# Phase Shift Measurement with Dual Trace Oscilloscope

The phase shift between the voltages of a network or between the voltages and currents owf a network can be found using a dual-trace (two signals displayed at the same time) oscilloscope.
In Fig. 1, channel 1 of the dual-trace oscilloscope is hooked up to display the applied voltage e. Channel 2 is connected to display the voltage across the inductor $v_L$. Of particular importance is the fact that the ground of the scope is connected to the ground of the oscilloscope for both channels. In other words, there is only one common ground for the circuit and oscilloscope. The resulting waveforms may appear as shown in Fig. 2.
Fig. 1: Determining the phase relationship between e and $v_L$
Fig. 2: Determining the phase angle between e and $v_L$
For the chosen horizontal sensitivity, each waveform of Fig. 2 has a period T defined by eight horizontal divisions, Using the fact that each period of a sinusoidal waveform encompasses $360^\circ$, the following ratios can be set up to determine the phase angle $\theta$: $${8 div \over 360^\circ } = {1.6 div \over \theta }$$ and $$\theta = ({1.6 \over 8}) 360^\circ = 72^\circ$$ In general, $$\theta = {(\text{div for} \theta) \over ( \text{div for} T)} \times 360^\circ$$ If the phase relationship between e and $v_R$ is required, the oscilloscope must not be hooked up as shown in Fig. 3. Points a and b have a common ground that will establish a zero-volt drop between the two points; this drop will have the same effect as a short-circuit connection between a and b. The resulting short circuit will "short out" the inductive element, and the current will increase due to the drop in impedance for the circuit. A dangerous situation can arise if the inductive element has a high impedance and the resistor has a relatively low impedance. The current, controlled solely by the resistance R, could jump to dangerous levels and damage the equipment.
Fig. 3: An improper phase-measurement connection.
Fig. 4: Determining the phase relationship between e and $v_R$ or e and i using a sensing resistor.
The phase relationship between e and $v_R$ can be determined by simply interchanging the positions of the coil and resistor or by introducing a sensing resistor, as shown in Fig. 4.
A sensing resistor is exactly that: introduced to "sense" a quantity without adversely affecting the behavior of the network. In other words, the sensing resistor must be small enough compared to the other impedances of the network not to cause a significant change in the voltage and current levels or phase relationships.
Note that the sensing resistor is introduced in a way that will result in one end being connected to the common ground of the network. In Fig. 4, channel 2 will display the voltage $v_{R_{s}}$, which is in phase with the current i. However, the current i is also in phase with the voltage $v_R$ across the resistor R. The net result is that the voltages $v_{R_{s}}$ and $v_R$ are in phase and the phase relationship between e and $v_R$ can be determined from the waveforms e and $v_{R_{s}}$. Since $v_R$ and i are in phase, the above procedure will also determine the phase angle between the applied voltage e and the source current i.
If the magnitude of $R_s$ is sufficiently small compared to$R$ or $X_L$, the phase measurements of Fig. 1 can be performed with $R_s$ in place. That is, channel 2 can be connected to the top of the inductor and to ground, and the effect of $R_s$ can be ignored. In the above application, the sensing resistor will not reveal the magnitude of the voltage $v_R$ but simply the phase relationship between e and $v_R$.
Fig. 5: Determining the phase relationship between $i_R$ and $i_L$.
Fig. 6: Determining the phase relationship between e and $i_s$.
For the parallel network of Fig. 5, the phase relationship between two of the branch currents, $i_R$ and $i_L$, can be determined using a sensing resistor, as shown in the figure. Channel 1 will display the voltage $V_R$, and channel 2 will display the voltage $v_{R_{s}}$ . Since $V_R$ is in phase with $i_R$, and $V_R$ is in phase with the current $i_L$, the phase relationship between $V_R$ and $v_{R_{s}}$ will be the same as that between $i_R$ and $i_L$.
In this case, the magnitudes of the current levels can be determined using Ohm's law and the resistance levels R and $R_s$ respectively.
If the phase relationship between e and $i_s$ of Fig. 5 is required, a sensing resistor can be employed, as shown in Fig. 6.
In general, therefore, for dual-trace measurements of phase relationships, be particularly careful of the grounding arrangement, and fully utilize the in-phase relationship between the voltage and current of a resistor.