X C = Z sin φ Application Note Written by David Lloyd Computer Engineering Program Humber College 5